[{"data":1,"prerenderedAt":24712},["ShallowReactive",2],{"blog-post-machine-learning-fundamentals-en":3,"blog-post-adjacent-machine-learning-fundamentals-en":695},{"id":4,"title":5,"author":6,"body":7,"date":677,"description":13,"extension":678,"image":679,"lastmod":677,"meta":680,"navigation":681,"order":670,"path":682,"seo":683,"sitemap":684,"slug":687,"stem":688,"summary":689,"tags":690,"__hash__":694},"content_en\u002Fblog\u002Fblog\u002Fmachine-learning-fundamentals.md","Machine Learning Fundamentals","David Deras",{"type":8,"value":9,"toc":665},"minimark",[10,14,17,20,25,45,48,74,77,94,96,100,103,171,173,177,180,303,305,411,413,462,483,485,489,492,546,550,553,562,565,576,589,591,595,620,631,633,637,640,660,662],[11,12,13],"p",{},"This is the first part of a series of articles where we will explore machine learning, from its basic concepts to neural networks and the creation of a machine learning model. In this first part, we will focus on the fundamentals of machine learning, including what it is, its types and some common algorithms.",[15,16],"table-of-contents",{},[18,19],"hr",{},[21,22,24],"h2",{"id":23},"artificial-intelligence-and-machine-learning","Artificial Intelligence and Machine Learning",[11,26,27,28,32,33,36,37,40,41,44],{},"You wake up one day, open Netflix and find exactly the series you wanted to watch. Then, you open Google and type something, and the search engine completes your sentence before you finish typing. How do they do this? The answer is: Artificial Intelligence. AI is a discipline focused on developing systems capable of performing tasks that normally require human intelligence, such as ",[29,30,31],"strong",{},"learning",", ",[29,34,35],{},"reasoning"," and ",[29,38,39],{},"perceiving"," our environment, as well as ",[29,42,43],{},"making decisions",".",[11,46,47],{},"There are three types of AI:",[49,50,51,58,64],"ul",{},[52,53,54,57],"li",{},[29,55,56],{},"Weak or Narrow AI (Artificial Narrow Intelligence or ANI)",": This is the AI we have today. It is designed to perform specific tasks, such as voice recognition, product recommendations, or automatic translation. It does not have consciousness or real understanding; it simply follows predefined algorithms and patterns. Large Language Models like GPT, Claude, or Gemini are examples of weak AI, as they are designed to process and generate text, \"only\" being statistical models that predict text, and not having a deep understanding of the world or being able to perform tasks outside their specific domain. Although they may seem intelligent, in reality, they are only mimicking language patterns based on the data with which they were trained.",[52,59,60,63],{},[29,61,62],{},"General AI (Artificial General Intelligence or AGI)",": This is a hypothetical AI that would have the ability to understand, learn, and apply knowledge across a wide range of tasks, similar to human intelligence. It does not exist yet, but it is a long-term goal in the field of AI.",[52,65,66,69,70,73],{},[29,67,68],{},"Superintelligence (Artificial Superintelligence or ASI)",": This is an AI that would surpass human intelligence in all aspects, including creativity, problem-solving, and decision-making. It is a ",[29,71,72],{},"theoretical"," concept that raises many ethical and philosophical questions about the future of humanity.",[11,75,76],{},"Where does Machine Learning fit in? It's very common to confuse AI with Machine Learning, but they aren't exactly the same. We can say that Artificial Intelligence is a broad conceptual umbrella, and under that umbrella lies Machine Learning, a specific subfield that allows computers to learn automatically from data, without the need for a human to program them step by step.",[78,79,80,83],"blockquote",{},[11,81,82],{},"Recommended resources:",[49,84,85],{},[52,86,87],{},[88,89,93],"a",{"href":90,"rel":91},"https:\u002F\u002Fyoutu.be\u002FdKqwnCKrpVI?si=g6qqFa_1G_M5P3LS",[92],"nofollow","Artificial Intelligence vs Machine Learning vs Deep Learning | Machine Learning 101",[18,95],{},[21,97,99],{"id":98},"ai-subfields","AI Subfields",[11,101,102],{},"Within the broad umbrella of AI, there are several subfields that specialize in different aspects of artificial intelligence:",[49,104,105,111,117,123,129,135,141,147,153,159,165],{},[52,106,107,110],{},[29,108,109],{},"Machine Learning",": Focuses on developing algorithms that allow machines to learn from data and improve their performance over time without being explicitly programmed for each specific task.",[52,112,113,116],{},[29,114,115],{},"Deep Learning",": Is a branch of machine learning that uses deep neural networks to model and solve complex problems. It is especially effective in tasks such as speech recognition, computer vision, and natural language processing.",[52,118,119,122],{},[29,120,121],{},"Natural Language Processing (NLP)",": Focuses on the interaction between computers and human language, allowing machines to understand, interpret, and generate text naturally. This is what makes it possible for chatbots like ChatGPT to maintain coherent conversations with users.",[52,124,125,128],{},[29,126,127],{},"Computer Vision",": This field enables machines to understand and process images and videos. It is fundamental for applications such as facial recognition, autonomous driving, and object detection.",[52,130,131,134],{},[29,132,133],{},"Robotics",": This field focuses on the design and construction of robots that can perform physical tasks in the real world, from manufacturing to medical assistance.",[52,136,137,140],{},[29,138,139],{},"Expert Systems",": These are programs that mimic the decision-making of a human expert in a specific domain, using rules and logic to solve complex problems.",[52,142,143,146],{},[29,144,145],{},"Automatic Reasoning",": This field focuses on inferring logical conclusions from formal rules. It is not the same as machine learning. This field includes symbolic logic, problem-solving, and automated planning.",[52,148,149,152],{},[29,150,151],{},"Intelligent Agents",": These are systems that can perceive their environment, reason about it, and make decisions to achieve specific goals. They can be as simple as a chatbot or as complex as an autonomous driving system (they can use machine learning, simple rules, logical reasoning, etc.).",[52,154,155,158],{},[29,156,157],{},"Distributed AI",": This refers to AI systems that operate across multiple devices or nodes, collaborating to solve problems more efficiently. This is especially relevant in applications such as the Internet of Things (IoT) and cloud computing.",[52,160,161,164],{},[29,162,163],{},"Explainable AI (XAI)",": This focuses on developing AI models that are transparent and understandable to humans, allowing users to understand how and why the AI ​​makes certain decisions.",[52,166,167,170],{},[29,168,169],{},"AI Ethics and Governance",": This deals with the ethical, legal, and social implications of AI development and use, addressing issues such as privacy, fairness, and transparency.",[18,172],{},[21,174,176],{"id":175},"how-a-machine-learns","How a Machine Learns",[11,178,179],{},"Everything depends on the data we give it:",[49,181,182],{},[52,183,184,187,188,191,192,195,196,199,200,254,256,257,259,260,274,276,277,285,286,288,289],{},[29,185,186],{},"Supervised Learning",": Here we give the machine clear examples with correct answers already \"labeled\". For example, if we want the AI to help with a medical diagnosis, we give it thousands of medical histories where we already know which patient was ",[29,189,190],{},"sick"," and which patient was ",[29,193,194],{},"healthy",". The machine learns to recognize patterns in this data to be able to predict the diagnosis of new patients based on what it has learned.",[197,198],"br",{},"A simple example would be something like this:",[201,202,203,222],"table",{},[204,205,206],"thead",{},[207,208,209,213,216,219],"tr",{},[210,211,212],"th",{},"Age",[210,214,215],{},"Symptoms",[210,217,218],{},"Test Results",[210,220,221],{},"Diagnosis",[223,224,225,240],"tbody",{},[207,226,227,231,234,237],{},[228,229,230],"td",{},"45",[228,232,233],{},"Fever, Cough",[228,235,236],{},"Positive",[228,238,239],{},"Sick",[207,241,242,245,248,251],{},[228,243,244],{},"30",[228,246,247],{},"Headache",[228,249,250],{},"Negative",[228,252,253],{},"Healthy",[197,255],{},"The label here is the \"Diagnosis\", and the machine learns to associate the features (Age, Symptoms, Test Results) with that label to make future predictions.",[197,258],{},"Within supervised learning, there are two main types of tasks:",[49,261,262,268],{},[52,263,264,267],{},[29,265,266],{},"Classification",": Where the machine assigns a label to each example. As in classifying emails as \"spam\" or \"not spam\".",[52,269,270,273],{},[29,271,272],{},"Regression",": Where the machine predicts a continuous value. For example, predicting the price of a house based on features like size, location, and number of rooms.",[197,275],{},"Basically, if the response we want to predict is a category (or a ",[88,278,284],{"href":279,"target":280,"rel":281,"ariaLabel":284},"https:\u002F\u002Fwww.geeksforgeeks.org\u002Fmaths\u002Fdifference-between-discrete-and-continuous-variable\u002F","_blank",[282,283],"noopener","noreferrer","discrete variable","), it's classification, if the response represents a measurable amount on a continuous scale, it's regression.\nAnd the requirement for supervised learning to work well is to have a large and representative dataset, with accurate labels. If the data is scarce or the labels are incorrect, the machine will not be able to learn correctly and its predictions will be inaccurate.",[197,287],{},"Examples of ML applications are:",[49,290,291,294,297,300],{},[52,292,293],{},"Fraud detection in financial transactions (classification)",[52,295,296],{},"Stock price prediction (regression)",[52,298,299],{},"Image recognition (classification)",[52,301,302],{},"Sentiment analysis in social media (classification)",[18,304],{},[49,306,307],{},[52,308,309,312,313,315,316,356,358,359,408,410],{},[29,310,311],{},"Unsupervised Learning",": Imagine being dropped in an unfamiliar country and having to deduce how society works simply by observing; it's similar. Here, the machine receives unlabeled data and must find hidden patterns on its own. The system will have to analyze similarities, differences, and behaviors to find unusual groupings or patterns. There's no \"teacher\" to tell it if it's right or wrong.",[197,314],{},"Main techniques:",[317,318,319,336,350],"ol",{},[52,320,321,324,325],{},[29,322,323],{},"Clustering (Grouping)",": Groups similar data points together.\nIt's typically used for customer segmentation, grouping documents by topic, and automatic image organization, among others. Some algorithms:\n",[49,326,327,330,333],{},[52,328,329],{},"K-means clustering",[52,331,332],{},"DBSCAN",[52,334,335],{},"Hierarchical clustering",[52,337,338,341,342],{},[29,339,340],{},"Dimensionality Reduction",": This seeks to reduce the number of variables while maintaining important information. Used for visualizing complex data and preparing data for other models.\n",[49,343,344,347],{},[52,345,346],{},"Principal Component Analysis (PCA)",[52,348,349],{},"t-SNE",[52,351,352,355],{},[29,353,354],{},"Anomaly Detection",": Identifies data points that behave differently from the rest. Useful for detecting fraud, system failures, and suspicious behaviors.",[197,357],{},"A simple example would be:",[201,360,361,373],{},[204,362,363],{},[207,364,365,367,370],{},[210,366,212],{},[210,368,369],{},"Annual Income",[210,371,372],{},"Monthly Expenses",[223,374,375,386,397],{},[207,376,377,380,383],{},[228,378,379],{},"25",[228,381,382],{},"$30,000",[228,384,385],{},"$1,000",[207,387,388,391,394],{},[228,389,390],{},"40",[228,392,393],{},"$80,000",[228,395,396],{},"$3,000",[207,398,399,402,405],{},[228,400,401],{},"60",[228,403,404],{},"$50,000",[228,406,407],{},"$2,000",[197,409],{},"The machine could group customers into segments based on their income and expenses, without explicitly telling it what groups exist.\nWith this, we can identify consumption patterns, such as young customers tending to spend less than middle-aged customers, or there being a group of customers with high income but low expenses, which could indicate a savings segment.",[18,412],{},[49,414,415],{},[52,416,417,420,421,44,426,428,429,455,457,458],{},[29,418,419],{},"Reinforcement Learning",": Think about how you train a pet with treats. The machine (the agent) takes decisions in an environment and receives \"rewards\" or \"penalties\". This is how Tesla's autonomous driving systems or robots learn to navigate the physical world. A example I like is a video where ",[88,422,425],{"href":423,"target":280,"rel":424},"https:\u002F\u002Fyoutu.be\u002FPKDMGPf-PEA?si=tAEMO3cETdPrvi_t",[282,283],"they train an agent to play Geometry Dash",[197,427],{},"In this type of learning, there are four main components:",[317,430,431,437,443,449],{},[52,432,433,436],{},[29,434,435],{},"Agent",": It is the system that takes decisions and learns through interaction with the environment. It can be a robot, a computer program, or any system that can perceive its environment and act upon it.",[52,438,439,442],{},[29,440,441],{},"Environment",": It is the world in which the agent operates. It can be a physical environment, such as a robot in a room, or a virtual environment, such as a video game.",[52,444,445,448],{},[29,446,447],{},"Reward",": It is the signal that the agent receives after taking an action. It can be positive (reward) or negative (penalty) and serves to guide the agent's learning.",[52,450,451,454],{},[29,452,453],{},"Policy",": It is the strategy that the agent uses to decide what action to take based on its current state and the rewards it has received in the past.",[197,456],{},"Basically they follow a flow like the following:",[459,460],"mermaid-diagram",{"content":461},"graph TD \nA[Agent] -->|Take action| B(Environment)\nB -->|Provide reward| C[Reward]\nC -->|Update policy| A",[78,463,464,467],{},[11,465,466],{},"Recommended Resources:",[49,468,469,476],{},[52,470,471],{},[88,472,475],{"href":473,"rel":474},"https:\u002F\u002Fyoutu.be\u002FoT3arRRB2Cw?si=ykU9KQjQLxdn9ggj",[92],"What is Supervised and Unsupervised Learning? | DotCSV",[52,477,478],{},[88,479,482],{"href":480,"rel":481},"https:\u002F\u002Fyoutu.be\u002FqBtB-xcJp4c?si=c2GuJBCFPorKGN44",[92],"Reinforcement Learning: The Definitive Guide",[18,484],{},[21,486,488],{"id":487},"ai-project-pipeline","AI Project Pipeline",[11,490,491],{},"An AI project typically follows a structured process that includes several key stages:",[317,493,494,500,506,512,518,524,530,540],{},[52,495,496,499],{},[29,497,498],{},"Problem Definition",": It is essential to clearly understand the problem to be solved and the project objectives. This includes identifying the questions to be answered, the expected results, and the success metrics.",[52,501,502,505],{},[29,503,504],{},"Data Collection",": The data needed to train the AI ​​model is collected. This can include structured data (such as databases) or unstructured data (such as text, images, or videos). It is important to ensure that the data is high-quality and representative of the problem to be solved.",[52,507,508,511],{},[29,509,510],{},"Data Preprocessing",": The collected data often needs to be cleaned and transformed before being used to train the model. This may include removing missing values, normalizing data, coding categorical variables, and splitting the data into training and test sets.",[52,513,514,517],{},[29,515,516],{},"Model Selection",": The most suitable machine learning algorithm is chosen for the problem at hand. This may depend on the nature of the data, the complexity of the problem, and the available resources.",[52,519,520,523],{},[29,521,522],{},"Model Training",": The training dataset is used to train the AI ​​model. During this stage, the model learns from the data and adjusts its parameters to minimize prediction errors.",[52,525,526,529],{},[29,527,528],{},"Model Evaluation",": The model's performance is evaluated using the test set. Specific metrics are used to measure the model's precision, accuracy, sensitivity, and other characteristics, depending on the type of problem (classification, regression, etc.).",[52,531,532,535,536,539],{},[29,533,534],{},"Hyperparameter Tuning",": If the model's performance is unsatisfactory, the model's hyperparameters can be tuned to improve its performance. This may include changing the model's architecture, adjusting the learning rate, or modifying other algorithm-specific parameters. 8. ",[29,537,538],{},"Implementation",": Once the model has been trained and evaluated, it is deployed in a production environment where it can be used to make real-time predictions or process new data.",[52,541,542,545],{},[29,543,544],{},"Maintenance and Updating",": After deployment, it is important to monitor the model's performance and update it regularly to ensure it remains effective as data and environmental conditions change.",[21,547,549],{"id":548},"memorizing-vs-learning","Memorizing vs. Learning",[11,551,552],{},"When we talk about learning, whether human or machine, there's a crucial concept we must understand: memorizing is not the same as learning.",[11,554,555,558,559,561],{},[29,556,557],{},"Memorizing"," is like copying and pasting information without truly understanding it. For example, if you memorize the formula for the area of ​​a circle (A = πr²) without understanding what each part means, you won't be able to apply it correctly in different contexts. Memory saves you in the short term, but it doesn't give you the ability to adapt to new situations or solve problems you haven't encountered before. In contrast, ",[29,560,31],{}," involves understanding the underlying concepts and being able to apply them to new situations.",[11,563,564],{},"In the world of machine learning, a machine that only memorizes training data might perform excellently on that specific data, but then fail spectacularly when used in real-world environments. Therefore, in machine learning, the true goal isn't to memorize specific patterns, but to generalize: to learn rules and relationships that work beyond the examples seen.",[11,566,567,568,571,572,575],{},"But learning isn't easy either. Sometimes, machines suffer from ",[29,569,570],{},"overfitting",", which occurs when a model perfectly \"memorizes\" the training data but fails miserably when faced with new, real-world data. It's exactly like a student who memorizes exam answers without truly understanding the concepts. Conversely, if the model is too simple and learns nothing, it suffers from ",[29,573,574],{},"underfitting",", like a student who didn't study enough.",[78,577,578,580],{},[11,579,466],{},[49,581,582],{},[52,583,584],{},[88,585,588],{"href":586,"rel":587},"https:\u002F\u002Fyoutu.be\u002Fo3DztvnfAJg?si=lorMlPZqLAMa-EV3",[92],"Underfitting and Overfitting: Explained",[18,590],{},[21,592,594],{"id":593},"the-components-of-an-ai-system","The Components of an AI System",[49,596,597,608,614],{},[52,598,599,602,603,44],{},[29,600,601],{},"Data",": This is the foundation of any AI system. Without data, there is no learning. It must be high-quality, relevant, and representative of the problem to be solved. It should be high-volume and have the least possible ",[88,604,607],{"href":605,"target":280,"rel":606},"https:\u002F\u002Fwww.innovatiana.com\u002Fen\u002Fpost\u002Fbias-estimation-in-machine-learning",[282,283],"bias",[52,609,610,613],{},[29,611,612],{},"Algorithms",": These are the recipes the machine follows to learn from the data. There are many types of algorithms, each with its own strengths and weaknesses, and they must be carefully selected according to the specific problem to be solved, configured, and fine-tuned to achieve the best possible performance.",[52,615,616,619],{},[29,617,618],{},"Infrastructure",": This is the hardware and software necessary to process the data and run the algorithms. This includes everything from servers (CPUs, GPUs, TPUs) to cloud computing platforms and development tools (AWS, Azure, GCP), as well as data storage and database management systems.",[11,621,622,623,626,627,630],{},"In addition to these components, we have ",[29,624,625],{},"evaluation",", which is the process of measuring the AI ​​model's performance to ensure it is functioning correctly and meeting the established objectives, as well as the role of ",[29,628,629],{},"ethics and governance",", which is crucial for considering the ethical and social implications.",[18,632],{},[21,634,636],{"id":635},"ethics-in-ai","Ethics in AI",[11,638,639],{},"As artificial intelligence becomes more ubiquitous in our lives, it is crucial to consider the ethical implications of its use. Machine learning models can perpetuate existing biases in the data, which can lead to unfair or discriminatory decisions. For example, if a hiring model is trained on historical data that reflects gender or racial bias, the model is likely to reproduce those biases in its recommendations. Furthermore, data privacy is a major concern. It is essential to ensure that the data used is collected and handled ethically, respecting people's privacy and rights.",[49,641,642,648,654],{},[52,643,644,647],{},[29,645,646],{},"Transparency",": Systems must be understandable and auditable so that users can understand how they work and why they make certain decisions. One solution to this is explainable AI (XAI).",[52,649,650,653],{},[29,651,652],{},"Explainability",": AI models must be able to explain their decisions clearly and comprehensibly to users, which helps build trust and allows users to understand the reasons behind the system's recommendations or actions.",[52,655,656,659],{},[29,657,658],{},"Accountability",": It must be clearly established who is responsible for the decisions made by AI systems, especially in cases where those decisions can have a significant impact on people's lives. This is where legal frameworks and regulations come into play, which must be developed to ensure that AI companies and developers are held accountable for their creations.",[18,661],{},[11,663,664],{},"This concludes the first part of this series of articles on machine learning. In the next part, we will explore some machine learning paradigms and certain mathematical foundations that are essential for understanding how the algorithms work.",{"title":666,"searchDepth":667,"depth":667,"links":668},"",3,[669,671,672,673,674,675,676],{"id":23,"depth":670,"text":24},2,{"id":98,"depth":670,"text":99},{"id":175,"depth":670,"text":176},{"id":487,"depth":670,"text":488},{"id":548,"depth":670,"text":549},{"id":593,"depth":670,"text":594},{"id":635,"depth":670,"text":636},"2026-03-30","md","\u002Fblog\u002Fmachine-learning-fundamentals\u002Fshared\u002Fml-fundamentals.webp",{},true,"\u002Fblog\u002Fblog\u002Fmachine-learning-fundamentals",{"title":5,"description":13},{"loc":685,"priority":686,"lastmod":677},"\u002Fes\u002Fblog\u002Fmachine-learning-fundamentals",0.7,"machine-learning-fundamentals","blog\u002Fblog\u002Fmachine-learning-fundamentals","Basic concepts for starting in the world of machine learning",[109,691,692,693,109],"Artificial Intelligence","Python","Data Science","pEcYe634ioebWgtBZVCgWgEWVuXQtl0dUEymnA919Tg",{"prev":696,"next":7351},{"id":697,"title":698,"author":6,"body":699,"date":7335,"description":703,"extension":678,"image":7336,"lastmod":7337,"meta":7338,"navigation":681,"order":808,"path":7339,"seo":7340,"sitemap":7341,"slug":7343,"stem":7344,"summary":7345,"tags":7346,"__hash__":7350},"content_en\u002Fblog\u002Fblog\u002Fgetting-started-vue-vite.md","Getting Started with Vue 3 and Vite",{"type":8,"value":700,"toc":7315},[701,704,706,708,712,715,754,757,767,769,773,781,784,787,789,793,796,819,830,841,844,853,856,865,868,946,949,958,978,981,984,987,995,1000,1014,1019,1031,1036,1051,1057,1071,1079,1086,1113,1116,1124,1133,1135,1139,1142,1411,1418,1426,1441,1446,1524,1528,1531,1541,1948,1958,1963,2463,2470,2477,2484,2495,2497,2501,2511,2719,2722,2783,2790,3045,3062,3068,3482,3495,3500,3754,3783,3786,3812,3820,3829,3840,3842,3845,3847,3851,3854,3857,3863,4088,4102,4447,4462,4465,4468,4488,4494,4496,4500,4503,4510,4516,4778,4793,4799,5021,5030,5347,5352,5355,5378,5382,5392,5402,5412,5416,5422,5467,5470,5493,5497,5504,5511,5514,5545,5548,5550,5554,5561,5573,5576,5652,5662,5668,5671,5694,5696,5700,5709,5718,5721,5739,5745,5994,6001,6042,6049,6319,6326,6338,6343,6948,6954,6961,6974,6976,6980,6983,6988,7001,7010,7017,7022,7037,7060,7072,7092,7110,7129,7134,7146,7156,7161,7196,7199,7202,7211,7214,7237,7240,7256,7258,7262,7265,7305,7307,7310,7312],[11,702,703],{},"In this guide, we will explore how to start a modern project with Vue 3 and Vite, explain the project structure, how a Vue application works, the router, Pinia, composables, and we'll see some best practices for developing scalable and maintainable applications. As a bonus, we'll publish the project to GitHub and see how to install Tailwind CSS v4 to get fast and responsive styles.",[15,705],{},[18,707],{},[21,709,711],{"id":710},"setting-up-the-environment","Setting up the environment",[11,713,714],{},"Before continuing, you should have:",[49,716,717,730,738,746],{},[52,718,719,720,723,724,729],{},"Node.js, any ",[29,721,722],{},"LTS"," version. I recommend getting used to using ",[88,725,728],{"target":280,"href":726,"rel":727},"https:\u002F\u002Fgithub.com\u002Fcoreybutler\u002Fnvm-windows",[282,283],"nvm"," to manage multiple Node versions.",[52,731,732,733,737],{},"A package manager (here we'll use ",[734,735,736],"code",{},"npm",", which is already installed with Node.js).",[52,739,740,745],{},[88,741,744],{"target":280,"href":742,"rel":743},"https:\u002F\u002Fcode.visualstudio.com\u002F",[282,283],"VS Code"," updated",[52,747,748,753],{},[88,749,752],{"target":280,"href":750,"rel":751},"https:\u002F\u002Fwww.desarrollolibre.net\u002Fblog\u002Fprogramacion-basica\u002Fsetting-up-git-for-the-first-time",[282,283],"Git configured"," correctly",[11,755,756],{},"Also, install the official Vue.js extension for VS Code:",[49,758,759],{},[52,760,761,766],{},[88,762,765],{"target":280,"href":763,"rel":764},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=Vue.volar",[282,283],"Vue.js (Volar)",".\nLater we'll explore other useful plugins like ESLint, Prettier, and Tailwind CSS IntelliSense.",[18,768],{},[21,770,772],{"id":771},"javascript-or-typescript","JavaScript or TypeScript?",[11,774,775,776,44],{},"Vue 3 has quite good support for TypeScript, you can use it keeping in mind the ",[88,777,780],{"target":280,"href":778,"rel":779},"https:\u002F\u002Fvuejs.org\u002Fguide\u002Ftypescript\u002Foverview.html",[282,283],"official usage guide",[11,782,783],{},"You don't need to use TypeScript in all your projects, especially if you're just starting out or just want to experiment. So my recommendation is that you assess your context, the application, and your team (if you're working in one).",[11,785,786],{},"In this guide we'll use JavaScript to keep things simple.",[18,788],{},[21,790,792],{"id":791},"creating-the-project-with-vite","Creating the project with Vite",[11,794,795],{},"The fastest way to start a Vue 3 project with Vite is using the command (you can run it in any folder where you want to create the project, use VS Code's integrated terminal or your favorite terminal):",[797,798,802],"pre",{"className":799,"code":800,"language":801,"meta":666,"style":666},"language-bash shiki shiki-themes vitesse-light vitesse-dark","npm create vue@latest\n","bash",[734,803,804],{"__ignoreMap":666},[805,806,809,812,816],"span",{"class":807,"line":808},"line",1,[805,810,736],{"class":811},"s_xSY",[805,813,815],{"class":814},"spP0B"," create",[805,817,818],{"class":814}," vue@latest\n",[11,820,821,822,825,826,829],{},"If it's the first time you use it, it will ask if you want to install the ",[734,823,824],{},"create-vue"," package. Answer yes (type ",[734,827,828],{},"y"," and press ENTER).",[11,831,832,837],{},[833,834],"img",{"alt":835,"src":836},"Installing create-vue","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Finstall-create-vue-package.webp",[838,839,840],"em",{},"Installing package: create-vue",[11,842,843],{},"It will ask you for the project name, type the name you want:",[11,845,846,850],{},[833,847],{"alt":848,"src":849},"Project name","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fproject-name.webp",[838,851,852],{},"Give your project a name",[11,854,855],{},"Then select the options you need:",[11,857,858,862],{},[833,859],{"alt":860,"src":861},"Project options","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fproject-options.webp",[838,863,864],{},"Available features when creating a Vue app",[11,866,867],{},"Let's go point by point (as shown in the image, we move with the arrow keys and select with the spacebar):",[317,869,870,876,887,899,905,911,934,940],{},[52,871,872,875],{},[29,873,874],{},"TypeScript",": as we mentioned before, we'll use JavaScript, so we skip this.",[52,877,878,881,882,44],{},[29,879,880],{},"JSX Support",": we won't use ",[88,883,886],{"target":280,"href":884,"rel":885},"https:\u002F\u002Fkinsta.com\u002Fes\u002Fblog\u002Fque-es-jsx\u002F",[282,283],"JSX",[52,888,889,892,893,898],{},[29,890,891],{},"Router (SPA development)",": Vue Router is essential for working with Vue as a SPA (Single Page Application). We won't go into detail about ",[88,894,897],{"target":280,"href":895,"rel":896},"https:\u002F\u002Fwww.geeksforgeeks.org\u002Fblogs\u002Fspa-vs-mpa-which-one-is-better-for-you\u002F",[282,283],"SPA vs MPA",", just keep in mind that Vue works as a SPA by default. Select this option.",[52,900,901,904],{},[29,902,903],{},"Pinia (state management)",": it's the official library for state management. It will help us manage shared data between components. Select this option.",[52,906,907,910],{},[29,908,909],{},"Vitest (unit testing)",": is the recommended testing framework for Vite projects. We won't use it in this guide.",[52,912,913,916,917,32,922,927,928,933],{},[29,914,915],{},"End-to-End Testing",": will allow you to choose and integrate a tool for E2E testing (",[88,918,921],{"target":280,"href":919,"rel":920},"https:\u002F\u002Fplaywright.dev\u002F",[282,283],"Playwright",[88,923,926],{"target":280,"href":924,"rel":925},"https:\u002F\u002Fwww.cypress.io\u002F",[282,283],"Cypress"," or ",[88,929,932],{"target":280,"href":930,"rel":931},"https:\u002F\u002Fnightwatchjs.org\u002F",[282,283],"Nightwatch","). We won't use it in this guide.",[52,935,936,939],{},[29,937,938],{},"ESLint (error prevention)",": is a tool to maintain code quality, helping to detect errors and maintain consistent style. I highly recommend its use, although it's important to customize it according to your team and project. We'll select it to explore it a bit more later.",[52,941,942,945],{},[29,943,944],{},"Prettier (code formatting)",": is a tool to automatically format code. In this case we'll select it too to explore it.",[11,947,948],{},"Press ENTER and now it will ask about some experimental Vite features:",[11,950,951,955],{},[833,952],{"alt":953,"src":954},"Experimental Vite features","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fproject-experimental-options.webp",[838,956,957],{},"Vite experimental features",[317,959,960,972],{},[52,961,962,965,966,971],{},[29,963,964],{},"Oxlint",": is a new linter, part of ",[88,967,970],{"target":280,"href":968,"rel":969},"https:\u002F\u002Foxc.rs\u002F",[282,283],"OXC",", a new collection of JavaScript tools written in Rust, Oxlint is extremely fast and shows promise, however as mentioned on its website, it still has some issues, so for now I recommend continuing to use ESLint. We ignore this option.",[52,973,974,977],{},[29,975,976],{},"rolldown vite (experimental)",": Rolldown-vite is a fork of Vite that uses Rolldown instead of Rollup and esbuild, with the goal of achieving maximum performance. It will soon become the default package, for now let's ignore this option.",[11,979,980],{},"We continue pressing ENTER.",[11,982,983],{},"It will ask if you want to start with a completely blank project or with examples. If it's your first time, I recommend selecting \"No\" so you can see a basic structure with examples. That's the option we'll select here.",[11,985,986],{},"Vite will create the initial project structure and give us some useful commands, let's run them one by one:",[11,988,989,993],{},[833,990],{"alt":991,"src":992},"Project created","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fproject-created.webp",[838,994,991],{},[317,996,997],{},[52,998,999],{},"First we navigate to the project folder:",[797,1001,1003],{"className":799,"code":1002,"language":801,"meta":666,"style":666},"cd project-name\n",[734,1004,1005],{"__ignoreMap":666},[805,1006,1007,1011],{"class":807,"line":808},[805,1008,1010],{"class":1009},"sHLBJ","cd",[805,1012,1013],{"class":814}," project-name\n",[317,1015,1016],{"start":670},[52,1017,1018],{},"We install the dependencies (this may take a few minutes depending on your connection):",[797,1020,1022],{"className":799,"code":1021,"language":801,"meta":666,"style":666},"npm install\n",[734,1023,1024],{"__ignoreMap":666},[805,1025,1026,1028],{"class":807,"line":808},[805,1027,736],{"class":811},[805,1029,1030],{"class":814}," install\n",[317,1032,1033],{"start":667},[52,1034,1035],{},"This command is to run Prettier, it won't have any effect because we haven't changed anything yet, but it's good to keep it in mind:",[797,1037,1039],{"className":799,"code":1038,"language":801,"meta":666,"style":666},"npm run format\n",[734,1040,1041],{"__ignoreMap":666},[805,1042,1043,1045,1048],{"class":807,"line":808},[805,1044,736],{"class":811},[805,1046,1047],{"class":814}," run",[805,1049,1050],{"class":814}," format\n",[317,1052,1054],{"start":1053},4,[52,1055,1056],{},"Finally, we start the development server:",[797,1058,1060],{"className":799,"code":1059,"language":801,"meta":666,"style":666},"npm run dev\n",[734,1061,1062],{"__ignoreMap":666},[805,1063,1064,1066,1068],{"class":807,"line":808},[805,1065,736],{"class":811},[805,1067,1047],{"class":814},[805,1069,1070],{"class":814}," dev\n",[11,1072,1073,1077],{},[833,1074],{"alt":1075,"src":1076},"Development server running","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fdev-server-running.webp",[838,1078,1075],{},[11,1080,1081,1082,1085],{},"Vite runs the development server at ",[734,1083,1084],{},"http:\u002F\u002Flocalhost:5173\u002F"," (the port may vary if 5173 is already in use (5174, 5175, etc.)).",[78,1087,1088,1091,1105],{},[11,1089,1090],{},"Port 5173 is a hint to Vite itself:",[49,1092,1093,1096,1099,1102],{},[52,1094,1095],{},"5 = V",[52,1097,1098],{},"1 = I",[52,1100,1101],{},"7 = T",[52,1103,1104],{},"3 = E",[11,1106,1107,1108,44],{},"Although beyond that, Vite uses port 5173 because it's uncommon in development environments, reducing the likelihood of conflicts with other local servers. Via ",[88,1109,1112],{"target":280,"href":1110,"rel":1111},"https:\u002F\u002Fmedium.com\u002F@bishakhghosh0\u002Fwhy-localhost-5173-is-every-frontend-developers-best-friend-b3bb5b6fb1db",[282,283],"Why localhost:5173 is Every Frontend Developer's Best Friend",[11,1114,1115],{},"If we go to that URL in our browser, we'll see the Vue app running:",[11,1117,1118,1122],{},[833,1119],{"alt":1120,"src":1121},"Vue app running","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fvue-app-running.webp",[838,1123,1120],{},[11,1125,1126,1127,1132],{},"The Vue team shares several official resources to learn more about the framework, starting with the ",[88,1128,1131],{"target":280,"href":1129,"rel":1130},"https:\u002F\u002Fvuejs.org\u002Fguide\u002Fintroduction.html",[282,283],"official documentation",". Explore each link to learn about the entire ecosystem.",[18,1134],{},[21,1136,1138],{"id":1137},"project-structure","Project Structure",[11,1140,1141],{},"Let's look at the initial structure, open the project in your favorite code editor (I recommend VS Code):",[797,1143,1145],{"className":799,"code":1144,"language":801,"meta":666,"style":666},"my-vue-app\u002F\n├── .vscode\u002F              # (If you're in VS Code) Visual Studio Code configuration\n├── node_modules\u002F         # Project dependencies, generated when installing packages\n├── public\u002F               # Static files\n├── src\u002F                  # Application source code\n│   ├── assets\u002F           # Resources like images and styles\n│   ├── components\u002F       # Reusable Vue components\n│   ├── router\u002F           # Vue Router configuration\n│   ├── store\u002F            # Pinia configuration\n│   ├── views\u002F            # Views for the routes\n│   ├── App.vue           # Root component of the application\n│   └── main.js           # Application entry point\n├── .editorconfig         # EditorConfig configuration\n├── .gitattributes        # Git configuration\n├── .gitignore            # Files and folders ignored by Git\n├── .prettierrc.json      # Prettier configuration\n├── eslint.config.js      # ESLint configuration\n├── index.html            # Main HTML file\n├── jsconfig.json         # JavaScript configuration for the editor\n├── package-lock.json     # Exact versions of dependencies (autogenerated)\n├── package.json          # Project information and scripts\n├── README.md             # Project documentation\n└── vite.config.js        # Vite configuration\n",[734,1146,1147,1152,1164,1174,1184,1195,1210,1223,1236,1249,1262,1275,1289,1300,1311,1322,1333,1344,1355,1366,1377,1388,1399],{"__ignoreMap":666},[805,1148,1149],{"class":807,"line":808},[805,1150,1151],{"class":811},"my-vue-app\u002F\n",[805,1153,1154,1157,1160],{"class":807,"line":670},[805,1155,1156],{"class":811},"├──",[805,1158,1159],{"class":814}," .vscode\u002F",[805,1161,1163],{"class":1162},"snYqZ","              # (If you're in VS Code) Visual Studio Code configuration\n",[805,1165,1166,1168,1171],{"class":807,"line":667},[805,1167,1156],{"class":811},[805,1169,1170],{"class":814}," node_modules\u002F",[805,1172,1173],{"class":1162},"         # Project dependencies, generated when installing packages\n",[805,1175,1176,1178,1181],{"class":807,"line":1053},[805,1177,1156],{"class":811},[805,1179,1180],{"class":814}," public\u002F",[805,1182,1183],{"class":1162},"               # Static files\n",[805,1185,1187,1189,1192],{"class":807,"line":1186},5,[805,1188,1156],{"class":811},[805,1190,1191],{"class":814}," src\u002F",[805,1193,1194],{"class":1162},"                  # Application source code\n",[805,1196,1198,1201,1204,1207],{"class":807,"line":1197},6,[805,1199,1200],{"class":811},"│",[805,1202,1203],{"class":814},"   ├──",[805,1205,1206],{"class":814}," assets\u002F",[805,1208,1209],{"class":1162},"           # Resources like images and styles\n",[805,1211,1213,1215,1217,1220],{"class":807,"line":1212},7,[805,1214,1200],{"class":811},[805,1216,1203],{"class":814},[805,1218,1219],{"class":814}," components\u002F",[805,1221,1222],{"class":1162},"       # Reusable Vue components\n",[805,1224,1226,1228,1230,1233],{"class":807,"line":1225},8,[805,1227,1200],{"class":811},[805,1229,1203],{"class":814},[805,1231,1232],{"class":814}," router\u002F",[805,1234,1235],{"class":1162},"           # Vue Router configuration\n",[805,1237,1239,1241,1243,1246],{"class":807,"line":1238},9,[805,1240,1200],{"class":811},[805,1242,1203],{"class":814},[805,1244,1245],{"class":814}," store\u002F",[805,1247,1248],{"class":1162},"            # Pinia configuration\n",[805,1250,1252,1254,1256,1259],{"class":807,"line":1251},10,[805,1253,1200],{"class":811},[805,1255,1203],{"class":814},[805,1257,1258],{"class":814}," views\u002F",[805,1260,1261],{"class":1162},"            # Views for the routes\n",[805,1263,1265,1267,1269,1272],{"class":807,"line":1264},11,[805,1266,1200],{"class":811},[805,1268,1203],{"class":814},[805,1270,1271],{"class":814}," App.vue",[805,1273,1274],{"class":1162},"           # Root component of the application\n",[805,1276,1278,1280,1283,1286],{"class":807,"line":1277},12,[805,1279,1200],{"class":811},[805,1281,1282],{"class":814},"   └──",[805,1284,1285],{"class":814}," main.js",[805,1287,1288],{"class":1162},"           # Application entry point\n",[805,1290,1292,1294,1297],{"class":807,"line":1291},13,[805,1293,1156],{"class":811},[805,1295,1296],{"class":814}," .editorconfig",[805,1298,1299],{"class":1162},"         # EditorConfig configuration\n",[805,1301,1303,1305,1308],{"class":807,"line":1302},14,[805,1304,1156],{"class":811},[805,1306,1307],{"class":814}," .gitattributes",[805,1309,1310],{"class":1162},"        # Git configuration\n",[805,1312,1314,1316,1319],{"class":807,"line":1313},15,[805,1315,1156],{"class":811},[805,1317,1318],{"class":814}," .gitignore",[805,1320,1321],{"class":1162},"            # Files and folders ignored by Git\n",[805,1323,1325,1327,1330],{"class":807,"line":1324},16,[805,1326,1156],{"class":811},[805,1328,1329],{"class":814}," .prettierrc.json",[805,1331,1332],{"class":1162},"      # Prettier configuration\n",[805,1334,1336,1338,1341],{"class":807,"line":1335},17,[805,1337,1156],{"class":811},[805,1339,1340],{"class":814}," eslint.config.js",[805,1342,1343],{"class":1162},"      # ESLint configuration\n",[805,1345,1347,1349,1352],{"class":807,"line":1346},18,[805,1348,1156],{"class":811},[805,1350,1351],{"class":814}," index.html",[805,1353,1354],{"class":1162},"            # Main HTML file\n",[805,1356,1358,1360,1363],{"class":807,"line":1357},19,[805,1359,1156],{"class":811},[805,1361,1362],{"class":814}," jsconfig.json",[805,1364,1365],{"class":1162},"         # JavaScript configuration for the editor\n",[805,1367,1369,1371,1374],{"class":807,"line":1368},20,[805,1370,1156],{"class":811},[805,1372,1373],{"class":814}," package-lock.json",[805,1375,1376],{"class":1162},"     # Exact versions of dependencies (autogenerated)\n",[805,1378,1380,1382,1385],{"class":807,"line":1379},21,[805,1381,1156],{"class":811},[805,1383,1384],{"class":814}," package.json",[805,1386,1387],{"class":1162},"          # Project information and scripts\n",[805,1389,1391,1393,1396],{"class":807,"line":1390},22,[805,1392,1156],{"class":811},[805,1394,1395],{"class":814}," README.md",[805,1397,1398],{"class":1162},"             # Project documentation\n",[805,1400,1402,1405,1408],{"class":807,"line":1401},23,[805,1403,1404],{"class":811},"└──",[805,1406,1407],{"class":814}," vite.config.js",[805,1409,1410],{"class":1162},"        # Vite configuration\n",[11,1412,1413,1414,1417],{},"The structure may vary slightly depending on the options selected when creating the project (for example, whether or not ESLint or Pinia is included). Also, if you're using VS Code you probably see the ",[734,1415,1416],{},".vscode\u002F"," folder with specific editor settings and some files you see \"grouped\":",[11,1419,1420,1424],{},[833,1421],{"alt":1422,"src":1423},"Project structure in VS Code","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fproject-structure-nested.webp",[838,1425,1422],{},[11,1427,1428,1429,1432,1433,1436,1437,1440],{},"This is just a visual way that VS Code has to organize files, you can display them individually (my preference) by changing the value of ",[734,1430,1431],{},"\"explorer.fileNesting.enabled\""," to ",[734,1434,1435],{},"false"," in the ",[734,1438,1439],{},".vscode\\settings.json"," file.",[1442,1443,1445],"h3",{"id":1444},"understanding-the-main-folders","Understanding the main folders",[49,1447,1448,1454],{},[52,1449,1450,1453],{},[734,1451,1452],{},"public\u002F",": Here go the static files that will not be processed by Vite. You can place images, fonts or other resources that you need to serve directly.",[52,1455,1456,1459,1460],{},[734,1457,1458],{},"src\u002F",": This is the most important folder, it contains all the source code for your application. Here we have:\n",[49,1461,1462,1468,1474,1486,1502,1512,1518],{},[52,1463,1464,1467],{},[734,1465,1466],{},"assets\u002F",": Here go the resources like images, styles, fonts, etc. that you will use in your application.",[52,1469,1470,1473],{},[734,1471,1472],{},"components\u002F",": Here go the global, reusable Vue components that you can use in different parts of your application, for example, buttons, cards, modals, etc.",[52,1475,1476,1479,1480,1485],{},[734,1477,1478],{},"router\u002F",": Here is the configuration of ",[88,1481,1484],{"href":1482,"target":280,"rel":1483},"https:\u002F\u002Frouter.vuejs.org\u002F",[282,283],"Vue Router",", where you define the routes of your application and how to navigate between them, as well as possible configuration of guards, lazy loading, etc.",[52,1487,1488,1479,1491,1496,1497],{},[734,1489,1490],{},"store\u002F",[88,1492,1495],{"href":1493,"target":280,"rel":1494},"https:\u002F\u002Fpinia.vuejs.org\u002F",[282,283],"Pinia",", where you define the ",[88,1498,1501],{"href":1499,"target":280,"rel":1500},"https:\u002F\u002Fkinsta.com\u002Fes\u002Fblog\u002Fvue-pinia\u002F",[282,283],"global state of your application. ",[52,1503,1504,1507,1508,1511],{},[734,1505,1506],{},"views\u002F",": Here go the main views that correspond to the routes defined in Vue Router. Each view generally represents a full page, unlike components which are smaller, reusable parts. In the end, all are ",[734,1509,1510],{},".vue"," files, so the difference lies in their purpose and use.",[52,1513,1514,1517],{},[734,1515,1516],{},"App.vue",": This is the root component of your application, where everything is mounted (we will explain more later).",[52,1519,1520,1523],{},[734,1521,1522],{},"main.js",": This is the entry point of your application, where Vue is initialized, plugins are configured (like Vue Router and Pinia) and the application is mounted on the DOM.",[1442,1525,1527],{"id":1526},"is-this-structure-enough","Is this structure enough?",[11,1529,1530],{},"It's a basic structure that for now is fine. As your application grows, it's normal to want to organize the code better. Here are two proposals:",[49,1532,1533],{},[52,1534,1535,1536,36,1538,1540],{},"Create subfolders by module\u002Ffunctionality within the different folders, like ",[734,1537,1472],{},[734,1539,1506],{},". For example:",[797,1542,1544],{"className":799,"code":1543,"language":801,"meta":666,"style":666},"my-vue-app\u002F\n├── src\u002F\n│   ├── assets\u002F\n│   ├── components\u002F\n│   │   ├── auth\u002F\n│   │   │   ├── LoginForm.vue\n│   │   │   └── RegisterForm.vue\n│   │   ├── dashboard\u002F\n│   │   │   ├── StatsCard.vue\n│   │   │   └── ChartWidget.vue\n│   │   └── shared\u002F\n│   │       ├── BaseButton.vue\n│   │       └── BaseModal.vue\n│   │\n│   ├── views\u002F\n│   │   ├── auth\u002F\n│   │   │   ├── LoginView.vue\n│   │   │   └── RegisterView.vue\n│   │   ├── dashboard\u002F\n│   │   │   └── DashboardView.vue\n│   │   └── home\u002F\n│   │       └── HomeView.vue\n│   │\n│   ├── store\u002F\n│   │   ├── auth\u002F\n│   │   │   └── auth.store.js\n│   │   ├── dashboard\u002F\n│   │   │   └── dashboard.store.js\n│   │   └── shared\u002F\n│   │       └── ui.store.js\n│   │\n│   ├── router\u002F\n│   │   ├── auth.routes.js\n│   │   ├── dashboard.routes.js\n│   │   └── index.js\n│   │\n│   ├── ...\n├── ...\n",[734,1545,1546,1550,1557,1566,1575,1587,1600,1613,1624,1637,1650,1661,1673,1685,1692,1701,1711,1724,1737,1747,1760,1771,1782,1788,1798,1809,1823,1834,1848,1859,1871,1878,1888,1900,1912,1924,1931,1941],{"__ignoreMap":666},[805,1547,1548],{"class":807,"line":808},[805,1549,1151],{"class":811},[805,1551,1552,1554],{"class":807,"line":670},[805,1553,1156],{"class":811},[805,1555,1556],{"class":814}," src\u002F\n",[805,1558,1559,1561,1563],{"class":807,"line":667},[805,1560,1200],{"class":811},[805,1562,1203],{"class":814},[805,1564,1565],{"class":814}," assets\u002F\n",[805,1567,1568,1570,1572],{"class":807,"line":1053},[805,1569,1200],{"class":811},[805,1571,1203],{"class":814},[805,1573,1574],{"class":814}," components\u002F\n",[805,1576,1577,1579,1582,1584],{"class":807,"line":1186},[805,1578,1200],{"class":811},[805,1580,1581],{"class":814},"   │",[805,1583,1203],{"class":814},[805,1585,1586],{"class":814}," auth\u002F\n",[805,1588,1589,1591,1593,1595,1597],{"class":807,"line":1197},[805,1590,1200],{"class":811},[805,1592,1581],{"class":814},[805,1594,1581],{"class":814},[805,1596,1203],{"class":814},[805,1598,1599],{"class":814}," LoginForm.vue\n",[805,1601,1602,1604,1606,1608,1610],{"class":807,"line":1212},[805,1603,1200],{"class":811},[805,1605,1581],{"class":814},[805,1607,1581],{"class":814},[805,1609,1282],{"class":814},[805,1611,1612],{"class":814}," RegisterForm.vue\n",[805,1614,1615,1617,1619,1621],{"class":807,"line":1225},[805,1616,1200],{"class":811},[805,1618,1581],{"class":814},[805,1620,1203],{"class":814},[805,1622,1623],{"class":814}," dashboard\u002F\n",[805,1625,1626,1628,1630,1632,1634],{"class":807,"line":1238},[805,1627,1200],{"class":811},[805,1629,1581],{"class":814},[805,1631,1581],{"class":814},[805,1633,1203],{"class":814},[805,1635,1636],{"class":814}," StatsCard.vue\n",[805,1638,1639,1641,1643,1645,1647],{"class":807,"line":1251},[805,1640,1200],{"class":811},[805,1642,1581],{"class":814},[805,1644,1581],{"class":814},[805,1646,1282],{"class":814},[805,1648,1649],{"class":814}," ChartWidget.vue\n",[805,1651,1652,1654,1656,1658],{"class":807,"line":1264},[805,1653,1200],{"class":811},[805,1655,1581],{"class":814},[805,1657,1282],{"class":814},[805,1659,1660],{"class":814}," shared\u002F\n",[805,1662,1663,1665,1667,1670],{"class":807,"line":1277},[805,1664,1200],{"class":811},[805,1666,1581],{"class":814},[805,1668,1669],{"class":814},"       ├──",[805,1671,1672],{"class":814}," BaseButton.vue\n",[805,1674,1675,1677,1679,1682],{"class":807,"line":1291},[805,1676,1200],{"class":811},[805,1678,1581],{"class":814},[805,1680,1681],{"class":814},"       └──",[805,1683,1684],{"class":814}," BaseModal.vue\n",[805,1686,1687,1689],{"class":807,"line":1302},[805,1688,1200],{"class":811},[805,1690,1691],{"class":814},"   │\n",[805,1693,1694,1696,1698],{"class":807,"line":1313},[805,1695,1200],{"class":811},[805,1697,1203],{"class":814},[805,1699,1700],{"class":814}," views\u002F\n",[805,1702,1703,1705,1707,1709],{"class":807,"line":1324},[805,1704,1200],{"class":811},[805,1706,1581],{"class":814},[805,1708,1203],{"class":814},[805,1710,1586],{"class":814},[805,1712,1713,1715,1717,1719,1721],{"class":807,"line":1335},[805,1714,1200],{"class":811},[805,1716,1581],{"class":814},[805,1718,1581],{"class":814},[805,1720,1203],{"class":814},[805,1722,1723],{"class":814}," LoginView.vue\n",[805,1725,1726,1728,1730,1732,1734],{"class":807,"line":1346},[805,1727,1200],{"class":811},[805,1729,1581],{"class":814},[805,1731,1581],{"class":814},[805,1733,1282],{"class":814},[805,1735,1736],{"class":814}," RegisterView.vue\n",[805,1738,1739,1741,1743,1745],{"class":807,"line":1357},[805,1740,1200],{"class":811},[805,1742,1581],{"class":814},[805,1744,1203],{"class":814},[805,1746,1623],{"class":814},[805,1748,1749,1751,1753,1755,1757],{"class":807,"line":1368},[805,1750,1200],{"class":811},[805,1752,1581],{"class":814},[805,1754,1581],{"class":814},[805,1756,1282],{"class":814},[805,1758,1759],{"class":814}," DashboardView.vue\n",[805,1761,1762,1764,1766,1768],{"class":807,"line":1379},[805,1763,1200],{"class":811},[805,1765,1581],{"class":814},[805,1767,1282],{"class":814},[805,1769,1770],{"class":814}," home\u002F\n",[805,1772,1773,1775,1777,1779],{"class":807,"line":1390},[805,1774,1200],{"class":811},[805,1776,1581],{"class":814},[805,1778,1681],{"class":814},[805,1780,1781],{"class":814}," HomeView.vue\n",[805,1783,1784,1786],{"class":807,"line":1401},[805,1785,1200],{"class":811},[805,1787,1691],{"class":814},[805,1789,1791,1793,1795],{"class":807,"line":1790},24,[805,1792,1200],{"class":811},[805,1794,1203],{"class":814},[805,1796,1797],{"class":814}," store\u002F\n",[805,1799,1801,1803,1805,1807],{"class":807,"line":1800},25,[805,1802,1200],{"class":811},[805,1804,1581],{"class":814},[805,1806,1203],{"class":814},[805,1808,1586],{"class":814},[805,1810,1812,1814,1816,1818,1820],{"class":807,"line":1811},26,[805,1813,1200],{"class":811},[805,1815,1581],{"class":814},[805,1817,1581],{"class":814},[805,1819,1282],{"class":814},[805,1821,1822],{"class":814}," auth.store.js\n",[805,1824,1826,1828,1830,1832],{"class":807,"line":1825},27,[805,1827,1200],{"class":811},[805,1829,1581],{"class":814},[805,1831,1203],{"class":814},[805,1833,1623],{"class":814},[805,1835,1837,1839,1841,1843,1845],{"class":807,"line":1836},28,[805,1838,1200],{"class":811},[805,1840,1581],{"class":814},[805,1842,1581],{"class":814},[805,1844,1282],{"class":814},[805,1846,1847],{"class":814}," dashboard.store.js\n",[805,1849,1851,1853,1855,1857],{"class":807,"line":1850},29,[805,1852,1200],{"class":811},[805,1854,1581],{"class":814},[805,1856,1282],{"class":814},[805,1858,1660],{"class":814},[805,1860,1862,1864,1866,1868],{"class":807,"line":1861},30,[805,1863,1200],{"class":811},[805,1865,1581],{"class":814},[805,1867,1681],{"class":814},[805,1869,1870],{"class":814}," ui.store.js\n",[805,1872,1874,1876],{"class":807,"line":1873},31,[805,1875,1200],{"class":811},[805,1877,1691],{"class":814},[805,1879,1881,1883,1885],{"class":807,"line":1880},32,[805,1882,1200],{"class":811},[805,1884,1203],{"class":814},[805,1886,1887],{"class":814}," router\u002F\n",[805,1889,1891,1893,1895,1897],{"class":807,"line":1890},33,[805,1892,1200],{"class":811},[805,1894,1581],{"class":814},[805,1896,1203],{"class":814},[805,1898,1899],{"class":814}," auth.routes.js\n",[805,1901,1903,1905,1907,1909],{"class":807,"line":1902},34,[805,1904,1200],{"class":811},[805,1906,1581],{"class":814},[805,1908,1203],{"class":814},[805,1910,1911],{"class":814}," dashboard.routes.js\n",[805,1913,1915,1917,1919,1921],{"class":807,"line":1914},35,[805,1916,1200],{"class":811},[805,1918,1581],{"class":814},[805,1920,1282],{"class":814},[805,1922,1923],{"class":814}," index.js\n",[805,1925,1927,1929],{"class":807,"line":1926},36,[805,1928,1200],{"class":811},[805,1930,1691],{"class":814},[805,1932,1934,1936,1938],{"class":807,"line":1933},37,[805,1935,1200],{"class":811},[805,1937,1203],{"class":814},[805,1939,1940],{"class":814}," ...\n",[805,1942,1944,1946],{"class":807,"line":1943},38,[805,1945,1156],{"class":811},[805,1947,1940],{"class":814},[11,1949,1950,1953,1954,1957],{},[29,1951,1952],{},"You can use an approach similar to this"," in small or medium apps, with few people working, where you want a bit more order without overcomplicating things and without changing too much the original structure proposed by Vue. ",[29,1955,1956],{},"The limitation"," is that as the app grows, it can become difficult to manage dependencies between modules and keep the code organized, because everything is scattered across multiple folders.",[49,1959,1960],{},[52,1961,1962],{},"In other cases, you can opt for a structure based on features or modules, where each one has its own folder that contains components, views, and related logic. For example:",[797,1964,1966],{"className":799,"code":1965,"language":801,"meta":666,"style":666},"my-vue-app\u002F\n├── src\u002F\n│   ├── modules\u002F\n│   │   ├── auth\u002F                     # If the module is small, you can omit the division into folders\n│   │   │   ├── components\u002F\n│   │   │   │   └── LoginForm.vue\n│   │   │   ├── views\u002F\n│   │   │   │   └── LoginView.vue\n│   │   │   ├── store\u002F\n│   │   │   │   └── auth.store.js\n│   │   │   ├── router\u002F\n│   │   │   │   └── auth.routes.js\n│   │   │   └── index.js\n│   │   │\n│   │   ├── dashboard\u002F\n│   │   │   ├── components\u002F\n│   │   │   ├── views\u002F\n│   │   │   ├── store\u002F\n│   │   │   ├── router\u002F\n│   │   │   └── index.js\n│   │   │\n│   │   └── users\u002F\n│   │       ├── components\u002F\n│   │       ├── views\u002F\n│   │       ├── store\u002F\n│   │       ├── router\u002F\n│   │       └── index.js\n│   │\n│   ├── shared\u002F\n│   │   ├── components\u002F\n│   │   │   ├── BaseButton.vue\n│   │   │   └── BaseModal.vue\n│   │   ├── composables\u002F\n│   │   │   └── useFetch.js\n│   │   ├── store\u002F\n│   │   │   └── ui.store.js\n│   │   ├── utils\u002F\n│   │   │   └── formatDate.js\n│   │   └── constants\u002F\n│   │       └── roles.js\n│   │\n│   ├── router\u002F\n│   │   └── index.js                # Main router file that imports routes from modules\n│   │                               # (guards go here)\n│   ├── ...\n├── ...\n",[734,1967,1968,1972,1978,1987,2001,2013,2027,2039,2053,2065,2079,2091,2105,2117,2125,2135,2147,2159,2171,2183,2195,2203,2214,2224,2234,2244,2254,2264,2270,2278,2288,2300,2312,2323,2336,2346,2358,2369,2382,2394,2406,2413,2422,2437,2447,2456],{"__ignoreMap":666},[805,1969,1970],{"class":807,"line":808},[805,1971,1151],{"class":811},[805,1973,1974,1976],{"class":807,"line":670},[805,1975,1156],{"class":811},[805,1977,1556],{"class":814},[805,1979,1980,1982,1984],{"class":807,"line":667},[805,1981,1200],{"class":811},[805,1983,1203],{"class":814},[805,1985,1986],{"class":814}," modules\u002F\n",[805,1988,1989,1991,1993,1995,1998],{"class":807,"line":1053},[805,1990,1200],{"class":811},[805,1992,1581],{"class":814},[805,1994,1203],{"class":814},[805,1996,1997],{"class":814}," auth\u002F",[805,1999,2000],{"class":1162},"                     # If the module is small, you can omit the division into folders\n",[805,2002,2003,2005,2007,2009,2011],{"class":807,"line":1186},[805,2004,1200],{"class":811},[805,2006,1581],{"class":814},[805,2008,1581],{"class":814},[805,2010,1203],{"class":814},[805,2012,1574],{"class":814},[805,2014,2015,2017,2019,2021,2023,2025],{"class":807,"line":1197},[805,2016,1200],{"class":811},[805,2018,1581],{"class":814},[805,2020,1581],{"class":814},[805,2022,1581],{"class":814},[805,2024,1282],{"class":814},[805,2026,1599],{"class":814},[805,2028,2029,2031,2033,2035,2037],{"class":807,"line":1212},[805,2030,1200],{"class":811},[805,2032,1581],{"class":814},[805,2034,1581],{"class":814},[805,2036,1203],{"class":814},[805,2038,1700],{"class":814},[805,2040,2041,2043,2045,2047,2049,2051],{"class":807,"line":1225},[805,2042,1200],{"class":811},[805,2044,1581],{"class":814},[805,2046,1581],{"class":814},[805,2048,1581],{"class":814},[805,2050,1282],{"class":814},[805,2052,1723],{"class":814},[805,2054,2055,2057,2059,2061,2063],{"class":807,"line":1238},[805,2056,1200],{"class":811},[805,2058,1581],{"class":814},[805,2060,1581],{"class":814},[805,2062,1203],{"class":814},[805,2064,1797],{"class":814},[805,2066,2067,2069,2071,2073,2075,2077],{"class":807,"line":1251},[805,2068,1200],{"class":811},[805,2070,1581],{"class":814},[805,2072,1581],{"class":814},[805,2074,1581],{"class":814},[805,2076,1282],{"class":814},[805,2078,1822],{"class":814},[805,2080,2081,2083,2085,2087,2089],{"class":807,"line":1264},[805,2082,1200],{"class":811},[805,2084,1581],{"class":814},[805,2086,1581],{"class":814},[805,2088,1203],{"class":814},[805,2090,1887],{"class":814},[805,2092,2093,2095,2097,2099,2101,2103],{"class":807,"line":1277},[805,2094,1200],{"class":811},[805,2096,1581],{"class":814},[805,2098,1581],{"class":814},[805,2100,1581],{"class":814},[805,2102,1282],{"class":814},[805,2104,1899],{"class":814},[805,2106,2107,2109,2111,2113,2115],{"class":807,"line":1291},[805,2108,1200],{"class":811},[805,2110,1581],{"class":814},[805,2112,1581],{"class":814},[805,2114,1282],{"class":814},[805,2116,1923],{"class":814},[805,2118,2119,2121,2123],{"class":807,"line":1302},[805,2120,1200],{"class":811},[805,2122,1581],{"class":814},[805,2124,1691],{"class":814},[805,2126,2127,2129,2131,2133],{"class":807,"line":1313},[805,2128,1200],{"class":811},[805,2130,1581],{"class":814},[805,2132,1203],{"class":814},[805,2134,1623],{"class":814},[805,2136,2137,2139,2141,2143,2145],{"class":807,"line":1324},[805,2138,1200],{"class":811},[805,2140,1581],{"class":814},[805,2142,1581],{"class":814},[805,2144,1203],{"class":814},[805,2146,1574],{"class":814},[805,2148,2149,2151,2153,2155,2157],{"class":807,"line":1335},[805,2150,1200],{"class":811},[805,2152,1581],{"class":814},[805,2154,1581],{"class":814},[805,2156,1203],{"class":814},[805,2158,1700],{"class":814},[805,2160,2161,2163,2165,2167,2169],{"class":807,"line":1346},[805,2162,1200],{"class":811},[805,2164,1581],{"class":814},[805,2166,1581],{"class":814},[805,2168,1203],{"class":814},[805,2170,1797],{"class":814},[805,2172,2173,2175,2177,2179,2181],{"class":807,"line":1357},[805,2174,1200],{"class":811},[805,2176,1581],{"class":814},[805,2178,1581],{"class":814},[805,2180,1203],{"class":814},[805,2182,1887],{"class":814},[805,2184,2185,2187,2189,2191,2193],{"class":807,"line":1368},[805,2186,1200],{"class":811},[805,2188,1581],{"class":814},[805,2190,1581],{"class":814},[805,2192,1282],{"class":814},[805,2194,1923],{"class":814},[805,2196,2197,2199,2201],{"class":807,"line":1379},[805,2198,1200],{"class":811},[805,2200,1581],{"class":814},[805,2202,1691],{"class":814},[805,2204,2205,2207,2209,2211],{"class":807,"line":1390},[805,2206,1200],{"class":811},[805,2208,1581],{"class":814},[805,2210,1282],{"class":814},[805,2212,2213],{"class":814}," users\u002F\n",[805,2215,2216,2218,2220,2222],{"class":807,"line":1401},[805,2217,1200],{"class":811},[805,2219,1581],{"class":814},[805,2221,1669],{"class":814},[805,2223,1574],{"class":814},[805,2225,2226,2228,2230,2232],{"class":807,"line":1790},[805,2227,1200],{"class":811},[805,2229,1581],{"class":814},[805,2231,1669],{"class":814},[805,2233,1700],{"class":814},[805,2235,2236,2238,2240,2242],{"class":807,"line":1800},[805,2237,1200],{"class":811},[805,2239,1581],{"class":814},[805,2241,1669],{"class":814},[805,2243,1797],{"class":814},[805,2245,2246,2248,2250,2252],{"class":807,"line":1811},[805,2247,1200],{"class":811},[805,2249,1581],{"class":814},[805,2251,1669],{"class":814},[805,2253,1887],{"class":814},[805,2255,2256,2258,2260,2262],{"class":807,"line":1825},[805,2257,1200],{"class":811},[805,2259,1581],{"class":814},[805,2261,1681],{"class":814},[805,2263,1923],{"class":814},[805,2265,2266,2268],{"class":807,"line":1836},[805,2267,1200],{"class":811},[805,2269,1691],{"class":814},[805,2271,2272,2274,2276],{"class":807,"line":1850},[805,2273,1200],{"class":811},[805,2275,1203],{"class":814},[805,2277,1660],{"class":814},[805,2279,2280,2282,2284,2286],{"class":807,"line":1861},[805,2281,1200],{"class":811},[805,2283,1581],{"class":814},[805,2285,1203],{"class":814},[805,2287,1574],{"class":814},[805,2289,2290,2292,2294,2296,2298],{"class":807,"line":1873},[805,2291,1200],{"class":811},[805,2293,1581],{"class":814},[805,2295,1581],{"class":814},[805,2297,1203],{"class":814},[805,2299,1672],{"class":814},[805,2301,2302,2304,2306,2308,2310],{"class":807,"line":1880},[805,2303,1200],{"class":811},[805,2305,1581],{"class":814},[805,2307,1581],{"class":814},[805,2309,1282],{"class":814},[805,2311,1684],{"class":814},[805,2313,2314,2316,2318,2320],{"class":807,"line":1890},[805,2315,1200],{"class":811},[805,2317,1581],{"class":814},[805,2319,1203],{"class":814},[805,2321,2322],{"class":814}," composables\u002F\n",[805,2324,2325,2327,2329,2331,2333],{"class":807,"line":1902},[805,2326,1200],{"class":811},[805,2328,1581],{"class":814},[805,2330,1581],{"class":814},[805,2332,1282],{"class":814},[805,2334,2335],{"class":814}," useFetch.js\n",[805,2337,2338,2340,2342,2344],{"class":807,"line":1914},[805,2339,1200],{"class":811},[805,2341,1581],{"class":814},[805,2343,1203],{"class":814},[805,2345,1797],{"class":814},[805,2347,2348,2350,2352,2354,2356],{"class":807,"line":1926},[805,2349,1200],{"class":811},[805,2351,1581],{"class":814},[805,2353,1581],{"class":814},[805,2355,1282],{"class":814},[805,2357,1870],{"class":814},[805,2359,2360,2362,2364,2366],{"class":807,"line":1933},[805,2361,1200],{"class":811},[805,2363,1581],{"class":814},[805,2365,1203],{"class":814},[805,2367,2368],{"class":814}," utils\u002F\n",[805,2370,2371,2373,2375,2377,2379],{"class":807,"line":1943},[805,2372,1200],{"class":811},[805,2374,1581],{"class":814},[805,2376,1581],{"class":814},[805,2378,1282],{"class":814},[805,2380,2381],{"class":814}," formatDate.js\n",[805,2383,2385,2387,2389,2391],{"class":807,"line":2384},39,[805,2386,1200],{"class":811},[805,2388,1581],{"class":814},[805,2390,1282],{"class":814},[805,2392,2393],{"class":814}," constants\u002F\n",[805,2395,2397,2399,2401,2403],{"class":807,"line":2396},40,[805,2398,1200],{"class":811},[805,2400,1581],{"class":814},[805,2402,1681],{"class":814},[805,2404,2405],{"class":814}," roles.js\n",[805,2407,2409,2411],{"class":807,"line":2408},41,[805,2410,1200],{"class":811},[805,2412,1691],{"class":814},[805,2414,2416,2418,2420],{"class":807,"line":2415},42,[805,2417,1200],{"class":811},[805,2419,1203],{"class":814},[805,2421,1887],{"class":814},[805,2423,2425,2427,2429,2431,2434],{"class":807,"line":2424},43,[805,2426,1200],{"class":811},[805,2428,1581],{"class":814},[805,2430,1282],{"class":814},[805,2432,2433],{"class":814}," index.js",[805,2435,2436],{"class":1162},"                # Main router file that imports routes from modules\n",[805,2438,2440,2442,2444],{"class":807,"line":2439},44,[805,2441,1200],{"class":811},[805,2443,1581],{"class":814},[805,2445,2446],{"class":1162},"                               # (guards go here)\n",[805,2448,2450,2452,2454],{"class":807,"line":2449},45,[805,2451,1200],{"class":811},[805,2453,1203],{"class":814},[805,2455,1940],{"class":814},[805,2457,2459,2461],{"class":807,"line":2458},46,[805,2460,1156],{"class":811},[805,2462,1940],{"class":814},[11,2464,2465,2466,2469],{},"Each module or feature concentrates ",[29,2467,2468],{},"everything it needs in one place",", which simplifies navigation and facilitates code maintenance. This approach is especially beneficial in large or complex applications.",[11,2471,2472,2473,2476],{},"The ",[734,2474,2475],{},"shared\u002F"," folder is useful for storing components, composables, utilities, and constants that are reusable throughout the application (just be careful not to overload it).",[11,2478,2479,2480,2483],{},"The basis of this approach is to divide the application into independent modules within ",[734,2481,2482],{},"\u002Fsrc\u002Fmodules",", each with its own internal structure.",[78,2485,2486],{},[11,2487,2488,2489,2494],{},"Ultimately, the choice depends on the size and complexity of your project, as well as the ",[29,2490,2491],{},[838,2492,2493],{},"preferences of your team",", so there's no single correct way to do it. These are just examples to inspire you.",[18,2496],{},[21,2498,2500],{"id":2499},"understanding-how-the-vue-app-is-mounted","Understanding how the Vue app is mounted",[11,2502,2503,2504,2507,2508,2510],{},"The entry point of the Vue application itself is the ",[734,2505,2506],{},"src\u002Fmain.js"," file, this is where it's initialized, plugins are configured and it's mounted on the DOM. If we look at the ",[734,2509,2506],{}," file:",[797,2512,2516],{"className":2513,"code":2514,"language":2515,"meta":666,"style":666},"language-javascript shiki shiki-themes vitesse-light vitesse-dark","import '.\u002Fassets\u002Fmain.css';\n\nimport { createApp } from 'vue';\nimport { createPinia } from 'pinia';\n\nimport App from '.\u002FApp.vue';\nimport router from '.\u002Frouter';\n\nconst app = createApp(App);\n\napp.use(createPinia());\napp.use(router);\n\napp.mount('#app');\n","javascript",[734,2517,2518,2538,2543,2569,2591,2595,2613,2631,2635,2658,2662,2680,2695,2699],{"__ignoreMap":666},[805,2519,2520,2524,2528,2531,2534],{"class":807,"line":808},[805,2521,2523],{"class":2522},"sTPum","import",[805,2525,2527],{"class":2526},"scnC2"," '",[805,2529,2530],{"class":814},".\u002Fassets\u002Fmain.css",[805,2532,2533],{"class":2526},"'",[805,2535,2537],{"class":2536},"si6no",";\n",[805,2539,2540],{"class":807,"line":670},[805,2541,2542],{"emptyLinePlaceholder":681},"\n",[805,2544,2545,2547,2550,2554,2557,2560,2562,2565,2567],{"class":807,"line":667},[805,2546,2523],{"class":2522},[805,2548,2549],{"class":2536}," {",[805,2551,2553],{"class":2552},"s9nN2"," createApp",[805,2555,2556],{"class":2536}," }",[805,2558,2559],{"class":2522}," from",[805,2561,2527],{"class":2526},[805,2563,2564],{"class":814},"vue",[805,2566,2533],{"class":2526},[805,2568,2537],{"class":2536},[805,2570,2571,2573,2575,2578,2580,2582,2584,2587,2589],{"class":807,"line":1053},[805,2572,2523],{"class":2522},[805,2574,2549],{"class":2536},[805,2576,2577],{"class":2552}," createPinia",[805,2579,2556],{"class":2536},[805,2581,2559],{"class":2522},[805,2583,2527],{"class":2526},[805,2585,2586],{"class":814},"pinia",[805,2588,2533],{"class":2526},[805,2590,2537],{"class":2536},[805,2592,2593],{"class":807,"line":1186},[805,2594,2542],{"emptyLinePlaceholder":681},[805,2596,2597,2599,2602,2604,2606,2609,2611],{"class":807,"line":1197},[805,2598,2523],{"class":2522},[805,2600,2601],{"class":2552}," App",[805,2603,2559],{"class":2522},[805,2605,2527],{"class":2526},[805,2607,2608],{"class":814},".\u002FApp.vue",[805,2610,2533],{"class":2526},[805,2612,2537],{"class":2536},[805,2614,2615,2617,2620,2622,2624,2627,2629],{"class":807,"line":1212},[805,2616,2523],{"class":2522},[805,2618,2619],{"class":2552}," router",[805,2621,2559],{"class":2522},[805,2623,2527],{"class":2526},[805,2625,2626],{"class":814},".\u002Frouter",[805,2628,2533],{"class":2526},[805,2630,2537],{"class":2536},[805,2632,2633],{"class":807,"line":1225},[805,2634,2542],{"emptyLinePlaceholder":681},[805,2636,2637,2641,2644,2647,2649,2652,2655],{"class":807,"line":1238},[805,2638,2640],{"class":2639},"s5TCs","const",[805,2642,2643],{"class":2552}," app",[805,2645,2646],{"class":2536}," =",[805,2648,2553],{"class":811},[805,2650,2651],{"class":2536},"(",[805,2653,2654],{"class":2552},"App",[805,2656,2657],{"class":2536},");\n",[805,2659,2660],{"class":807,"line":1251},[805,2661,2542],{"emptyLinePlaceholder":681},[805,2663,2664,2667,2669,2672,2674,2677],{"class":807,"line":1264},[805,2665,2666],{"class":2552},"app",[805,2668,44],{"class":2536},[805,2670,2671],{"class":811},"use",[805,2673,2651],{"class":2536},[805,2675,2676],{"class":811},"createPinia",[805,2678,2679],{"class":2536},"());\n",[805,2681,2682,2684,2686,2688,2690,2693],{"class":807,"line":1277},[805,2683,2666],{"class":2552},[805,2685,44],{"class":2536},[805,2687,2671],{"class":811},[805,2689,2651],{"class":2536},[805,2691,2692],{"class":2552},"router",[805,2694,2657],{"class":2536},[805,2696,2697],{"class":807,"line":1291},[805,2698,2542],{"emptyLinePlaceholder":681},[805,2700,2701,2703,2705,2708,2710,2712,2715,2717],{"class":807,"line":1302},[805,2702,2666],{"class":2552},[805,2704,44],{"class":2536},[805,2706,2707],{"class":811},"mount",[805,2709,2651],{"class":2536},[805,2711,2533],{"class":2526},[805,2713,2714],{"class":814},"#app",[805,2716,2533],{"class":2526},[805,2718,2657],{"class":2536},[11,2720,2721],{},"Here's what we're doing:",[317,2723,2724,2731,2740,2745,2751,2761,2768,2774],{},[52,2725,2726,2727,2730],{},"We import the global styles from ",[734,2728,2729],{},"main.css",". This file contains any styles you want to apply globally.",[52,2732,2733,2734,36,2737,2739],{},"We import the ",[734,2735,2736],{},"createApp",[734,2738,2676],{}," functions to create new instances of Vue and Pinia.",[52,2741,2742,2743,44],{},"We import the root component ",[734,2744,1516],{},[52,2746,2747,2748,44],{},"We import the route configuration from ",[734,2749,2750],{},"src\u002Frouter\u002Findex.js",[52,2752,2753,2754,2757,2758,2760],{},"We create the application instance with ",[734,2755,2756],{},"createApp(App)",", passing the root component ",[734,2759,2654],{}," as an argument, indicating that this will be the starting point of our Vue application, basically we're telling Vue: \"Here's the main component, you'll build the entire application from it\".",[52,2762,2763,2764,2767],{},"We use ",[734,2765,2766],{},"app.use(createPinia())"," to register the Pinia instance, enabling global state management in our application.",[52,2769,2763,2770,2773],{},[734,2771,2772],{},"app.use(router)"," to register the Vue Router instance and enable navigation between views.",[52,2775,2776,2777,2779,2780,44],{},"Finally, we mount the Vue application on the DOM element with the id ",[734,2778,2666],{}," using ",[734,2781,2782],{},"app.mount('#app')",[11,2784,2785,2786,2789],{},"Now, open the ",[734,2787,2788],{},"index.html"," file located at the root of the project:",[797,2791,2795],{"className":2792,"code":2793,"language":2794,"meta":666,"style":666},"language-html shiki shiki-themes vitesse-light vitesse-dark","\u003C!DOCTYPE html>\n\u003Chtml lang=\"\">\n  \u003Chead>\n    \u003Cmeta charset=\"UTF-8\" \u002F>\n    \u003Clink rel=\"icon\" href=\"\u002Ffavicon.ico\" \u002F>\n    \u003Cmeta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\" \u002F>\n    \u003Ctitle>Vite App\u003C\u002Ftitle>\n  \u003C\u002Fhead>\n  \u003Cbody>\n    \u003Cdiv id=\"app\">\u003C\u002Fdiv>\n    \u003Cscript type=\"module\" src=\"\u002Fsrc\u002Fmain.js\">\u003C\u002Fscript>\n  \u003C\u002Fbody>\n\u003C\u002Fhtml>\n","html",[734,2796,2797,2811,2829,2839,2863,2896,2928,2949,2958,2967,2992,3029,3037],{"__ignoreMap":666},[805,2798,2799,2802,2805,2808],{"class":807,"line":808},[805,2800,2801],{"class":2536},"\u003C!",[805,2803,2804],{"class":2522},"DOCTYPE",[805,2806,2807],{"class":2552}," html",[805,2809,2810],{"class":2536},">\n",[805,2812,2813,2816,2818,2821,2824,2827],{"class":807,"line":670},[805,2814,2815],{"class":2536},"\u003C",[805,2817,2794],{"class":2522},[805,2819,2820],{"class":2552}," lang",[805,2822,2823],{"class":2536},"=",[805,2825,2826],{"class":2526},"\"\"",[805,2828,2810],{"class":2536},[805,2830,2831,2834,2837],{"class":807,"line":667},[805,2832,2833],{"class":2536},"  \u003C",[805,2835,2836],{"class":2522},"head",[805,2838,2810],{"class":2536},[805,2840,2841,2844,2847,2850,2852,2855,2858,2860],{"class":807,"line":1053},[805,2842,2843],{"class":2536},"    \u003C",[805,2845,2846],{"class":2522},"meta",[805,2848,2849],{"class":2552}," charset",[805,2851,2823],{"class":2536},[805,2853,2854],{"class":2526},"\"",[805,2856,2857],{"class":814},"UTF-8",[805,2859,2854],{"class":2526},[805,2861,2862],{"class":2536}," \u002F>\n",[805,2864,2865,2867,2870,2873,2875,2877,2880,2882,2885,2887,2889,2892,2894],{"class":807,"line":1186},[805,2866,2843],{"class":2536},[805,2868,2869],{"class":2522},"link",[805,2871,2872],{"class":2552}," rel",[805,2874,2823],{"class":2536},[805,2876,2854],{"class":2526},[805,2878,2879],{"class":814},"icon",[805,2881,2854],{"class":2526},[805,2883,2884],{"class":2552}," href",[805,2886,2823],{"class":2536},[805,2888,2854],{"class":2526},[805,2890,2891],{"class":814},"\u002Ffavicon.ico",[805,2893,2854],{"class":2526},[805,2895,2862],{"class":2536},[805,2897,2898,2900,2902,2905,2907,2909,2912,2914,2917,2919,2921,2924,2926],{"class":807,"line":1197},[805,2899,2843],{"class":2536},[805,2901,2846],{"class":2522},[805,2903,2904],{"class":2552}," name",[805,2906,2823],{"class":2536},[805,2908,2854],{"class":2526},[805,2910,2911],{"class":814},"viewport",[805,2913,2854],{"class":2526},[805,2915,2916],{"class":2552}," content",[805,2918,2823],{"class":2536},[805,2920,2854],{"class":2526},[805,2922,2923],{"class":814},"width=device-width, initial-scale=1.0",[805,2925,2854],{"class":2526},[805,2927,2862],{"class":2536},[805,2929,2930,2932,2935,2938,2942,2945,2947],{"class":807,"line":1212},[805,2931,2843],{"class":2536},[805,2933,2934],{"class":2522},"title",[805,2936,2937],{"class":2536},">",[805,2939,2941],{"class":2940},"s8w-G","Vite App",[805,2943,2944],{"class":2536},"\u003C\u002F",[805,2946,2934],{"class":2522},[805,2948,2810],{"class":2536},[805,2950,2951,2954,2956],{"class":807,"line":1225},[805,2952,2953],{"class":2536},"  \u003C\u002F",[805,2955,2836],{"class":2522},[805,2957,2810],{"class":2536},[805,2959,2960,2962,2965],{"class":807,"line":1238},[805,2961,2833],{"class":2536},[805,2963,2964],{"class":2522},"body",[805,2966,2810],{"class":2536},[805,2968,2969,2971,2974,2977,2979,2981,2983,2985,2988,2990],{"class":807,"line":1251},[805,2970,2843],{"class":2536},[805,2972,2973],{"class":2522},"div",[805,2975,2976],{"class":2552}," id",[805,2978,2823],{"class":2536},[805,2980,2854],{"class":2526},[805,2982,2666],{"class":814},[805,2984,2854],{"class":2526},[805,2986,2987],{"class":2536},">\u003C\u002F",[805,2989,2973],{"class":2522},[805,2991,2810],{"class":2536},[805,2993,2994,2996,2999,3002,3004,3006,3009,3011,3014,3016,3018,3021,3023,3025,3027],{"class":807,"line":1264},[805,2995,2843],{"class":2536},[805,2997,2998],{"class":2522},"script",[805,3000,3001],{"class":2552}," type",[805,3003,2823],{"class":2536},[805,3005,2854],{"class":2526},[805,3007,3008],{"class":814},"module",[805,3010,2854],{"class":2526},[805,3012,3013],{"class":2552}," src",[805,3015,2823],{"class":2536},[805,3017,2854],{"class":2526},[805,3019,3020],{"class":814},"\u002Fsrc\u002Fmain.js",[805,3022,2854],{"class":2526},[805,3024,2987],{"class":2536},[805,3026,2998],{"class":2522},[805,3028,2810],{"class":2536},[805,3030,3031,3033,3035],{"class":807,"line":1277},[805,3032,2953],{"class":2536},[805,3034,2964],{"class":2522},[805,3036,2810],{"class":2536},[805,3038,3039,3041,3043],{"class":807,"line":1291},[805,3040,2944],{"class":2536},[805,3042,2794],{"class":2522},[805,3044,2810],{"class":2536},[11,3046,3047,3048,3051,3052,3054,3055,3058,3059,3061],{},"Here we see that the HTML file has a ",[734,3049,3050],{},"\u003Cdiv>"," with the id ",[734,3053,2666],{},", which is where Vue will mount our application. The ",[734,3056,3057],{},"\u003Cscript>"," that follows loads our ",[734,3060,1522],{}," file, which is where we initialize everything.",[11,3063,3064,3065,2510],{},"Open the ",[734,3066,3067],{},"src\u002FApp.vue",[797,3069,3072],{"className":3070,"code":3071,"language":2564,"meta":666,"style":666},"language-vue shiki shiki-themes vitesse-light vitesse-dark","\u003Cscript setup>\nimport { RouterLink, RouterView } from 'vue-router';\nimport HelloWorld from '.\u002Fcomponents\u002FHelloWorld.vue';\n\u003C\u002Fscript>\n\n\u003Cscript setup>\nimport { RouterLink, RouterView } from 'vue-router';\nimport HelloWorld from '.\u002Fcomponents\u002FHelloWorld.vue';\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n  \u003Cheader>\n    \u003Cimg alt=\"Vue logo\" class=\"logo\" src=\"@\u002Fassets\u002Flogo.svg\" width=\"125\" height=\"125\" \u002F>\n\n    \u003Cdiv class=\"wrapper\">\n      \u003CHelloWorld msg=\"You did it!\" \u002F>\n\n      \u003Cnav>\n        \u003CRouterLink to=\"\u002F\">Home\u003C\u002FRouterLink>\n        \u003CRouterLink to=\"\u002Fabout\">About\u003C\u002FRouterLink>\n      \u003C\u002Fnav>\n    \u003C\u002Fdiv>\n  \u003C\u002Fheader>\n\n  \u003CRouterView \u002F>\n\u003C\u002Ftemplate>\n\n\u003Cstyle scoped>\n\u002F* Estilos CSS *\u002F\n\u003C\u002Fstyle>\n",[734,3073,3074,3085,3113,3131,3139,3143,3153,3177,3193,3201,3205,3214,3223,3289,3293,3312,3334,3338,3347,3378,3406,3415,3424,3432,3436,3445,3453,3457,3469,3474],{"__ignoreMap":666},[805,3075,3076,3078,3080,3083],{"class":807,"line":808},[805,3077,2815],{"class":2536},[805,3079,2998],{"class":2522},[805,3081,3082],{"class":2552}," setup",[805,3084,2810],{"class":2536},[805,3086,3087,3089,3091,3094,3097,3100,3102,3104,3106,3109,3111],{"class":807,"line":670},[805,3088,2523],{"class":2522},[805,3090,2549],{"class":2536},[805,3092,3093],{"class":2552}," RouterLink",[805,3095,3096],{"class":2536},",",[805,3098,3099],{"class":2552}," RouterView",[805,3101,2556],{"class":2536},[805,3103,2559],{"class":2522},[805,3105,2527],{"class":2526},[805,3107,3108],{"class":814},"vue-router",[805,3110,2533],{"class":2526},[805,3112,2537],{"class":2536},[805,3114,3115,3117,3120,3122,3124,3127,3129],{"class":807,"line":667},[805,3116,2523],{"class":2522},[805,3118,3119],{"class":2552}," HelloWorld",[805,3121,2559],{"class":2522},[805,3123,2527],{"class":2526},[805,3125,3126],{"class":814},".\u002Fcomponents\u002FHelloWorld.vue",[805,3128,2533],{"class":2526},[805,3130,2537],{"class":2536},[805,3132,3133,3135,3137],{"class":807,"line":1053},[805,3134,2944],{"class":2536},[805,3136,2998],{"class":2522},[805,3138,2810],{"class":2536},[805,3140,3141],{"class":807,"line":1186},[805,3142,2542],{"emptyLinePlaceholder":681},[805,3144,3145,3147,3149,3151],{"class":807,"line":1197},[805,3146,2815],{"class":2536},[805,3148,2998],{"class":2522},[805,3150,3082],{"class":2552},[805,3152,2810],{"class":2536},[805,3154,3155,3157,3159,3161,3163,3165,3167,3169,3171,3173,3175],{"class":807,"line":1212},[805,3156,2523],{"class":2522},[805,3158,2549],{"class":2536},[805,3160,3093],{"class":2552},[805,3162,3096],{"class":2536},[805,3164,3099],{"class":2552},[805,3166,2556],{"class":2536},[805,3168,2559],{"class":2522},[805,3170,2527],{"class":2526},[805,3172,3108],{"class":814},[805,3174,2533],{"class":2526},[805,3176,2537],{"class":2536},[805,3178,3179,3181,3183,3185,3187,3189,3191],{"class":807,"line":1225},[805,3180,2523],{"class":2522},[805,3182,3119],{"class":2552},[805,3184,2559],{"class":2522},[805,3186,2527],{"class":2526},[805,3188,3126],{"class":814},[805,3190,2533],{"class":2526},[805,3192,2537],{"class":2536},[805,3194,3195,3197,3199],{"class":807,"line":1238},[805,3196,2944],{"class":2536},[805,3198,2998],{"class":2522},[805,3200,2810],{"class":2536},[805,3202,3203],{"class":807,"line":1251},[805,3204,2542],{"emptyLinePlaceholder":681},[805,3206,3207,3209,3212],{"class":807,"line":1264},[805,3208,2815],{"class":2536},[805,3210,3211],{"class":2522},"template",[805,3213,2810],{"class":2536},[805,3215,3216,3218,3221],{"class":807,"line":1277},[805,3217,2833],{"class":2536},[805,3219,3220],{"class":2522},"header",[805,3222,2810],{"class":2536},[805,3224,3225,3227,3229,3232,3234,3236,3239,3241,3244,3246,3248,3251,3253,3255,3257,3259,3262,3264,3267,3269,3271,3274,3276,3279,3281,3283,3285,3287],{"class":807,"line":1291},[805,3226,2843],{"class":2536},[805,3228,833],{"class":2522},[805,3230,3231],{"class":2552}," alt",[805,3233,2823],{"class":2536},[805,3235,2854],{"class":2526},[805,3237,3238],{"class":814},"Vue logo",[805,3240,2854],{"class":2526},[805,3242,3243],{"class":2552}," class",[805,3245,2823],{"class":2536},[805,3247,2854],{"class":2526},[805,3249,3250],{"class":814},"logo",[805,3252,2854],{"class":2526},[805,3254,3013],{"class":2552},[805,3256,2823],{"class":2536},[805,3258,2854],{"class":2526},[805,3260,3261],{"class":814},"@\u002Fassets\u002Flogo.svg",[805,3263,2854],{"class":2526},[805,3265,3266],{"class":2552}," width",[805,3268,2823],{"class":2536},[805,3270,2854],{"class":2526},[805,3272,3273],{"class":814},"125",[805,3275,2854],{"class":2526},[805,3277,3278],{"class":2552}," height",[805,3280,2823],{"class":2536},[805,3282,2854],{"class":2526},[805,3284,3273],{"class":814},[805,3286,2854],{"class":2526},[805,3288,2862],{"class":2536},[805,3290,3291],{"class":807,"line":1302},[805,3292,2542],{"emptyLinePlaceholder":681},[805,3294,3295,3297,3299,3301,3303,3305,3308,3310],{"class":807,"line":1313},[805,3296,2843],{"class":2536},[805,3298,2973],{"class":2522},[805,3300,3243],{"class":2552},[805,3302,2823],{"class":2536},[805,3304,2854],{"class":2526},[805,3306,3307],{"class":814},"wrapper",[805,3309,2854],{"class":2526},[805,3311,2810],{"class":2536},[805,3313,3314,3317,3320,3323,3325,3327,3330,3332],{"class":807,"line":1324},[805,3315,3316],{"class":2536},"      \u003C",[805,3318,3319],{"class":2522},"HelloWorld",[805,3321,3322],{"class":2552}," msg",[805,3324,2823],{"class":2536},[805,3326,2854],{"class":2526},[805,3328,3329],{"class":814},"You did it!",[805,3331,2854],{"class":2526},[805,3333,2862],{"class":2536},[805,3335,3336],{"class":807,"line":1335},[805,3337,2542],{"emptyLinePlaceholder":681},[805,3339,3340,3342,3345],{"class":807,"line":1346},[805,3341,3316],{"class":2536},[805,3343,3344],{"class":2522},"nav",[805,3346,2810],{"class":2536},[805,3348,3349,3352,3355,3358,3360,3362,3365,3367,3369,3372,3374,3376],{"class":807,"line":1357},[805,3350,3351],{"class":2536},"        \u003C",[805,3353,3354],{"class":2522},"RouterLink",[805,3356,3357],{"class":2552}," to",[805,3359,2823],{"class":2536},[805,3361,2854],{"class":2526},[805,3363,3364],{"class":814},"\u002F",[805,3366,2854],{"class":2526},[805,3368,2937],{"class":2536},[805,3370,3371],{"class":2940},"Home",[805,3373,2944],{"class":2536},[805,3375,3354],{"class":2522},[805,3377,2810],{"class":2536},[805,3379,3380,3382,3384,3386,3388,3390,3393,3395,3397,3400,3402,3404],{"class":807,"line":1368},[805,3381,3351],{"class":2536},[805,3383,3354],{"class":2522},[805,3385,3357],{"class":2552},[805,3387,2823],{"class":2536},[805,3389,2854],{"class":2526},[805,3391,3392],{"class":814},"\u002Fabout",[805,3394,2854],{"class":2526},[805,3396,2937],{"class":2536},[805,3398,3399],{"class":2940},"About",[805,3401,2944],{"class":2536},[805,3403,3354],{"class":2522},[805,3405,2810],{"class":2536},[805,3407,3408,3411,3413],{"class":807,"line":1379},[805,3409,3410],{"class":2536},"      \u003C\u002F",[805,3412,3344],{"class":2522},[805,3414,2810],{"class":2536},[805,3416,3417,3420,3422],{"class":807,"line":1390},[805,3418,3419],{"class":2536},"    \u003C\u002F",[805,3421,2973],{"class":2522},[805,3423,2810],{"class":2536},[805,3425,3426,3428,3430],{"class":807,"line":1401},[805,3427,2953],{"class":2536},[805,3429,3220],{"class":2522},[805,3431,2810],{"class":2536},[805,3433,3434],{"class":807,"line":1790},[805,3435,2542],{"emptyLinePlaceholder":681},[805,3437,3438,3440,3443],{"class":807,"line":1800},[805,3439,2833],{"class":2536},[805,3441,3442],{"class":2522},"RouterView",[805,3444,2862],{"class":2536},[805,3446,3447,3449,3451],{"class":807,"line":1811},[805,3448,2944],{"class":2536},[805,3450,3211],{"class":2522},[805,3452,2810],{"class":2536},[805,3454,3455],{"class":807,"line":1825},[805,3456,2542],{"emptyLinePlaceholder":681},[805,3458,3459,3461,3464,3467],{"class":807,"line":1836},[805,3460,2815],{"class":2536},[805,3462,3463],{"class":2522},"style",[805,3465,3466],{"class":2552}," scoped",[805,3468,2810],{"class":2536},[805,3470,3471],{"class":807,"line":1850},[805,3472,3473],{"class":1162},"\u002F* Estilos CSS *\u002F\n",[805,3475,3476,3478,3480],{"class":807,"line":1861},[805,3477,2944],{"class":2536},[805,3479,3463],{"class":2522},[805,3481,2810],{"class":2536},[11,3483,3484,3485,3487,3488,3490,3491,3494],{},"Here we see that ",[734,3486,1516],{}," contains a header with a logo, a ",[734,3489,3319],{}," component, and a navigation menu with links to the routes defined. The ",[734,3492,3493],{},"\u003CRouterView \u002F>"," is where the corresponding views will be rendered based on the current route, thanks to Vue Router.",[11,3496,3497,3498,2510],{},"Let's explore the router configuration. Open the ",[734,3499,2750],{},[797,3501,3503],{"className":2513,"code":3502,"language":2515,"meta":666,"style":666},"import { createRouter, createWebHistory } from 'vue-router';\nimport HomeView from '..\u002Fviews\u002FHomeView.vue';\n\nconst router = createRouter({\n  history: createWebHistory(import.meta.env.BASE_URL),\n  routes: [\n    {\n      path: '\u002F',\n      name: 'home',\n      component: HomeView,\n    },\n    {\n      path: '\u002Fabout',\n      name: 'about',\n      \u002F* This route uses lazy loading *\u002F\n      component: () => import('..\u002Fviews\u002FAboutView.vue'),\n    },\n  ],\n});\n\nexport default router;\n",[734,3504,3505,3531,3549,3553,3566,3597,3607,3612,3628,3644,3655,3660,3664,3678,3693,3698,3724,3728,3733,3738,3742],{"__ignoreMap":666},[805,3506,3507,3509,3511,3514,3516,3519,3521,3523,3525,3527,3529],{"class":807,"line":808},[805,3508,2523],{"class":2522},[805,3510,2549],{"class":2536},[805,3512,3513],{"class":2552}," createRouter",[805,3515,3096],{"class":2536},[805,3517,3518],{"class":2552}," createWebHistory",[805,3520,2556],{"class":2536},[805,3522,2559],{"class":2522},[805,3524,2527],{"class":2526},[805,3526,3108],{"class":814},[805,3528,2533],{"class":2526},[805,3530,2537],{"class":2536},[805,3532,3533,3535,3538,3540,3542,3545,3547],{"class":807,"line":670},[805,3534,2523],{"class":2522},[805,3536,3537],{"class":2552}," HomeView",[805,3539,2559],{"class":2522},[805,3541,2527],{"class":2526},[805,3543,3544],{"class":814},"..\u002Fviews\u002FHomeView.vue",[805,3546,2533],{"class":2526},[805,3548,2537],{"class":2536},[805,3550,3551],{"class":807,"line":667},[805,3552,2542],{"emptyLinePlaceholder":681},[805,3554,3555,3557,3559,3561,3563],{"class":807,"line":1053},[805,3556,2640],{"class":2639},[805,3558,2619],{"class":2552},[805,3560,2646],{"class":2536},[805,3562,3513],{"class":811},[805,3564,3565],{"class":2536},"({\n",[805,3567,3568,3571,3574,3576,3578,3580,3582,3584,3586,3589,3591,3594],{"class":807,"line":1186},[805,3569,3570],{"class":1009},"  history",[805,3572,3573],{"class":2536},":",[805,3575,3518],{"class":811},[805,3577,2651],{"class":2536},[805,3579,2523],{"class":2522},[805,3581,44],{"class":2536},[805,3583,2846],{"class":1009},[805,3585,44],{"class":2536},[805,3587,3588],{"class":2552},"env",[805,3590,44],{"class":2536},[805,3592,3593],{"class":2552},"BASE_URL",[805,3595,3596],{"class":2536},"),\n",[805,3598,3599,3602,3604],{"class":807,"line":1197},[805,3600,3601],{"class":1009},"  routes",[805,3603,3573],{"class":2536},[805,3605,3606],{"class":2536}," [\n",[805,3608,3609],{"class":807,"line":1212},[805,3610,3611],{"class":2536},"    {\n",[805,3613,3614,3617,3619,3621,3623,3625],{"class":807,"line":1225},[805,3615,3616],{"class":1009},"      path",[805,3618,3573],{"class":2536},[805,3620,2527],{"class":2526},[805,3622,3364],{"class":814},[805,3624,2533],{"class":2526},[805,3626,3627],{"class":2536},",\n",[805,3629,3630,3633,3635,3637,3640,3642],{"class":807,"line":1238},[805,3631,3632],{"class":1009},"      name",[805,3634,3573],{"class":2536},[805,3636,2527],{"class":2526},[805,3638,3639],{"class":814},"home",[805,3641,2533],{"class":2526},[805,3643,3627],{"class":2536},[805,3645,3646,3649,3651,3653],{"class":807,"line":1251},[805,3647,3648],{"class":1009},"      component",[805,3650,3573],{"class":2536},[805,3652,3537],{"class":2552},[805,3654,3627],{"class":2536},[805,3656,3657],{"class":807,"line":1264},[805,3658,3659],{"class":2536},"    },\n",[805,3661,3662],{"class":807,"line":1277},[805,3663,3611],{"class":2536},[805,3665,3666,3668,3670,3672,3674,3676],{"class":807,"line":1291},[805,3667,3616],{"class":1009},[805,3669,3573],{"class":2536},[805,3671,2527],{"class":2526},[805,3673,3392],{"class":814},[805,3675,2533],{"class":2526},[805,3677,3627],{"class":2536},[805,3679,3680,3682,3684,3686,3689,3691],{"class":807,"line":1302},[805,3681,3632],{"class":1009},[805,3683,3573],{"class":2536},[805,3685,2527],{"class":2526},[805,3687,3688],{"class":814},"about",[805,3690,2533],{"class":2526},[805,3692,3627],{"class":2536},[805,3694,3695],{"class":807,"line":1313},[805,3696,3697],{"class":1162},"      \u002F* This route uses lazy loading *\u002F\n",[805,3699,3700,3702,3704,3707,3710,3713,3715,3717,3720,3722],{"class":807,"line":1324},[805,3701,3648],{"class":811},[805,3703,3573],{"class":2536},[805,3705,3706],{"class":2536}," ()",[805,3708,3709],{"class":2536}," =>",[805,3711,3712],{"class":2639}," import",[805,3714,2651],{"class":2536},[805,3716,2533],{"class":2526},[805,3718,3719],{"class":814},"..\u002Fviews\u002FAboutView.vue",[805,3721,2533],{"class":2526},[805,3723,3596],{"class":2536},[805,3725,3726],{"class":807,"line":1335},[805,3727,3659],{"class":2536},[805,3729,3730],{"class":807,"line":1346},[805,3731,3732],{"class":2536},"  ],\n",[805,3734,3735],{"class":807,"line":1357},[805,3736,3737],{"class":2536},"});\n",[805,3739,3740],{"class":807,"line":1368},[805,3741,2542],{"emptyLinePlaceholder":681},[805,3743,3744,3747,3750,3752],{"class":807,"line":1379},[805,3745,3746],{"class":2522},"export",[805,3748,3749],{"class":2522}," default",[805,3751,2619],{"class":2552},[805,3753,2537],{"class":2536},[11,3755,3756,3757,3760,3761,3764,3765,3767,3768,3770,3771,3773,3774,3776,3777,3782],{},"Here we're importing the necessary functions from Vue Router and defining a basic route for the ",[734,3758,3759],{},"HomeView"," view and another for ",[734,3762,3763],{},"AboutView",". The ",[734,3766,3364],{}," route renders ",[734,3769,3759],{},", and the ",[734,3772,3392],{}," route loads ",[734,3775,3763],{}," using lazy loading (",[88,3778,3781],{"href":3779,"target":280,"rel":3780},"https:\u002F\u002Fmedium.com\u002F@drewcauchi\u002Flazy-loading-in-vue-js-bb32018d2c2d",[282,283],"lazy loading",").",[11,3784,3785],{},"Let's review the complete flow:",[317,3787,3788,3795,3804],{},[52,3789,2472,3790,3792,3793,44],{},[734,3791,2788],{}," file is what the browser shows us, and the first thing it does is load ",[734,3794,1522],{},[52,3796,3797,3798,3800,3801,3803],{},"In ",[734,3799,1522],{},", the Vue application is created with ",[734,3802,1516],{}," as the root component (apart from importing styles, configuring Pinia, Vue Router and other configurations).",[52,3805,3806,3808,3809,3811],{},[734,3807,1516],{}," defines the main structure of the application (like a main layout) and uses ",[734,3810,3493],{}," to render views based on the current route.",[11,3813,3814,3818],{},[833,3815],{"alt":3816,"src":3817},"Flow of a Vue app","\u002Fblog\u002Fgetting-started-vue-vite\u002Fen\u002Fexplaining-vue-app.webp",[838,3819,3816],{},[11,3821,3822,3823,3825,3826,3828],{},"This means that everything you see inside ",[734,3824,1516],{}," will always be present (like the header and navigation menu), while the main content will be displayed inside ",[734,3827,3493],{}," and will change depending on the current route, thanks to our router.",[11,3830,3831,3832,3834,3835,44],{},"Normally, in more complex applications, ",[734,3833,1516],{}," will also contain other common elements like a footer, sidebars, global modals, etc.\nAnd when needed, specific layouts are created for different application modules, for example, a layout for admin views and another for public views like login. You can explore more at ",[88,3836,3839],{"href":3837,"target":280,"rel":3838},"https:\u002F\u002Fvueschool.io\u002Farticles\u002Fvuejs-tutorials\u002Fcomposing-layouts-with-vue-router\u002F",[282,283],"this Vue School resource",[18,3841],{},[11,3843,3844],{},"At this point we already have a clear idea of how a Vue 3 application with Vite is structured and mounted. You can start to explore and modify the components, views and routes to become more familiar with the framework. Now we'll continue exploring other important concepts and tools in the ecosystem.",[18,3846],{},[21,3848,3850],{"id":3849},"pinia-for-global-state-management","Pinia for global state management",[11,3852,3853],{},"Pinia helps us manage the global state of our application: data that should be accessible from any part of the application, data that we need to share between multiple components or views.",[11,3855,3856],{},"For example, when we need access to authenticated user information in different parts of the app, or if we want to manage a shopping cart that can be consulted from different components.",[11,3858,3859,3860,3573],{},"Let's see a basic example. Create the file ",[734,3861,3862],{},"src\u002Fstore\u002Fauth.store.js",[797,3864,3866],{"className":2513,"code":3865,"language":2515,"meta":666,"style":666},"import { defineStore } from 'pinia';\n\nexport const useAuthStore = defineStore('auth', {\n  state: () => ({\n    user: null,\n    token: null,\n  }),\n\n  actions: {\n    login(userData, token) {\n      this.user = userData;\n      this.token = token;\n    },\n\n    logout() {\n      this.user = null;\n      this.token = null;\n    },\n  },\n});\n",[734,3867,3868,3889,3893,3921,3935,3947,3958,3963,3967,3976,3996,4014,4029,4033,4037,4047,4061,4075,4079,4084],{"__ignoreMap":666},[805,3869,3870,3872,3874,3877,3879,3881,3883,3885,3887],{"class":807,"line":808},[805,3871,2523],{"class":2522},[805,3873,2549],{"class":2536},[805,3875,3876],{"class":2552}," defineStore",[805,3878,2556],{"class":2536},[805,3880,2559],{"class":2522},[805,3882,2527],{"class":2526},[805,3884,2586],{"class":814},[805,3886,2533],{"class":2526},[805,3888,2537],{"class":2536},[805,3890,3891],{"class":807,"line":670},[805,3892,2542],{"emptyLinePlaceholder":681},[805,3894,3895,3897,3900,3903,3905,3907,3909,3911,3914,3916,3918],{"class":807,"line":667},[805,3896,3746],{"class":2522},[805,3898,3899],{"class":2639}," const",[805,3901,3902],{"class":2552}," useAuthStore",[805,3904,2646],{"class":2536},[805,3906,3876],{"class":811},[805,3908,2651],{"class":2536},[805,3910,2533],{"class":2526},[805,3912,3913],{"class":814},"auth",[805,3915,2533],{"class":2526},[805,3917,3096],{"class":2536},[805,3919,3920],{"class":2536}," {\n",[805,3922,3923,3926,3928,3930,3932],{"class":807,"line":1053},[805,3924,3925],{"class":811},"  state",[805,3927,3573],{"class":2536},[805,3929,3706],{"class":2536},[805,3931,3709],{"class":2536},[805,3933,3934],{"class":2536}," ({\n",[805,3936,3937,3940,3942,3945],{"class":807,"line":1186},[805,3938,3939],{"class":1009},"    user",[805,3941,3573],{"class":2536},[805,3943,3944],{"class":2639}," null",[805,3946,3627],{"class":2536},[805,3948,3949,3952,3954,3956],{"class":807,"line":1197},[805,3950,3951],{"class":1009},"    token",[805,3953,3573],{"class":2536},[805,3955,3944],{"class":2639},[805,3957,3627],{"class":2536},[805,3959,3960],{"class":807,"line":1212},[805,3961,3962],{"class":2536},"  }),\n",[805,3964,3965],{"class":807,"line":1225},[805,3966,2542],{"emptyLinePlaceholder":681},[805,3968,3969,3972,3974],{"class":807,"line":1238},[805,3970,3971],{"class":1009},"  actions",[805,3973,3573],{"class":2536},[805,3975,3920],{"class":2536},[805,3977,3978,3981,3983,3986,3988,3991,3994],{"class":807,"line":1251},[805,3979,3980],{"class":811},"    login",[805,3982,2651],{"class":2536},[805,3984,3985],{"class":2552},"userData",[805,3987,3096],{"class":2536},[805,3989,3990],{"class":2552}," token",[805,3992,3993],{"class":2536},")",[805,3995,3920],{"class":2536},[805,3997,3998,4002,4004,4007,4009,4012],{"class":807,"line":1264},[805,3999,4001],{"class":4000},"sfsYZ","      this",[805,4003,44],{"class":2536},[805,4005,4006],{"class":2552},"user",[805,4008,2646],{"class":2536},[805,4010,4011],{"class":2552}," userData",[805,4013,2537],{"class":2536},[805,4015,4016,4018,4020,4023,4025,4027],{"class":807,"line":1277},[805,4017,4001],{"class":4000},[805,4019,44],{"class":2536},[805,4021,4022],{"class":2552},"token",[805,4024,2646],{"class":2536},[805,4026,3990],{"class":2552},[805,4028,2537],{"class":2536},[805,4030,4031],{"class":807,"line":1291},[805,4032,3659],{"class":2536},[805,4034,4035],{"class":807,"line":1302},[805,4036,2542],{"emptyLinePlaceholder":681},[805,4038,4039,4042,4045],{"class":807,"line":1313},[805,4040,4041],{"class":811},"    logout",[805,4043,4044],{"class":2536},"()",[805,4046,3920],{"class":2536},[805,4048,4049,4051,4053,4055,4057,4059],{"class":807,"line":1324},[805,4050,4001],{"class":4000},[805,4052,44],{"class":2536},[805,4054,4006],{"class":2552},[805,4056,2646],{"class":2536},[805,4058,3944],{"class":2639},[805,4060,2537],{"class":2536},[805,4062,4063,4065,4067,4069,4071,4073],{"class":807,"line":1335},[805,4064,4001],{"class":4000},[805,4066,44],{"class":2536},[805,4068,4022],{"class":2552},[805,4070,2646],{"class":2536},[805,4072,3944],{"class":2639},[805,4074,2537],{"class":2536},[805,4076,4077],{"class":807,"line":1346},[805,4078,3659],{"class":2536},[805,4080,4081],{"class":807,"line":1357},[805,4082,4083],{"class":2536},"  },\n",[805,4085,4086],{"class":807,"line":1368},[805,4087,3737],{"class":2536},[11,4089,4090,4091,4093,4094,36,4096,4098,4099,3573],{},"Here we're defining a store called ",[734,4092,3913],{}," that has a state with the properties ",[734,4095,4006],{},[734,4097,4022],{},", and two actions to log in and log out.\nNow, let's see how to use this store in a component. Create a Vue component, for example ",[734,4100,4101],{},"src\u002Fcomponents\u002FAuth.vue",[797,4103,4105],{"className":3070,"code":4104,"language":2564,"meta":666,"style":666},"\u003Ctemplate>\n  \u003Cdiv>\n    \u003Cdiv v-if=\"authStore.user\">\n      \u003Cp>Welcome, {{ authStore.user.name }}!\u003C\u002Fp>\n      \u003Cbutton @click=\"logout\">Logout\u003C\u002Fbutton>\n    \u003C\u002Fdiv>\n    \u003Cdiv v-else>\n      \u003Cbutton @click=\"login\">Login\u003C\u002Fbutton>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n\u003C\u002Ftemplate>\n\u003Cscript setup>\nimport { useAuthStore } from '..\u002Fstore\u002Fauth.store';\nconst authStore = useAuthStore();\nconst login = () => {\n  \u002F\u002F Simulate a login\n  const userData = { name: 'John Doe', email: 'john.doe@example.com' };\n  const token = 'fake-jwt-token';\n  authStore.login(userData, token);\n};\n\nconst logout = () => {\n  authStore.logout();\n};\n\u003C\u002Fscript>\n",[734,4106,4107,4115,4123,4143,4160,4190,4198,4209,4237,4245,4253,4261,4271,4292,4306,4321,4326,4365,4382,4401,4406,4410,4425,4435,4439],{"__ignoreMap":666},[805,4108,4109,4111,4113],{"class":807,"line":808},[805,4110,2815],{"class":2536},[805,4112,3211],{"class":2522},[805,4114,2810],{"class":2536},[805,4116,4117,4119,4121],{"class":807,"line":670},[805,4118,2833],{"class":2536},[805,4120,2973],{"class":2522},[805,4122,2810],{"class":2536},[805,4124,4125,4127,4129,4132,4134,4136,4139,4141],{"class":807,"line":667},[805,4126,2843],{"class":2536},[805,4128,2973],{"class":2522},[805,4130,4131],{"class":2552}," v-if",[805,4133,2823],{"class":2536},[805,4135,2854],{"class":2526},[805,4137,4138],{"class":814},"authStore.user",[805,4140,2854],{"class":2526},[805,4142,2810],{"class":2536},[805,4144,4145,4147,4149,4151,4154,4156,4158],{"class":807,"line":1053},[805,4146,3316],{"class":2536},[805,4148,11],{"class":2522},[805,4150,2937],{"class":2536},[805,4152,4153],{"class":2940},"Welcome, {{ authStore.user.name }}!",[805,4155,2944],{"class":2536},[805,4157,11],{"class":2522},[805,4159,2810],{"class":2536},[805,4161,4162,4164,4167,4170,4172,4174,4177,4179,4181,4184,4186,4188],{"class":807,"line":1186},[805,4163,3316],{"class":2536},[805,4165,4166],{"class":2522},"button",[805,4168,4169],{"class":2552}," @click",[805,4171,2823],{"class":2536},[805,4173,2854],{"class":2526},[805,4175,4176],{"class":814},"logout",[805,4178,2854],{"class":2526},[805,4180,2937],{"class":2536},[805,4182,4183],{"class":2940},"Logout",[805,4185,2944],{"class":2536},[805,4187,4166],{"class":2522},[805,4189,2810],{"class":2536},[805,4191,4192,4194,4196],{"class":807,"line":1197},[805,4193,3419],{"class":2536},[805,4195,2973],{"class":2522},[805,4197,2810],{"class":2536},[805,4199,4200,4202,4204,4207],{"class":807,"line":1212},[805,4201,2843],{"class":2536},[805,4203,2973],{"class":2522},[805,4205,4206],{"class":2552}," v-else",[805,4208,2810],{"class":2536},[805,4210,4211,4213,4215,4217,4219,4221,4224,4226,4228,4231,4233,4235],{"class":807,"line":1225},[805,4212,3316],{"class":2536},[805,4214,4166],{"class":2522},[805,4216,4169],{"class":2552},[805,4218,2823],{"class":2536},[805,4220,2854],{"class":2526},[805,4222,4223],{"class":814},"login",[805,4225,2854],{"class":2526},[805,4227,2937],{"class":2536},[805,4229,4230],{"class":2940},"Login",[805,4232,2944],{"class":2536},[805,4234,4166],{"class":2522},[805,4236,2810],{"class":2536},[805,4238,4239,4241,4243],{"class":807,"line":1238},[805,4240,3419],{"class":2536},[805,4242,2973],{"class":2522},[805,4244,2810],{"class":2536},[805,4246,4247,4249,4251],{"class":807,"line":1251},[805,4248,2953],{"class":2536},[805,4250,2973],{"class":2522},[805,4252,2810],{"class":2536},[805,4254,4255,4257,4259],{"class":807,"line":1264},[805,4256,2944],{"class":2536},[805,4258,3211],{"class":2522},[805,4260,2810],{"class":2536},[805,4262,4263,4265,4267,4269],{"class":807,"line":1277},[805,4264,2815],{"class":2536},[805,4266,2998],{"class":2522},[805,4268,3082],{"class":2552},[805,4270,2810],{"class":2536},[805,4272,4273,4275,4277,4279,4281,4283,4285,4288,4290],{"class":807,"line":1291},[805,4274,2523],{"class":2522},[805,4276,2549],{"class":2536},[805,4278,3902],{"class":2552},[805,4280,2556],{"class":2536},[805,4282,2559],{"class":2522},[805,4284,2527],{"class":2526},[805,4286,4287],{"class":814},"..\u002Fstore\u002Fauth.store",[805,4289,2533],{"class":2526},[805,4291,2537],{"class":2536},[805,4293,4294,4296,4299,4301,4303],{"class":807,"line":1302},[805,4295,2640],{"class":2639},[805,4297,4298],{"class":2552}," authStore",[805,4300,2646],{"class":2536},[805,4302,3902],{"class":811},[805,4304,4305],{"class":2536},"();\n",[805,4307,4308,4310,4313,4315,4317,4319],{"class":807,"line":1313},[805,4309,2640],{"class":2639},[805,4311,4312],{"class":811}," login",[805,4314,2646],{"class":2536},[805,4316,3706],{"class":2536},[805,4318,3709],{"class":2536},[805,4320,3920],{"class":2536},[805,4322,4323],{"class":807,"line":1324},[805,4324,4325],{"class":1162},"  \u002F\u002F Simulate a login\n",[805,4327,4328,4331,4333,4335,4337,4339,4341,4343,4346,4348,4350,4353,4355,4357,4360,4362],{"class":807,"line":1335},[805,4329,4330],{"class":2639},"  const",[805,4332,4011],{"class":2552},[805,4334,2646],{"class":2536},[805,4336,2549],{"class":2536},[805,4338,2904],{"class":1009},[805,4340,3573],{"class":2536},[805,4342,2527],{"class":2526},[805,4344,4345],{"class":814},"John Doe",[805,4347,2533],{"class":2526},[805,4349,3096],{"class":2536},[805,4351,4352],{"class":1009}," email",[805,4354,3573],{"class":2536},[805,4356,2527],{"class":2526},[805,4358,4359],{"class":814},"john.doe@example.com",[805,4361,2533],{"class":2526},[805,4363,4364],{"class":2536}," };\n",[805,4366,4367,4369,4371,4373,4375,4378,4380],{"class":807,"line":1346},[805,4368,4330],{"class":2639},[805,4370,3990],{"class":2552},[805,4372,2646],{"class":2536},[805,4374,2527],{"class":2526},[805,4376,4377],{"class":814},"fake-jwt-token",[805,4379,2533],{"class":2526},[805,4381,2537],{"class":2536},[805,4383,4384,4387,4389,4391,4393,4395,4397,4399],{"class":807,"line":1357},[805,4385,4386],{"class":2552},"  authStore",[805,4388,44],{"class":2536},[805,4390,4223],{"class":811},[805,4392,2651],{"class":2536},[805,4394,3985],{"class":2552},[805,4396,3096],{"class":2536},[805,4398,3990],{"class":2552},[805,4400,2657],{"class":2536},[805,4402,4403],{"class":807,"line":1368},[805,4404,4405],{"class":2536},"};\n",[805,4407,4408],{"class":807,"line":1379},[805,4409,2542],{"emptyLinePlaceholder":681},[805,4411,4412,4414,4417,4419,4421,4423],{"class":807,"line":1390},[805,4413,2640],{"class":2639},[805,4415,4416],{"class":811}," logout",[805,4418,2646],{"class":2536},[805,4420,3706],{"class":2536},[805,4422,3709],{"class":2536},[805,4424,3920],{"class":2536},[805,4426,4427,4429,4431,4433],{"class":807,"line":1401},[805,4428,4386],{"class":2552},[805,4430,44],{"class":2536},[805,4432,4176],{"class":811},[805,4434,4305],{"class":2536},[805,4436,4437],{"class":807,"line":1790},[805,4438,4405],{"class":2536},[805,4440,4441,4443,4445],{"class":807,"line":1800},[805,4442,2944],{"class":2536},[805,4444,2998],{"class":2522},[805,4446,2810],{"class":2536},[11,4448,4449,4450,4453,4454,4456,4457,36,4459,4461],{},"Here we're importing the ",[734,4451,4452],{},"useAuthStore"," store and using it to access the ",[734,4455,4006],{}," state and the ",[734,4458,4223],{},[734,4460,4176],{}," actions.",[11,4463,4464],{},"Thanks to Pinia, the user state is kept consistent everywhere, and any change (like logging in or out) is automatically reflected in all components that use this store.",[11,4466,4467],{},"Some real-world use cases:",[49,4469,4470,4476,4482],{},[52,4471,4472,4475],{},[29,4473,4474],{},"User Authentication",": You create a store to manage the state of the authenticated user, with data like the token, name, roles, etc. You can create actions to log in, log out, and update user information.",[52,4477,4478,4481],{},[29,4479,4480],{},"Shopping Cart",": You create a store to manage the products in the cart, with actions to add, remove, and update products. This state can be accessed from any part of the application, like the products page and the cart page.",[52,4483,4484,4487],{},[29,4485,4486],{},"User Preferences",": You create a store to manage user preferences, like theme (light\u002Fdark), language, etc. You can create actions to update these preferences and reflect them throughout the application.",[11,4489,4490,4491,44],{},"You can explore more about Pinia in the ",[88,4492,1131],{"href":1493,"target":280,"rel":4493},[282,283],[18,4495],{},[21,4497,4499],{"id":4498},"composables","Composables",[11,4501,4502],{},"Composables are reusable functions that encapsulate specific logic, with state, and can be used across different components.",[11,4504,4505,4506,4509],{},"They allow us to organize our code better, promoting reuse and separation of concerns. Normally all logic with state we would put inside a ",[734,4507,4508],{},"\u003Cscript setup>"," in a Vue component, but if that logic is something that could be useful in multiple components, we can extract it to a composable.",[11,4511,4512,4513,3573],{},"Let's see a fairly basic example. Create a file called ",[734,4514,4515],{},"src\u002Fcomposables\u002FuseClipboard.js",[797,4517,4519],{"className":2513,"code":4518,"language":2515,"meta":666,"style":666},"import { ref } from 'vue';\n\nexport function useClipboard() {\n  const copied = ref(false);\n\n  const copyToClipboard = async (text) => {\n    try {\n      await navigator.clipboard.writeText(text);\n      copied.value = true;\n      setTimeout(() => {\n        copied.value = false;\n      }, 2000);\n    } catch (error) {\n      console.error('Failed to copy:', error);\n    }\n  };\n\n  return {\n    copied,\n    copyToClipboard,\n  };\n}\n",[734,4520,4521,4542,4546,4560,4577,4581,4605,4612,4636,4653,4665,4681,4692,4709,4734,4739,4744,4748,4755,4762,4769,4773],{"__ignoreMap":666},[805,4522,4523,4525,4527,4530,4532,4534,4536,4538,4540],{"class":807,"line":808},[805,4524,2523],{"class":2522},[805,4526,2549],{"class":2536},[805,4528,4529],{"class":2552}," ref",[805,4531,2556],{"class":2536},[805,4533,2559],{"class":2522},[805,4535,2527],{"class":2526},[805,4537,2564],{"class":814},[805,4539,2533],{"class":2526},[805,4541,2537],{"class":2536},[805,4543,4544],{"class":807,"line":670},[805,4545,2542],{"emptyLinePlaceholder":681},[805,4547,4548,4550,4553,4556,4558],{"class":807,"line":667},[805,4549,3746],{"class":2522},[805,4551,4552],{"class":2639}," function",[805,4554,4555],{"class":811}," useClipboard",[805,4557,4044],{"class":2536},[805,4559,3920],{"class":2536},[805,4561,4562,4564,4567,4569,4571,4573,4575],{"class":807,"line":1053},[805,4563,4330],{"class":2639},[805,4565,4566],{"class":2552}," copied",[805,4568,2646],{"class":2536},[805,4570,4529],{"class":811},[805,4572,2651],{"class":2536},[805,4574,1435],{"class":2522},[805,4576,2657],{"class":2536},[805,4578,4579],{"class":807,"line":1186},[805,4580,2542],{"emptyLinePlaceholder":681},[805,4582,4583,4585,4588,4590,4593,4596,4599,4601,4603],{"class":807,"line":1197},[805,4584,4330],{"class":2639},[805,4586,4587],{"class":811}," copyToClipboard",[805,4589,2646],{"class":2536},[805,4591,4592],{"class":2639}," async",[805,4594,4595],{"class":2536}," (",[805,4597,4598],{"class":2552},"text",[805,4600,3993],{"class":2536},[805,4602,3709],{"class":2536},[805,4604,3920],{"class":2536},[805,4606,4607,4610],{"class":807,"line":1212},[805,4608,4609],{"class":2522},"    try",[805,4611,3920],{"class":2536},[805,4613,4614,4617,4620,4622,4625,4627,4630,4632,4634],{"class":807,"line":1225},[805,4615,4616],{"class":2522},"      await",[805,4618,4619],{"class":2552}," navigator",[805,4621,44],{"class":2536},[805,4623,4624],{"class":2552},"clipboard",[805,4626,44],{"class":2536},[805,4628,4629],{"class":811},"writeText",[805,4631,2651],{"class":2536},[805,4633,4598],{"class":2552},[805,4635,2657],{"class":2536},[805,4637,4638,4641,4643,4646,4648,4651],{"class":807,"line":1238},[805,4639,4640],{"class":2552},"      copied",[805,4642,44],{"class":2536},[805,4644,4645],{"class":2552},"value",[805,4647,2646],{"class":2536},[805,4649,4650],{"class":2522}," true",[805,4652,2537],{"class":2536},[805,4654,4655,4658,4661,4663],{"class":807,"line":1251},[805,4656,4657],{"class":811},"      setTimeout",[805,4659,4660],{"class":2536},"(()",[805,4662,3709],{"class":2536},[805,4664,3920],{"class":2536},[805,4666,4667,4670,4672,4674,4676,4679],{"class":807,"line":1264},[805,4668,4669],{"class":2552},"        copied",[805,4671,44],{"class":2536},[805,4673,4645],{"class":2552},[805,4675,2646],{"class":2536},[805,4677,4678],{"class":2522}," false",[805,4680,2537],{"class":2536},[805,4682,4683,4686,4690],{"class":807,"line":1277},[805,4684,4685],{"class":2536},"      },",[805,4687,4689],{"class":4688},"sqbOQ"," 2000",[805,4691,2657],{"class":2536},[805,4693,4694,4697,4700,4702,4705,4707],{"class":807,"line":1291},[805,4695,4696],{"class":2536},"    }",[805,4698,4699],{"class":2522}," catch",[805,4701,4595],{"class":2536},[805,4703,4704],{"class":2552},"error",[805,4706,3993],{"class":2536},[805,4708,3920],{"class":2536},[805,4710,4711,4714,4716,4718,4720,4722,4725,4727,4729,4732],{"class":807,"line":1302},[805,4712,4713],{"class":2552},"      console",[805,4715,44],{"class":2536},[805,4717,4704],{"class":811},[805,4719,2651],{"class":2536},[805,4721,2533],{"class":2526},[805,4723,4724],{"class":814},"Failed to copy:",[805,4726,2533],{"class":2526},[805,4728,3096],{"class":2536},[805,4730,4731],{"class":2552}," error",[805,4733,2657],{"class":2536},[805,4735,4736],{"class":807,"line":1313},[805,4737,4738],{"class":2536},"    }\n",[805,4740,4741],{"class":807,"line":1324},[805,4742,4743],{"class":2536},"  };\n",[805,4745,4746],{"class":807,"line":1335},[805,4747,2542],{"emptyLinePlaceholder":681},[805,4749,4750,4753],{"class":807,"line":1346},[805,4751,4752],{"class":2522},"  return",[805,4754,3920],{"class":2536},[805,4756,4757,4760],{"class":807,"line":1357},[805,4758,4759],{"class":2552},"    copied",[805,4761,3627],{"class":2536},[805,4763,4764,4767],{"class":807,"line":1368},[805,4765,4766],{"class":2552},"    copyToClipboard",[805,4768,3627],{"class":2536},[805,4770,4771],{"class":807,"line":1379},[805,4772,4743],{"class":2536},[805,4774,4775],{"class":807,"line":1390},[805,4776,4777],{"class":2536},"}\n",[11,4779,4780,4781,4784,4785,4788,4789,4792],{},"Here we're defining a composable called ",[734,4782,4783],{},"useClipboard"," that provides a function to copy text to the clipboard and a ",[29,4786,4787],{},"reactive"," state ",[734,4790,4791],{},"copied"," that indicates whether the text was copied successfully.",[11,4794,4795,4796,3573],{},"When implementing it we could have something like this. Create a component called ",[734,4797,4798],{},"src\u002Fcomponents\u002FClipboardExample.vue",[797,4800,4802],{"className":3070,"code":4801,"language":2564,"meta":666,"style":666},"\u003Ctemplate>\n  \u003Cdiv>\n    \u003Cinput v-model=\"textToCopy\" placeholder=\"Type something to copy\" \u002F>\n    \u003Cbutton @click=\"copyToClipboard(textToCopy)\">Copy to Clipboard\u003C\u002Fbutton>\n    \u003Cp v-if=\"copied\">Text copied!\u003C\u002Fp>\n  \u003C\u002Fdiv>\n\u003C\u002Ftemplate>\n\u003Cscript setup>\nimport { ref } from 'vue';\nimport { useClipboard } from '..\u002Fcomposables\u002FuseClipboard';\nconst { copied, copyToClipboard } = useClipboard();\nconst textToCopy = ref('');\n\u003C\u002Fscript>\n",[734,4803,4804,4812,4820,4853,4881,4908,4916,4924,4934,4954,4975,4995,5013],{"__ignoreMap":666},[805,4805,4806,4808,4810],{"class":807,"line":808},[805,4807,2815],{"class":2536},[805,4809,3211],{"class":2522},[805,4811,2810],{"class":2536},[805,4813,4814,4816,4818],{"class":807,"line":670},[805,4815,2833],{"class":2536},[805,4817,2973],{"class":2522},[805,4819,2810],{"class":2536},[805,4821,4822,4824,4827,4830,4832,4834,4837,4839,4842,4844,4846,4849,4851],{"class":807,"line":667},[805,4823,2843],{"class":2536},[805,4825,4826],{"class":2522},"input",[805,4828,4829],{"class":2552}," v-model",[805,4831,2823],{"class":2536},[805,4833,2854],{"class":2526},[805,4835,4836],{"class":814},"textToCopy",[805,4838,2854],{"class":2526},[805,4840,4841],{"class":2552}," placeholder",[805,4843,2823],{"class":2536},[805,4845,2854],{"class":2526},[805,4847,4848],{"class":814},"Type something to copy",[805,4850,2854],{"class":2526},[805,4852,2862],{"class":2536},[805,4854,4855,4857,4859,4861,4863,4865,4868,4870,4872,4875,4877,4879],{"class":807,"line":1053},[805,4856,2843],{"class":2536},[805,4858,4166],{"class":2522},[805,4860,4169],{"class":2552},[805,4862,2823],{"class":2536},[805,4864,2854],{"class":2526},[805,4866,4867],{"class":814},"copyToClipboard(textToCopy)",[805,4869,2854],{"class":2526},[805,4871,2937],{"class":2536},[805,4873,4874],{"class":2940},"Copy to Clipboard",[805,4876,2944],{"class":2536},[805,4878,4166],{"class":2522},[805,4880,2810],{"class":2536},[805,4882,4883,4885,4887,4889,4891,4893,4895,4897,4899,4902,4904,4906],{"class":807,"line":1186},[805,4884,2843],{"class":2536},[805,4886,11],{"class":2522},[805,4888,4131],{"class":2552},[805,4890,2823],{"class":2536},[805,4892,2854],{"class":2526},[805,4894,4791],{"class":814},[805,4896,2854],{"class":2526},[805,4898,2937],{"class":2536},[805,4900,4901],{"class":2940},"Text copied!",[805,4903,2944],{"class":2536},[805,4905,11],{"class":2522},[805,4907,2810],{"class":2536},[805,4909,4910,4912,4914],{"class":807,"line":1197},[805,4911,2953],{"class":2536},[805,4913,2973],{"class":2522},[805,4915,2810],{"class":2536},[805,4917,4918,4920,4922],{"class":807,"line":1212},[805,4919,2944],{"class":2536},[805,4921,3211],{"class":2522},[805,4923,2810],{"class":2536},[805,4925,4926,4928,4930,4932],{"class":807,"line":1225},[805,4927,2815],{"class":2536},[805,4929,2998],{"class":2522},[805,4931,3082],{"class":2552},[805,4933,2810],{"class":2536},[805,4935,4936,4938,4940,4942,4944,4946,4948,4950,4952],{"class":807,"line":1238},[805,4937,2523],{"class":2522},[805,4939,2549],{"class":2536},[805,4941,4529],{"class":2552},[805,4943,2556],{"class":2536},[805,4945,2559],{"class":2522},[805,4947,2527],{"class":2526},[805,4949,2564],{"class":814},[805,4951,2533],{"class":2526},[805,4953,2537],{"class":2536},[805,4955,4956,4958,4960,4962,4964,4966,4968,4971,4973],{"class":807,"line":1251},[805,4957,2523],{"class":2522},[805,4959,2549],{"class":2536},[805,4961,4555],{"class":2552},[805,4963,2556],{"class":2536},[805,4965,2559],{"class":2522},[805,4967,2527],{"class":2526},[805,4969,4970],{"class":814},"..\u002Fcomposables\u002FuseClipboard",[805,4972,2533],{"class":2526},[805,4974,2537],{"class":2536},[805,4976,4977,4979,4981,4983,4985,4987,4989,4991,4993],{"class":807,"line":1264},[805,4978,2640],{"class":2639},[805,4980,2549],{"class":2536},[805,4982,4566],{"class":2552},[805,4984,3096],{"class":2536},[805,4986,4587],{"class":2552},[805,4988,2556],{"class":2536},[805,4990,2646],{"class":2536},[805,4992,4555],{"class":811},[805,4994,4305],{"class":2536},[805,4996,4997,4999,5002,5004,5006,5008,5011],{"class":807,"line":1277},[805,4998,2640],{"class":2639},[805,5000,5001],{"class":2552}," textToCopy",[805,5003,2646],{"class":2536},[805,5005,4529],{"class":811},[805,5007,2651],{"class":2536},[805,5009,5010],{"class":2526},"''",[805,5012,2657],{"class":2536},[805,5014,5015,5017,5019],{"class":807,"line":1291},[805,5016,2944],{"class":2536},[805,5018,2998],{"class":2522},[805,5020,2810],{"class":2536},[11,5022,5023,5024,5026,5027,5029],{},"And then import this component in ",[734,5025,1516],{}," and add it below the ",[734,5028,3319],{}," component to test it:",[797,5031,5033],{"className":3070,"code":5032,"language":2564,"meta":666,"style":666},"\u003Cscript setup>\nimport { RouterLink, RouterView } from 'vue-router';\nimport HelloWorld from '.\u002Fcomponents\u002FHelloWorld.vue';\nimport ClipboardExample from '.\u002Fcomponents\u002FClipboardExample.vue';\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n  \u003Cheader>\n    \u003Cimg alt=\"Vue logo\" class=\"logo\" src=\"@\u002Fassets\u002Flogo.svg\" width=\"125\" height=\"125\" \u002F>\n\n    \u003Cdiv class=\"wrapper\">\n      \u003CHelloWorld msg=\"You did it!\" \u002F>\n      \u003CClipboardExample \u002F>\n      \u003C!-- Aquí -->\n      \u003Cnav>\n        \u003CRouterLink to=\"\u002F\">Home\u003C\u002FRouterLink>\n        \u003CRouterLink to=\"\u002Fabout\">About\u003C\u002FRouterLink>\n      \u003C\u002Fnav>\n    \u003C\u002Fdiv>\n  \u003C\u002Fheader>\n\n  \u003CRouterView \u002F>\n\u003C\u002Ftemplate>\n",[734,5034,5035,5045,5069,5085,5103,5111,5115,5123,5131,5189,5193,5211,5229,5238,5243,5251,5277,5303,5311,5319,5327,5331,5339],{"__ignoreMap":666},[805,5036,5037,5039,5041,5043],{"class":807,"line":808},[805,5038,2815],{"class":2536},[805,5040,2998],{"class":2522},[805,5042,3082],{"class":2552},[805,5044,2810],{"class":2536},[805,5046,5047,5049,5051,5053,5055,5057,5059,5061,5063,5065,5067],{"class":807,"line":670},[805,5048,2523],{"class":2522},[805,5050,2549],{"class":2536},[805,5052,3093],{"class":2552},[805,5054,3096],{"class":2536},[805,5056,3099],{"class":2552},[805,5058,2556],{"class":2536},[805,5060,2559],{"class":2522},[805,5062,2527],{"class":2526},[805,5064,3108],{"class":814},[805,5066,2533],{"class":2526},[805,5068,2537],{"class":2536},[805,5070,5071,5073,5075,5077,5079,5081,5083],{"class":807,"line":667},[805,5072,2523],{"class":2522},[805,5074,3119],{"class":2552},[805,5076,2559],{"class":2522},[805,5078,2527],{"class":2526},[805,5080,3126],{"class":814},[805,5082,2533],{"class":2526},[805,5084,2537],{"class":2536},[805,5086,5087,5089,5092,5094,5096,5099,5101],{"class":807,"line":1053},[805,5088,2523],{"class":2522},[805,5090,5091],{"class":2552}," ClipboardExample",[805,5093,2559],{"class":2522},[805,5095,2527],{"class":2526},[805,5097,5098],{"class":814},".\u002Fcomponents\u002FClipboardExample.vue",[805,5100,2533],{"class":2526},[805,5102,2537],{"class":2536},[805,5104,5105,5107,5109],{"class":807,"line":1186},[805,5106,2944],{"class":2536},[805,5108,2998],{"class":2522},[805,5110,2810],{"class":2536},[805,5112,5113],{"class":807,"line":1197},[805,5114,2542],{"emptyLinePlaceholder":681},[805,5116,5117,5119,5121],{"class":807,"line":1212},[805,5118,2815],{"class":2536},[805,5120,3211],{"class":2522},[805,5122,2810],{"class":2536},[805,5124,5125,5127,5129],{"class":807,"line":1225},[805,5126,2833],{"class":2536},[805,5128,3220],{"class":2522},[805,5130,2810],{"class":2536},[805,5132,5133,5135,5137,5139,5141,5143,5145,5147,5149,5151,5153,5155,5157,5159,5161,5163,5165,5167,5169,5171,5173,5175,5177,5179,5181,5183,5185,5187],{"class":807,"line":1238},[805,5134,2843],{"class":2536},[805,5136,833],{"class":2522},[805,5138,3231],{"class":2552},[805,5140,2823],{"class":2536},[805,5142,2854],{"class":2526},[805,5144,3238],{"class":814},[805,5146,2854],{"class":2526},[805,5148,3243],{"class":2552},[805,5150,2823],{"class":2536},[805,5152,2854],{"class":2526},[805,5154,3250],{"class":814},[805,5156,2854],{"class":2526},[805,5158,3013],{"class":2552},[805,5160,2823],{"class":2536},[805,5162,2854],{"class":2526},[805,5164,3261],{"class":814},[805,5166,2854],{"class":2526},[805,5168,3266],{"class":2552},[805,5170,2823],{"class":2536},[805,5172,2854],{"class":2526},[805,5174,3273],{"class":814},[805,5176,2854],{"class":2526},[805,5178,3278],{"class":2552},[805,5180,2823],{"class":2536},[805,5182,2854],{"class":2526},[805,5184,3273],{"class":814},[805,5186,2854],{"class":2526},[805,5188,2862],{"class":2536},[805,5190,5191],{"class":807,"line":1251},[805,5192,2542],{"emptyLinePlaceholder":681},[805,5194,5195,5197,5199,5201,5203,5205,5207,5209],{"class":807,"line":1264},[805,5196,2843],{"class":2536},[805,5198,2973],{"class":2522},[805,5200,3243],{"class":2552},[805,5202,2823],{"class":2536},[805,5204,2854],{"class":2526},[805,5206,3307],{"class":814},[805,5208,2854],{"class":2526},[805,5210,2810],{"class":2536},[805,5212,5213,5215,5217,5219,5221,5223,5225,5227],{"class":807,"line":1277},[805,5214,3316],{"class":2536},[805,5216,3319],{"class":2522},[805,5218,3322],{"class":2552},[805,5220,2823],{"class":2536},[805,5222,2854],{"class":2526},[805,5224,3329],{"class":814},[805,5226,2854],{"class":2526},[805,5228,2862],{"class":2536},[805,5230,5231,5233,5236],{"class":807,"line":1291},[805,5232,3316],{"class":2536},[805,5234,5235],{"class":2522},"ClipboardExample",[805,5237,2862],{"class":2536},[805,5239,5240],{"class":807,"line":1302},[805,5241,5242],{"class":1162},"      \u003C!-- Aquí -->\n",[805,5244,5245,5247,5249],{"class":807,"line":1313},[805,5246,3316],{"class":2536},[805,5248,3344],{"class":2522},[805,5250,2810],{"class":2536},[805,5252,5253,5255,5257,5259,5261,5263,5265,5267,5269,5271,5273,5275],{"class":807,"line":1324},[805,5254,3351],{"class":2536},[805,5256,3354],{"class":2522},[805,5258,3357],{"class":2552},[805,5260,2823],{"class":2536},[805,5262,2854],{"class":2526},[805,5264,3364],{"class":814},[805,5266,2854],{"class":2526},[805,5268,2937],{"class":2536},[805,5270,3371],{"class":2940},[805,5272,2944],{"class":2536},[805,5274,3354],{"class":2522},[805,5276,2810],{"class":2536},[805,5278,5279,5281,5283,5285,5287,5289,5291,5293,5295,5297,5299,5301],{"class":807,"line":1335},[805,5280,3351],{"class":2536},[805,5282,3354],{"class":2522},[805,5284,3357],{"class":2552},[805,5286,2823],{"class":2536},[805,5288,2854],{"class":2526},[805,5290,3392],{"class":814},[805,5292,2854],{"class":2526},[805,5294,2937],{"class":2536},[805,5296,3399],{"class":2940},[805,5298,2944],{"class":2536},[805,5300,3354],{"class":2522},[805,5302,2810],{"class":2536},[805,5304,5305,5307,5309],{"class":807,"line":1346},[805,5306,3410],{"class":2536},[805,5308,3344],{"class":2522},[805,5310,2810],{"class":2536},[805,5312,5313,5315,5317],{"class":807,"line":1357},[805,5314,3419],{"class":2536},[805,5316,2973],{"class":2522},[805,5318,2810],{"class":2536},[805,5320,5321,5323,5325],{"class":807,"line":1368},[805,5322,2953],{"class":2536},[805,5324,3220],{"class":2522},[805,5326,2810],{"class":2536},[805,5328,5329],{"class":807,"line":1379},[805,5330,2542],{"emptyLinePlaceholder":681},[805,5332,5333,5335,5337],{"class":807,"line":1390},[805,5334,2833],{"class":2536},[805,5336,3442],{"class":2522},[805,5338,2862],{"class":2536},[805,5340,5341,5343,5345],{"class":807,"line":1401},[805,5342,2944],{"class":2536},[805,5344,3211],{"class":2522},[805,5346,2810],{"class":2536},[11,5348,2472,5349,5351],{},[734,5350,4791],{}," state allows us to show a message when the text has been copied successfully.",[11,5353,5354],{},"What do we achieve with this?",[49,5356,5357,5366,5372],{},[52,5358,5359,5362,5363,5365],{},[29,5360,5361],{},"Reusability",": We can use ",[734,5364,4783],{}," in any component that needs clipboard copy functionality, without duplicating code.",[52,5367,5368,5371],{},[29,5369,5370],{},"Organization",": The logic related to the clipboard is encapsulated in a single place.",[52,5373,5374,5377],{},[29,5375,5376],{},"Maintenance",": If we need to change how we copy to the clipboard, we only have to modify the composable, not all the components that use it.",[1442,5379,5381],{"id":5380},"whats-the-difference-between-a-composable-and-a-pinia-store","What's the difference between a composable and a Pinia store?",[11,5383,5384,5387,5388,5391],{},[29,5385,5386],{},"Global state vs local state",": Pinia stores are designed to manage global application state, data that must be ",[29,5389,5390],{},"accessible and shared"," from anywhere.",[11,5393,5394,5395,5398,5399,5401],{},"Composables, on the other hand, handle logic and state that can be reused in multiple components, but ",[29,5396,5397],{},"none of that is shared",". For example, if I have multiple components that need clipboard copy functionality (and need some state control), I would use a composable, and the ",[734,5400,4791],{}," state would be local in each instance of the composable. On the other hand, if I need to know in multiple places if a copy to the clipboard has been performed at the application level, I would use a Pinia store, because here all components would be aware and could react to that change.",[11,5403,5404,5405,5407,5408,5411],{},"Let's put it as an example: thanks to the ",[734,5406,4791],{}," variable, I can display something within the same component, such as the message \"Text copied\". Now, if I wanted, for example, my application's layout to display an icon in the navigation bar every time something is copied to the clipboard, then it would be better to use a Pinia store to manage that state globally, so that when copying from my ",[734,5409,5410],{},"ClipboardExample.vue"," component, the layout would also notice the change and could display the icon.",[1442,5413,5415],{"id":5414},"whats-the-difference-between-a-composable-and-a-utility-file-utils","What's the difference between a composable and a utility file (utils)?",[11,5417,5418,5419,5421],{},"Utility functions are generally simpler and don't have state, they take inputs (some) and return outputs without side effects. For example, if we didn't need the ",[734,5420,4791],{}," state and only wanted a function to copy text, we could create a simple utility function instead of a composable:",[797,5423,5425],{"className":2513,"code":5424,"language":2515,"meta":666,"style":666},"export function copyToClipboard(text) {\n  return navigator.clipboard.writeText(text);\n}\n",[734,5426,5427,5443,5463],{"__ignoreMap":666},[805,5428,5429,5431,5433,5435,5437,5439,5441],{"class":807,"line":808},[805,5430,3746],{"class":2522},[805,5432,4552],{"class":2639},[805,5434,4587],{"class":811},[805,5436,2651],{"class":2536},[805,5438,4598],{"class":2552},[805,5440,3993],{"class":2536},[805,5442,3920],{"class":2536},[805,5444,5445,5447,5449,5451,5453,5455,5457,5459,5461],{"class":807,"line":670},[805,5446,4752],{"class":2522},[805,5448,4619],{"class":2552},[805,5450,44],{"class":2536},[805,5452,4624],{"class":2552},[805,5454,44],{"class":2536},[805,5456,4629],{"class":811},[805,5458,2651],{"class":2536},[805,5460,4598],{"class":2552},[805,5462,2657],{"class":2536},[805,5464,5465],{"class":807,"line":667},[805,5466,4777],{"class":2536},[11,5468,5469],{},"Recommended readings:",[49,5471,5472,5479,5486],{},[52,5473,5474],{},[88,5475,5478],{"href":5476,"target":280,"rel":5477},"https:\u002F\u002Fvuejs.org\u002Fguide\u002Freusability\u002Fcomposables.html",[282,283],"Composables - Vue.js Documentation",[52,5480,5481],{},[88,5482,5485],{"href":5483,"target":280,"rel":5484},"https:\u002F\u002Fdev.to\u002Fjacobandrewsky\u002Fgood-practices-and-design-patterns-for-vue-composables-24lk",[282,283],"Good Practices and Design Patterns for Vue Composables",[52,5487,5488],{},[88,5489,5492],{"href":5490,"target":280,"rel":5491},"https:\u002F\u002Frobconery.com\u002Ffrontend\u002Fwhat-should-be-a-plugin-vs-a-composable-vs-a-store-in-nuxt\u002F",[282,283],"What should be a Plugin vs a Composable vs a Store in Nuxt",[1442,5494,5496],{"id":5495},"vueuse","VueUse",[11,5498,5499,5503],{},[88,5500,5496],{"href":5501,"target":280,"rel":5502},"https:\u002F\u002Fvueuse.org\u002F",[282,283]," is a collection of composables for Vue 3. It provides a wide range of ready-to-use features, from state management to DOM interactions and browser APIs. You can explore the official documentation to see all available composables and how to use them in your projects.",[11,5505,5506,5509],{},[833,5507],{"alt":5496,"src":5508},"\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fvue-use.webp",[838,5510,5496],{},[11,5512,5513],{},"Some popular examples of composables in VueUse include:",[49,5515,5516,5522,5528,5534,5540],{},[52,5517,5518,5521],{},[734,5519,5520],{},"useBreakpoints",": to be able to manage breakpoints in your application and help with responsive design.",[52,5523,5524,5527],{},[734,5525,5526],{},"useFetch",": to make HTTP requests easily.",[52,5529,5530,5533],{},[734,5531,5532],{},"useLocalStorage",": to sync data with the browser's local storage.",[52,5535,5536,5539],{},[734,5537,5538],{},"useDark",": to handle dark and light themes in the application.",[52,5541,5542,5544],{},[734,5543,4783],{},": to copy text to the clipboard (similar to the one we created before).",[11,5546,5547],{},"It's a very useful collection that can save you time if you need any of these features.\nJust be careful not to overload your project with unnecessary dependencies, install only what you're really going to use and only if you really need a library for it.",[18,5549],{},[21,5551,5553],{"id":5552},"eslint-and-prettier","ESLint and Prettier",[11,5555,5556,5557,5560],{},"Get used to using ",[29,5558,5559],{},"ESLint",", it's too useful for maintaining code quality and avoiding common errors.\nThe configuration generated by Vite is a good starting point, but I always recommend personalizing it.",[11,5562,5563,5564,36,5568,44],{},"You can explore the available rules in the official documentation of ",[88,5565,5559],{"href":5566,"target":280,"rel":5567},"https:\u002F\u002Feslint.org\u002Fdocs\u002Frules\u002F",[282,283],[88,5569,5572],{"href":5570,"target":280,"rel":5571},"https:\u002F\u002Feslint.vuejs.org\u002Frules\u002F",[282,283],"eslint-plugin-vue",[11,5574,5575],{},"All rules can be configured as \"off\", \"warn\" or \"error\", depending on the severity you want to assign:",[797,5577,5581],{"className":5578,"code":5579,"language":5580,"meta":666,"style":666},"language-js shiki shiki-themes vitesse-light vitesse-dark","\"no-console\": \"warn\", \u002F\u002F Shows a warning if console.log is used\n\"eqeqeq\": \"error\", \u002F\u002F Forces the use of === and !== instead of == and !=\n\"vue\u002Fmulti-word-component-names\": \"off\", \u002F\u002F Disables the rule that forces multi-word component names\n","js",[734,5582,5583,5607,5629],{"__ignoreMap":666},[805,5584,5585,5587,5590,5592,5595,5597,5600,5602,5604],{"class":807,"line":808},[805,5586,2854],{"class":2526},[805,5588,5589],{"class":814},"no-console",[805,5591,2854],{"class":2526},[805,5593,5594],{"class":2940},": ",[805,5596,2854],{"class":2526},[805,5598,5599],{"class":814},"warn",[805,5601,2854],{"class":2526},[805,5603,3096],{"class":2536},[805,5605,5606],{"class":1162}," \u002F\u002F Shows a warning if console.log is used\n",[805,5608,5609,5611,5614,5616,5618,5620,5622,5624,5626],{"class":807,"line":670},[805,5610,2854],{"class":2526},[805,5612,5613],{"class":814},"eqeqeq",[805,5615,2854],{"class":2526},[805,5617,5594],{"class":2940},[805,5619,2854],{"class":2526},[805,5621,4704],{"class":814},[805,5623,2854],{"class":2526},[805,5625,3096],{"class":2536},[805,5627,5628],{"class":1162}," \u002F\u002F Forces the use of === and !== instead of == and !=\n",[805,5630,5631,5633,5636,5638,5640,5642,5645,5647,5649],{"class":807,"line":667},[805,5632,2854],{"class":2526},[805,5634,5635],{"class":814},"vue\u002Fmulti-word-component-names",[805,5637,2854],{"class":2526},[805,5639,5594],{"class":2940},[805,5641,2854],{"class":2526},[805,5643,5644],{"class":814},"off",[805,5646,2854],{"class":2526},[805,5648,3096],{"class":2536},[805,5650,5651],{"class":1162}," \u002F\u002F Disables the rule that forces multi-word component names\n",[11,5653,5654,5655,5658,5659,1440],{},"In the project we created, you can use the ",[734,5656,5657],{},"npm run lint"," command to review all your code according to the configuration in the ",[734,5660,5661],{},"eslint.config.js",[11,5663,5664,5667],{},[29,5665,5666],{},"About Prettier",", there's some debate about its use, many people don't like it because it can impose a style you don't like, so try it, explore it, and use it only if you're comfortable with it.\nInstead of Prettier, to try to always maintain consistent style, you can use alternatives like EditorConfig, ESLint's own rules, or your code editor's configuration.",[11,5669,5670],{},"If you use some of these tools you can explore their extensions:",[49,5672,5673,5680,5687],{},[52,5674,5675],{},[88,5676,5679],{"href":5677,"target":280,"rel":5678},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=dbaeumer.vscode-eslint",[282,283],"ESLint extension for VS Code",[52,5681,5682],{},[88,5683,5686],{"href":5684,"target":280,"rel":5685},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=esbenp.prettier-vscode",[282,283],"Prettier - Code formatter",[52,5688,5689],{},[88,5690,5693],{"href":5691,"target":280,"rel":5692},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=EditorConfig.EditorConfig",[282,283],"EditorConfig for VS Code",[18,5695],{},[21,5697,5699],{"id":5698},"extra-installing-tailwind-css-v4","EXTRA: Installing Tailwind CSS v4",[11,5701,5702,5703,5708],{},"Pure CSS is more than fine, you can achieve incredible results (sometimes even ",[88,5704,5707],{"href":5705,"target":280,"rel":5706},"https:\u002F\u002Fgithub.com\u002Fyou-dont-need\u002FYou-Dont-Need-JavaScript",[282,283],"without needing JavaScript","), but an option you have to speed up your development and maintain consistent style is Tailwind CSS.",[11,5710,5711,5712,5717],{},"Following the official steps in the ",[88,5713,5716],{"href":5714,"target":280,"rel":5715},"https:\u002F\u002Ftailwindcss.com\u002Fdocs\u002Finstallation\u002Fusing-vite",[282,283],"Tailwind CSS documentation",", let's install Tailwind CSS v4 in our Vue project with Vite.",[11,5719,5720],{},"First (being inside the project folder), install Tailwind CSS and its Vite plugin:",[797,5722,5724],{"className":799,"code":5723,"language":801,"meta":666,"style":666},"npm install tailwindcss @tailwindcss\u002Fvite\n",[734,5725,5726],{"__ignoreMap":666},[805,5727,5728,5730,5733,5736],{"class":807,"line":808},[805,5729,736],{"class":811},[805,5731,5732],{"class":814}," install",[805,5734,5735],{"class":814}," tailwindcss",[805,5737,5738],{"class":814}," @tailwindcss\u002Fvite\n",[11,5740,5741,5742,2510],{},"Then, we add the plugin to our ",[734,5743,5744],{},"vite.config.js",[797,5746,5748],{"className":2513,"code":5747,"language":2515,"meta":666,"style":666},"import { fileURLToPath, URL } from 'node:url';\n\nimport { defineConfig } from 'vite';\nimport vue from '@vitejs\u002Fplugin-vue';\nimport vueDevTools from 'vite-plugin-vue-devtools';\nimport tailwindcss from '@tailwindcss\u002Fvite'; \u002F\u002F Import the plugin\n\n\u002F\u002F https:\u002F\u002Fvite.dev\u002Fconfig\u002F\nexport default defineConfig({\n  plugins: [\n    vue(),\n    vueDevTools(),\n    tailwindcss(), \u002F\u002F Add to Vite configuration\n  ],\n  resolve: {\n    alias: {\n      '@': fileURLToPath(new URL('.\u002Fsrc', import.meta.url)),\n    },\n  },\n});\n",[734,5749,5750,5777,5781,5803,5821,5839,5860,5864,5869,5879,5888,5896,5903,5914,5918,5927,5936,5982,5986,5990],{"__ignoreMap":666},[805,5751,5752,5754,5756,5759,5761,5764,5766,5768,5770,5773,5775],{"class":807,"line":808},[805,5753,2523],{"class":2522},[805,5755,2549],{"class":2536},[805,5757,5758],{"class":2552}," fileURLToPath",[805,5760,3096],{"class":2536},[805,5762,5763],{"class":2552}," URL",[805,5765,2556],{"class":2536},[805,5767,2559],{"class":2522},[805,5769,2527],{"class":2526},[805,5771,5772],{"class":814},"node:url",[805,5774,2533],{"class":2526},[805,5776,2537],{"class":2536},[805,5778,5779],{"class":807,"line":670},[805,5780,2542],{"emptyLinePlaceholder":681},[805,5782,5783,5785,5787,5790,5792,5794,5796,5799,5801],{"class":807,"line":667},[805,5784,2523],{"class":2522},[805,5786,2549],{"class":2536},[805,5788,5789],{"class":2552}," defineConfig",[805,5791,2556],{"class":2536},[805,5793,2559],{"class":2522},[805,5795,2527],{"class":2526},[805,5797,5798],{"class":814},"vite",[805,5800,2533],{"class":2526},[805,5802,2537],{"class":2536},[805,5804,5805,5807,5810,5812,5814,5817,5819],{"class":807,"line":1053},[805,5806,2523],{"class":2522},[805,5808,5809],{"class":2552}," vue",[805,5811,2559],{"class":2522},[805,5813,2527],{"class":2526},[805,5815,5816],{"class":814},"@vitejs\u002Fplugin-vue",[805,5818,2533],{"class":2526},[805,5820,2537],{"class":2536},[805,5822,5823,5825,5828,5830,5832,5835,5837],{"class":807,"line":1186},[805,5824,2523],{"class":2522},[805,5826,5827],{"class":2552}," vueDevTools",[805,5829,2559],{"class":2522},[805,5831,2527],{"class":2526},[805,5833,5834],{"class":814},"vite-plugin-vue-devtools",[805,5836,2533],{"class":2526},[805,5838,2537],{"class":2536},[805,5840,5841,5843,5845,5847,5849,5852,5854,5857],{"class":807,"line":1197},[805,5842,2523],{"class":2522},[805,5844,5735],{"class":2552},[805,5846,2559],{"class":2522},[805,5848,2527],{"class":2526},[805,5850,5851],{"class":814},"@tailwindcss\u002Fvite",[805,5853,2533],{"class":2526},[805,5855,5856],{"class":2536},";",[805,5858,5859],{"class":1162}," \u002F\u002F Import the plugin\n",[805,5861,5862],{"class":807,"line":1212},[805,5863,2542],{"emptyLinePlaceholder":681},[805,5865,5866],{"class":807,"line":1225},[805,5867,5868],{"class":1162},"\u002F\u002F https:\u002F\u002Fvite.dev\u002Fconfig\u002F\n",[805,5870,5871,5873,5875,5877],{"class":807,"line":1238},[805,5872,3746],{"class":2522},[805,5874,3749],{"class":2522},[805,5876,5789],{"class":811},[805,5878,3565],{"class":2536},[805,5880,5881,5884,5886],{"class":807,"line":1251},[805,5882,5883],{"class":1009},"  plugins",[805,5885,3573],{"class":2536},[805,5887,3606],{"class":2536},[805,5889,5890,5893],{"class":807,"line":1264},[805,5891,5892],{"class":811},"    vue",[805,5894,5895],{"class":2536},"(),\n",[805,5897,5898,5901],{"class":807,"line":1277},[805,5899,5900],{"class":811},"    vueDevTools",[805,5902,5895],{"class":2536},[805,5904,5905,5908,5911],{"class":807,"line":1291},[805,5906,5907],{"class":811},"    tailwindcss",[805,5909,5910],{"class":2536},"(),",[805,5912,5913],{"class":1162}," \u002F\u002F Add to Vite configuration\n",[805,5915,5916],{"class":807,"line":1302},[805,5917,3732],{"class":2536},[805,5919,5920,5923,5925],{"class":807,"line":1313},[805,5921,5922],{"class":1009},"  resolve",[805,5924,3573],{"class":2536},[805,5926,3920],{"class":2536},[805,5928,5929,5932,5934],{"class":807,"line":1324},[805,5930,5931],{"class":1009},"    alias",[805,5933,3573],{"class":2536},[805,5935,3920],{"class":2536},[805,5937,5938,5941,5944,5946,5948,5950,5952,5955,5957,5959,5961,5964,5966,5968,5970,5972,5974,5976,5979],{"class":807,"line":1335},[805,5939,5940],{"class":2526},"      '",[805,5942,5943],{"class":814},"@",[805,5945,2533],{"class":2526},[805,5947,3573],{"class":2536},[805,5949,5758],{"class":811},[805,5951,2651],{"class":2536},[805,5953,5954],{"class":2639},"new",[805,5956,5763],{"class":811},[805,5958,2651],{"class":2536},[805,5960,2533],{"class":2526},[805,5962,5963],{"class":814},".\u002Fsrc",[805,5965,2533],{"class":2526},[805,5967,3096],{"class":2536},[805,5969,3712],{"class":2522},[805,5971,44],{"class":2536},[805,5973,2846],{"class":1009},[805,5975,44],{"class":2536},[805,5977,5978],{"class":2552},"url",[805,5980,5981],{"class":2536},")),\n",[805,5983,5984],{"class":807,"line":1346},[805,5985,3659],{"class":2536},[805,5987,5988],{"class":807,"line":1357},[805,5989,4083],{"class":2536},[805,5991,5992],{"class":807,"line":1368},[805,5993,3737],{"class":2536},[11,5995,5996,5997,6000],{},"Then in the ",[734,5998,5999],{},"src\u002Fassets\u002Fmain.css"," file, replace all content with the following:",[797,6002,6006],{"className":6003,"code":6004,"language":6005,"meta":666,"style":666},"language-css shiki shiki-themes vitesse-light vitesse-dark","@import '.\u002Fbase.css';\n\n@import 'tailwindcss';\n","css",[734,6007,6008,6023,6027],{"__ignoreMap":666},[805,6009,6010,6012,6014,6016,6019,6021],{"class":807,"line":808},[805,6011,5943],{"class":2536},[805,6013,2523],{"class":2522},[805,6015,2527],{"class":2526},[805,6017,6018],{"class":814},".\u002Fbase.css",[805,6020,2533],{"class":2526},[805,6022,2537],{"class":2536},[805,6024,6025],{"class":807,"line":670},[805,6026,2542],{"emptyLinePlaceholder":681},[805,6028,6029,6031,6033,6035,6038,6040],{"class":807,"line":667},[805,6030,5943],{"class":2536},[805,6032,2523],{"class":2522},[805,6034,2527],{"class":2526},[805,6036,6037],{"class":814},"tailwindcss",[805,6039,2533],{"class":2526},[805,6041,2537],{"class":2536},[11,6043,6044,6045,6048],{},"We take the opportunity to clean up the ",[734,6046,6047],{},"src\u002Fassets\u002Fbase.css"," file and leave only what's necessary:",[797,6050,6052],{"className":6003,"code":6051,"language":6005,"meta":666,"style":666},"*,\n*::before,\n*::after {\n  box-sizing: border-box;\n  margin: 0;\n  font-weight: normal;\n}\n\nbody {\n  min-height: 100vh;\n  color: var(--color-text);\n  background: var(--color-background);\n  transition:\n    color 0.5s,\n    background-color 0.5s;\n  line-height: 1.6;\n  font-family: Inter, system-ui, Avenir, Helvetica, Arial, sans-serif;\n  font-size: 15px;\n  text-rendering: optimizeLegibility;\n  -webkit-font-smoothing: antialiased;\n  -moz-osx-font-smoothing: grayscale;\n}\n",[734,6053,6054,6061,6073,6084,6096,6108,6120,6124,6128,6134,6149,6166,6182,6190,6203,6215,6227,6264,6279,6291,6303,6315],{"__ignoreMap":666},[805,6055,6056,6059],{"class":807,"line":808},[805,6057,6058],{"class":2522},"*",[805,6060,3627],{"class":2536},[805,6062,6063,6065,6068,6071],{"class":807,"line":670},[805,6064,6058],{"class":2522},[805,6066,6067],{"class":2536},"::",[805,6069,6070],{"class":2552},"before",[805,6072,3627],{"class":2536},[805,6074,6075,6077,6079,6082],{"class":807,"line":667},[805,6076,6058],{"class":2522},[805,6078,6067],{"class":2536},[805,6080,6081],{"class":2552},"after",[805,6083,3920],{"class":2536},[805,6085,6086,6089,6091,6094],{"class":807,"line":1053},[805,6087,6088],{"class":1009},"  box-sizing",[805,6090,3573],{"class":2536},[805,6092,6093],{"class":4000}," border-box",[805,6095,2537],{"class":2536},[805,6097,6098,6101,6103,6106],{"class":807,"line":1186},[805,6099,6100],{"class":1009},"  margin",[805,6102,3573],{"class":2536},[805,6104,6105],{"class":4688}," 0",[805,6107,2537],{"class":2536},[805,6109,6110,6113,6115,6118],{"class":807,"line":1197},[805,6111,6112],{"class":1009},"  font-weight",[805,6114,3573],{"class":2536},[805,6116,6117],{"class":4000}," normal",[805,6119,2537],{"class":2536},[805,6121,6122],{"class":807,"line":1212},[805,6123,4777],{"class":2536},[805,6125,6126],{"class":807,"line":1225},[805,6127,2542],{"emptyLinePlaceholder":681},[805,6129,6130,6132],{"class":807,"line":1238},[805,6131,2964],{"class":2522},[805,6133,3920],{"class":2536},[805,6135,6136,6139,6141,6144,6147],{"class":807,"line":1251},[805,6137,6138],{"class":1009},"  min-height",[805,6140,3573],{"class":2536},[805,6142,6143],{"class":4688}," 100",[805,6145,6146],{"class":2639},"vh",[805,6148,2537],{"class":2536},[805,6150,6151,6154,6156,6159,6161,6164],{"class":807,"line":1264},[805,6152,6153],{"class":1009},"  color",[805,6155,3573],{"class":2536},[805,6157,6158],{"class":1009}," var",[805,6160,2651],{"class":2536},[805,6162,6163],{"class":2552},"--color-text",[805,6165,2657],{"class":2536},[805,6167,6168,6171,6173,6175,6177,6180],{"class":807,"line":1277},[805,6169,6170],{"class":1009},"  background",[805,6172,3573],{"class":2536},[805,6174,6158],{"class":1009},[805,6176,2651],{"class":2536},[805,6178,6179],{"class":2552},"--color-background",[805,6181,2657],{"class":2536},[805,6183,6184,6187],{"class":807,"line":1291},[805,6185,6186],{"class":1009},"  transition",[805,6188,6189],{"class":2536},":\n",[805,6191,6192,6195,6198,6201],{"class":807,"line":1302},[805,6193,6194],{"class":4000},"    color",[805,6196,6197],{"class":4688}," 0.5",[805,6199,6200],{"class":2639},"s",[805,6202,3627],{"class":2536},[805,6204,6205,6208,6211,6213],{"class":807,"line":1313},[805,6206,6207],{"class":2940},"    background-color ",[805,6209,6210],{"class":4688},"0.5",[805,6212,6200],{"class":2639},[805,6214,2537],{"class":2536},[805,6216,6217,6220,6222,6225],{"class":807,"line":1324},[805,6218,6219],{"class":1009},"  line-height",[805,6221,3573],{"class":2536},[805,6223,6224],{"class":4688}," 1.6",[805,6226,2537],{"class":2536},[805,6228,6229,6232,6234,6237,6239,6242,6244,6247,6249,6252,6254,6257,6259,6262],{"class":807,"line":1335},[805,6230,6231],{"class":1009},"  font-family",[805,6233,3573],{"class":2536},[805,6235,6236],{"class":2940}," Inter",[805,6238,3096],{"class":2536},[805,6240,6241],{"class":4000}," system-ui",[805,6243,3096],{"class":2536},[805,6245,6246],{"class":2940}," Avenir",[805,6248,3096],{"class":2536},[805,6250,6251],{"class":4000}," Helvetica",[805,6253,3096],{"class":2536},[805,6255,6256],{"class":4000}," Arial",[805,6258,3096],{"class":2536},[805,6260,6261],{"class":4000}," sans-serif",[805,6263,2537],{"class":2536},[805,6265,6266,6269,6271,6274,6277],{"class":807,"line":1346},[805,6267,6268],{"class":1009},"  font-size",[805,6270,3573],{"class":2536},[805,6272,6273],{"class":4688}," 15",[805,6275,6276],{"class":2639},"px",[805,6278,2537],{"class":2536},[805,6280,6281,6284,6286,6289],{"class":807,"line":1357},[805,6282,6283],{"class":1009},"  text-rendering",[805,6285,3573],{"class":2536},[805,6287,6288],{"class":4000}," optimizeLegibility",[805,6290,2537],{"class":2536},[805,6292,6293,6296,6298,6301],{"class":807,"line":1368},[805,6294,6295],{"class":1009},"  -webkit-font-smoothing",[805,6297,3573],{"class":2536},[805,6299,6300],{"class":4000}," antialiased",[805,6302,2537],{"class":2536},[805,6304,6305,6308,6310,6313],{"class":807,"line":1379},[805,6306,6307],{"class":1009},"  -moz-osx-font-smoothing",[805,6309,3573],{"class":2536},[805,6311,6312],{"class":4000}," grayscale",[805,6314,2537],{"class":2536},[805,6316,6317],{"class":807,"line":1390},[805,6318,4777],{"class":2536},[11,6320,6321,6322,6325],{},"Now let's test that everything works correctly. First, stop the development server if it's running (",[734,6323,6324],{},"Ctrl + C"," in the terminal) and then start it again:",[797,6327,6328],{"className":799,"code":1059,"language":801,"meta":666,"style":666},[734,6329,6330],{"__ignoreMap":666},[805,6331,6332,6334,6336],{"class":807,"line":808},[805,6333,736],{"class":811},[805,6335,1047],{"class":814},[805,6337,1070],{"class":814},[11,6339,3064,6340,6342],{},[734,6341,3067],{}," file and replace all content with the following:",[797,6344,6346],{"className":3070,"code":6345,"language":2564,"meta":666,"style":666},"\u003Cscript setup>\nimport { RouterLink, RouterView } from 'vue-router';\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n  \u003Cdiv class=\"flex h-screen flex-col bg-gray-50\">\n    \u003C!-- App Bar -->\n    \u003Cheader class=\"flex h-16 items-center justify-between bg-white px-6 shadow-sm\">\n      \u003Cdiv class=\"text-xl font-bold text-gray-800\">My Project\u003C\u002Fdiv>\n      \u003Cnav class=\"flex gap-4\" aria-label=\"Main Navigation\">\n        \u003Ca href=\"#\" class=\"text-gray-600 hover:text-gray-900\">Home\u003C\u002Fa>\n        \u003Ca href=\"#\" class=\"text-gray-600 hover:text-gray-900\">Profile\u003C\u002Fa>\n      \u003C\u002Fnav>\n    \u003C\u002Fheader>\n\n    \u003Cdiv class=\"flex flex-1 overflow-hidden\">\n      \u003C!-- Sidebar -->\n      \u003Caside class=\"w-64 overflow-y-auto bg-white border-r border-gray-200\">\n        \u003Cnav class=\"p-4 space-y-2\" aria-label=\"Sidebar Navigation\">\n          \u003CRouterLink\n            to=\"\u002F\"\n            class=\"block rounded-md px-4 py-2 text-gray-700 hover:bg-gray-100\"\n            exact-active-class=\"text-indigo-700 font-bold\"\n          >\n            Home\n          \u003C\u002FRouterLink>\n          \u003CRouterLink\n            to=\"\u002Fabout\"\n            class=\"block rounded-md px-4 py-2 text-gray-700 hover:bg-gray-100\"\n            exact-active-class=\"text-indigo-700 font-bold\"\n          >\n            About\n          \u003C\u002FRouterLink>\n        \u003C\u002Fnav>\n      \u003C\u002Faside>\n\n      \u003C!-- Main Content -->\n      \u003Cmain class=\"flex-1 overflow-y-auto p-6\">\n        \u003CRouterView \u002F>\n      \u003C\u002Fmain>\n    \u003C\u002Fdiv>\n\n    \u003C!-- Footer -->\n    \u003Cfooter\n      class=\"flex h-12 items-center justify-center bg-white border-t border-gray-200 text-sm text-gray-500\"\n    >\n      © 2025 My Project. All rights reserved.\n    \u003C\u002Ffooter>\n  \u003C\u002Fdiv>\n\u003C\u002Ftemplate>\n",[734,6347,6348,6358,6382,6390,6394,6402,6421,6426,6445,6473,6504,6542,6579,6587,6595,6599,6618,6623,6643,6673,6681,6695,6709,6723,6728,6733,6742,6748,6760,6772,6784,6788,6793,6801,6810,6818,6822,6827,6847,6855,6863,6871,6875,6880,6887,6901,6906,6920,6930,6939],{"__ignoreMap":666},[805,6349,6350,6352,6354,6356],{"class":807,"line":808},[805,6351,2815],{"class":2536},[805,6353,2998],{"class":2522},[805,6355,3082],{"class":2552},[805,6357,2810],{"class":2536},[805,6359,6360,6362,6364,6366,6368,6370,6372,6374,6376,6378,6380],{"class":807,"line":670},[805,6361,2523],{"class":2522},[805,6363,2549],{"class":2536},[805,6365,3093],{"class":2552},[805,6367,3096],{"class":2536},[805,6369,3099],{"class":2552},[805,6371,2556],{"class":2536},[805,6373,2559],{"class":2522},[805,6375,2527],{"class":2526},[805,6377,3108],{"class":814},[805,6379,2533],{"class":2526},[805,6381,2537],{"class":2536},[805,6383,6384,6386,6388],{"class":807,"line":667},[805,6385,2944],{"class":2536},[805,6387,2998],{"class":2522},[805,6389,2810],{"class":2536},[805,6391,6392],{"class":807,"line":1053},[805,6393,2542],{"emptyLinePlaceholder":681},[805,6395,6396,6398,6400],{"class":807,"line":1186},[805,6397,2815],{"class":2536},[805,6399,3211],{"class":2522},[805,6401,2810],{"class":2536},[805,6403,6404,6406,6408,6410,6412,6414,6417,6419],{"class":807,"line":1197},[805,6405,2833],{"class":2536},[805,6407,2973],{"class":2522},[805,6409,3243],{"class":2552},[805,6411,2823],{"class":2536},[805,6413,2854],{"class":2526},[805,6415,6416],{"class":814},"flex h-screen flex-col bg-gray-50",[805,6418,2854],{"class":2526},[805,6420,2810],{"class":2536},[805,6422,6423],{"class":807,"line":1212},[805,6424,6425],{"class":1162},"    \u003C!-- App Bar -->\n",[805,6427,6428,6430,6432,6434,6436,6438,6441,6443],{"class":807,"line":1225},[805,6429,2843],{"class":2536},[805,6431,3220],{"class":2522},[805,6433,3243],{"class":2552},[805,6435,2823],{"class":2536},[805,6437,2854],{"class":2526},[805,6439,6440],{"class":814},"flex h-16 items-center justify-between bg-white px-6 shadow-sm",[805,6442,2854],{"class":2526},[805,6444,2810],{"class":2536},[805,6446,6447,6449,6451,6453,6455,6457,6460,6462,6464,6467,6469,6471],{"class":807,"line":1238},[805,6448,3316],{"class":2536},[805,6450,2973],{"class":2522},[805,6452,3243],{"class":2552},[805,6454,2823],{"class":2536},[805,6456,2854],{"class":2526},[805,6458,6459],{"class":814},"text-xl font-bold text-gray-800",[805,6461,2854],{"class":2526},[805,6463,2937],{"class":2536},[805,6465,6466],{"class":2940},"My Project",[805,6468,2944],{"class":2536},[805,6470,2973],{"class":2522},[805,6472,2810],{"class":2536},[805,6474,6475,6477,6479,6481,6483,6485,6488,6490,6493,6495,6497,6500,6502],{"class":807,"line":1251},[805,6476,3316],{"class":2536},[805,6478,3344],{"class":2522},[805,6480,3243],{"class":2552},[805,6482,2823],{"class":2536},[805,6484,2854],{"class":2526},[805,6486,6487],{"class":814},"flex gap-4",[805,6489,2854],{"class":2526},[805,6491,6492],{"class":2552}," aria-label",[805,6494,2823],{"class":2536},[805,6496,2854],{"class":2526},[805,6498,6499],{"class":814},"Main Navigation",[805,6501,2854],{"class":2526},[805,6503,2810],{"class":2536},[805,6505,6506,6508,6510,6512,6514,6516,6519,6521,6523,6525,6527,6530,6532,6534,6536,6538,6540],{"class":807,"line":1264},[805,6507,3351],{"class":2536},[805,6509,88],{"class":2522},[805,6511,2884],{"class":2552},[805,6513,2823],{"class":2536},[805,6515,2854],{"class":2526},[805,6517,6518],{"class":814},"#",[805,6520,2854],{"class":2526},[805,6522,3243],{"class":2552},[805,6524,2823],{"class":2536},[805,6526,2854],{"class":2526},[805,6528,6529],{"class":814},"text-gray-600 hover:text-gray-900",[805,6531,2854],{"class":2526},[805,6533,2937],{"class":2536},[805,6535,3371],{"class":2940},[805,6537,2944],{"class":2536},[805,6539,88],{"class":2522},[805,6541,2810],{"class":2536},[805,6543,6544,6546,6548,6550,6552,6554,6556,6558,6560,6562,6564,6566,6568,6570,6573,6575,6577],{"class":807,"line":1277},[805,6545,3351],{"class":2536},[805,6547,88],{"class":2522},[805,6549,2884],{"class":2552},[805,6551,2823],{"class":2536},[805,6553,2854],{"class":2526},[805,6555,6518],{"class":814},[805,6557,2854],{"class":2526},[805,6559,3243],{"class":2552},[805,6561,2823],{"class":2536},[805,6563,2854],{"class":2526},[805,6565,6529],{"class":814},[805,6567,2854],{"class":2526},[805,6569,2937],{"class":2536},[805,6571,6572],{"class":2940},"Profile",[805,6574,2944],{"class":2536},[805,6576,88],{"class":2522},[805,6578,2810],{"class":2536},[805,6580,6581,6583,6585],{"class":807,"line":1291},[805,6582,3410],{"class":2536},[805,6584,3344],{"class":2522},[805,6586,2810],{"class":2536},[805,6588,6589,6591,6593],{"class":807,"line":1302},[805,6590,3419],{"class":2536},[805,6592,3220],{"class":2522},[805,6594,2810],{"class":2536},[805,6596,6597],{"class":807,"line":1313},[805,6598,2542],{"emptyLinePlaceholder":681},[805,6600,6601,6603,6605,6607,6609,6611,6614,6616],{"class":807,"line":1324},[805,6602,2843],{"class":2536},[805,6604,2973],{"class":2522},[805,6606,3243],{"class":2552},[805,6608,2823],{"class":2536},[805,6610,2854],{"class":2526},[805,6612,6613],{"class":814},"flex flex-1 overflow-hidden",[805,6615,2854],{"class":2526},[805,6617,2810],{"class":2536},[805,6619,6620],{"class":807,"line":1335},[805,6621,6622],{"class":1162},"      \u003C!-- Sidebar -->\n",[805,6624,6625,6627,6630,6632,6634,6636,6639,6641],{"class":807,"line":1346},[805,6626,3316],{"class":2536},[805,6628,6629],{"class":2522},"aside",[805,6631,3243],{"class":2552},[805,6633,2823],{"class":2536},[805,6635,2854],{"class":2526},[805,6637,6638],{"class":814},"w-64 overflow-y-auto bg-white border-r border-gray-200",[805,6640,2854],{"class":2526},[805,6642,2810],{"class":2536},[805,6644,6645,6647,6649,6651,6653,6655,6658,6660,6662,6664,6666,6669,6671],{"class":807,"line":1357},[805,6646,3351],{"class":2536},[805,6648,3344],{"class":2522},[805,6650,3243],{"class":2552},[805,6652,2823],{"class":2536},[805,6654,2854],{"class":2526},[805,6656,6657],{"class":814},"p-4 space-y-2",[805,6659,2854],{"class":2526},[805,6661,6492],{"class":2552},[805,6663,2823],{"class":2536},[805,6665,2854],{"class":2526},[805,6667,6668],{"class":814},"Sidebar Navigation",[805,6670,2854],{"class":2526},[805,6672,2810],{"class":2536},[805,6674,6675,6678],{"class":807,"line":1368},[805,6676,6677],{"class":2536},"          \u003C",[805,6679,6680],{"class":2522},"RouterLink\n",[805,6682,6683,6686,6688,6690,6692],{"class":807,"line":1379},[805,6684,6685],{"class":2552},"            to",[805,6687,2823],{"class":2536},[805,6689,2854],{"class":2526},[805,6691,3364],{"class":814},[805,6693,6694],{"class":2526},"\"\n",[805,6696,6697,6700,6702,6704,6707],{"class":807,"line":1390},[805,6698,6699],{"class":2552},"            class",[805,6701,2823],{"class":2536},[805,6703,2854],{"class":2526},[805,6705,6706],{"class":814},"block rounded-md px-4 py-2 text-gray-700 hover:bg-gray-100",[805,6708,6694],{"class":2526},[805,6710,6711,6714,6716,6718,6721],{"class":807,"line":1401},[805,6712,6713],{"class":2552},"            exact-active-class",[805,6715,2823],{"class":2536},[805,6717,2854],{"class":2526},[805,6719,6720],{"class":814},"text-indigo-700 font-bold",[805,6722,6694],{"class":2526},[805,6724,6725],{"class":807,"line":1790},[805,6726,6727],{"class":2536},"          >\n",[805,6729,6730],{"class":807,"line":1800},[805,6731,6732],{"class":2940},"            Home\n",[805,6734,6735,6738,6740],{"class":807,"line":1811},[805,6736,6737],{"class":2536},"          \u003C\u002F",[805,6739,3354],{"class":2522},[805,6741,2810],{"class":2536},[805,6743,6744,6746],{"class":807,"line":1825},[805,6745,6677],{"class":2536},[805,6747,6680],{"class":2522},[805,6749,6750,6752,6754,6756,6758],{"class":807,"line":1836},[805,6751,6685],{"class":2552},[805,6753,2823],{"class":2536},[805,6755,2854],{"class":2526},[805,6757,3392],{"class":814},[805,6759,6694],{"class":2526},[805,6761,6762,6764,6766,6768,6770],{"class":807,"line":1850},[805,6763,6699],{"class":2552},[805,6765,2823],{"class":2536},[805,6767,2854],{"class":2526},[805,6769,6706],{"class":814},[805,6771,6694],{"class":2526},[805,6773,6774,6776,6778,6780,6782],{"class":807,"line":1861},[805,6775,6713],{"class":2552},[805,6777,2823],{"class":2536},[805,6779,2854],{"class":2526},[805,6781,6720],{"class":814},[805,6783,6694],{"class":2526},[805,6785,6786],{"class":807,"line":1873},[805,6787,6727],{"class":2536},[805,6789,6790],{"class":807,"line":1880},[805,6791,6792],{"class":2940},"            About\n",[805,6794,6795,6797,6799],{"class":807,"line":1890},[805,6796,6737],{"class":2536},[805,6798,3354],{"class":2522},[805,6800,2810],{"class":2536},[805,6802,6803,6806,6808],{"class":807,"line":1902},[805,6804,6805],{"class":2536},"        \u003C\u002F",[805,6807,3344],{"class":2522},[805,6809,2810],{"class":2536},[805,6811,6812,6814,6816],{"class":807,"line":1914},[805,6813,3410],{"class":2536},[805,6815,6629],{"class":2522},[805,6817,2810],{"class":2536},[805,6819,6820],{"class":807,"line":1926},[805,6821,2542],{"emptyLinePlaceholder":681},[805,6823,6824],{"class":807,"line":1933},[805,6825,6826],{"class":1162},"      \u003C!-- Main Content -->\n",[805,6828,6829,6831,6834,6836,6838,6840,6843,6845],{"class":807,"line":1943},[805,6830,3316],{"class":2536},[805,6832,6833],{"class":2522},"main",[805,6835,3243],{"class":2552},[805,6837,2823],{"class":2536},[805,6839,2854],{"class":2526},[805,6841,6842],{"class":814},"flex-1 overflow-y-auto p-6",[805,6844,2854],{"class":2526},[805,6846,2810],{"class":2536},[805,6848,6849,6851,6853],{"class":807,"line":2384},[805,6850,3351],{"class":2536},[805,6852,3442],{"class":2522},[805,6854,2862],{"class":2536},[805,6856,6857,6859,6861],{"class":807,"line":2396},[805,6858,3410],{"class":2536},[805,6860,6833],{"class":2522},[805,6862,2810],{"class":2536},[805,6864,6865,6867,6869],{"class":807,"line":2408},[805,6866,3419],{"class":2536},[805,6868,2973],{"class":2522},[805,6870,2810],{"class":2536},[805,6872,6873],{"class":807,"line":2415},[805,6874,2542],{"emptyLinePlaceholder":681},[805,6876,6877],{"class":807,"line":2424},[805,6878,6879],{"class":1162},"    \u003C!-- Footer -->\n",[805,6881,6882,6884],{"class":807,"line":2439},[805,6883,2843],{"class":2536},[805,6885,6886],{"class":2522},"footer\n",[805,6888,6889,6892,6894,6896,6899],{"class":807,"line":2449},[805,6890,6891],{"class":2552},"      class",[805,6893,2823],{"class":2536},[805,6895,2854],{"class":2526},[805,6897,6898],{"class":814},"flex h-12 items-center justify-center bg-white border-t border-gray-200 text-sm text-gray-500",[805,6900,6694],{"class":2526},[805,6902,6903],{"class":807,"line":2458},[805,6904,6905],{"class":2536},"    >\n",[805,6907,6909,6912,6915,6917],{"class":807,"line":6908},47,[805,6910,6911],{"class":2536},"      &",[805,6913,6914],{"class":4000},"copy",[805,6916,5856],{"class":2536},[805,6918,6919],{"class":2940}," 2025 My Project. All rights reserved.\n",[805,6921,6923,6925,6928],{"class":807,"line":6922},48,[805,6924,3419],{"class":2536},[805,6926,6927],{"class":2522},"footer",[805,6929,2810],{"class":2536},[805,6931,6933,6935,6937],{"class":807,"line":6932},49,[805,6934,2953],{"class":2536},[805,6936,2973],{"class":2522},[805,6938,2810],{"class":2536},[805,6940,6942,6944,6946],{"class":807,"line":6941},50,[805,6943,2944],{"class":2536},[805,6945,3211],{"class":2522},[805,6947,2810],{"class":2536},[11,6949,6950,6951,6953],{},"With this, we have created a basic structure with an app bar, a sidebar, a main content area, and a footer, all styled with Tailwind CSS classes.\nIdeally, each block (AppBar, Sidebar, Footer) should be a separate component to promote reuse and maintenance, but for this example, we leave everything in ",[734,6952,1516],{}," for simplicity.",[11,6955,6956,6957,6960],{},"Verify that everything works correctly by opening ",[734,6958,6959],{},"http:\u002F\u002Flocalhost:5173"," in your browser. You should see the basic structure with styles applied.",[11,6962,6963,6964,6968,6969,44],{},"With that, you now have Tailwind CSS v4 working in your project. Explore its ",[88,6965,1131],{"href":6966,"target":280,"rel":6967},"https:\u002F\u002Ftailwindcss.com",[282,283]," and install its extension for better autocompletion in VS Code: ",[88,6970,6973],{"href":6971,"target":280,"rel":6972},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=bradlc.vscode-tailwindcss",[282,283],"Tailwind CSS IntelliSense",[18,6975],{},[21,6977,6979],{"id":6978},"extra-uploading-the-project-to-github","EXTRA: Uploading the project to GitHub",[11,6981,6982],{},"For this, we have many options. Before starting, I assume you already have a GitHub account and Git installed on your machine.",[317,6984,6985],{},[52,6986,6987],{},"First, initialize a Git repository in the project folder (if you haven't already):",[797,6989,6991],{"className":799,"code":6990,"language":801,"meta":666,"style":666},"git init\n",[734,6992,6993],{"__ignoreMap":666},[805,6994,6995,6998],{"class":807,"line":808},[805,6996,6997],{"class":811},"git",[805,6999,7000],{"class":814}," init\n",[317,7002,7003],{"start":670},[52,7004,7005,7006,7009],{},"Then create a ",[734,7007,7008],{},".gitignore"," file in the root of the project (you probably already have one) and make sure it contains the following lines to ignore unnecessary files and folders:",[797,7011,7015],{"className":7012,"code":7014,"language":4598},[7013],"language-text","# Logs\nlogs\n*.log\nnpm-debug.log*\nyarn-debug.log*\nyarn-error.log*\npnpm-debug.log*\nlerna-debug.log*\n\nnode_modules\n.DS_Store\ndist\ndist-ssr\ncoverage\n*.local\n\n# Editor directories and files\n.vscode\u002F*\n!.vscode\u002Fextensions.json\n.idea\n*.suo\n*.ntvs*\n*.njsproj\n*.sln\n*.sw?\n\n*.tsbuildinfo\n\n.eslintcache\n\n# Cypress\n\u002Fcypress\u002Fvideos\u002F\n\u002Fcypress\u002Fscreenshots\u002F\n\n# Vitest\n__screenshots__\u002F\n\n",[734,7016,7014],{"__ignoreMap":666},[317,7018,7019],{"start":667},[52,7020,7021],{},"Now let's add all files and make the initial commit:",[797,7023,7025],{"className":799,"code":7024,"language":801,"meta":666,"style":666},"git add .\n",[734,7026,7027],{"__ignoreMap":666},[805,7028,7029,7031,7034],{"class":807,"line":808},[805,7030,6997],{"class":811},[805,7032,7033],{"class":814}," add",[805,7035,7036],{"class":814}," .\n",[797,7038,7040],{"className":799,"code":7039,"language":801,"meta":666,"style":666},"git commit -m \"Initial commit\"\n",[734,7041,7042],{"__ignoreMap":666},[805,7043,7044,7046,7049,7052,7055,7058],{"class":807,"line":808},[805,7045,6997],{"class":811},[805,7047,7048],{"class":814}," commit",[805,7050,7051],{"class":4000}," -m",[805,7053,7054],{"class":2526}," \"",[805,7056,7057],{"class":814},"Initial commit",[805,7059,6694],{"class":2526},[317,7061,7062,7069],{"start":1053},[52,7063,7064,7065,7068],{},"Now, go to GitHub and create a new repository. You can do this by clicking the \"+\" icon in the top right corner and selecting \"New repository\". Give it a name (for example, ",[734,7066,7067],{},"my-vue-app","), add a description if you want, and choose whether it will be public or private. Do not initialize it with a README, .gitignore, or license, since we already have those locally.",[52,7070,7071],{},"Then, follow the instructions GitHub gives you to connect your local repository with the remote one. They are usually something like this:",[797,7073,7075],{"className":799,"code":7074,"language":801,"meta":666,"style":666},"git remote add origin https:\u002F\u002Fexample-url.git\n",[734,7076,7077],{"__ignoreMap":666},[805,7078,7079,7081,7084,7086,7089],{"class":807,"line":808},[805,7080,6997],{"class":811},[805,7082,7083],{"class":814}," remote",[805,7085,7033],{"class":814},[805,7087,7088],{"class":814}," origin",[805,7090,7091],{"class":814}," https:\u002F\u002Fexample-url.git\n",[797,7093,7095],{"className":799,"code":7094,"language":801,"meta":666,"style":666},"git branch -M main\n",[734,7096,7097],{"__ignoreMap":666},[805,7098,7099,7101,7104,7107],{"class":807,"line":808},[805,7100,6997],{"class":811},[805,7102,7103],{"class":814}," branch",[805,7105,7106],{"class":4000}," -M",[805,7108,7109],{"class":814}," main\n",[797,7111,7113],{"className":799,"code":7112,"language":801,"meta":666,"style":666},"git push -u origin main\n",[734,7114,7115],{"__ignoreMap":666},[805,7116,7117,7119,7122,7125,7127],{"class":807,"line":808},[805,7118,6997],{"class":811},[805,7120,7121],{"class":814}," push",[805,7123,7124],{"class":4000}," -u",[805,7126,7088],{"class":814},[805,7128,7109],{"class":814},[317,7130,7131],{"start":1197},[52,7132,7133],{},"We check status",[797,7135,7137],{"className":799,"code":7136,"language":801,"meta":666,"style":666},"git status\n",[734,7138,7139],{"__ignoreMap":666},[805,7140,7141,7143],{"class":807,"line":808},[805,7142,6997],{"class":811},[805,7144,7145],{"class":814}," status\n",[11,7147,7148,7149,7151,7152,7155],{},"It should say that we are on the ",[734,7150,6833],{}," branch, we are synchronized with the remote repository ",[734,7153,7154],{},"origin\u002Fmain",", and there is nothing to commit.",[317,7157,7158],{"start":1212},[52,7159,7160],{},"Finally, whenever we make changes and want to push them to GitHub, we do:",[797,7162,7164],{"className":799,"code":7163,"language":801,"meta":666,"style":666},"git add .\ngit commit -m \"Description of the changes\"\ngit push\n",[734,7165,7166,7174,7189],{"__ignoreMap":666},[805,7167,7168,7170,7172],{"class":807,"line":808},[805,7169,6997],{"class":811},[805,7171,7033],{"class":814},[805,7173,7036],{"class":814},[805,7175,7176,7178,7180,7182,7184,7187],{"class":807,"line":670},[805,7177,6997],{"class":811},[805,7179,7048],{"class":814},[805,7181,7051],{"class":4000},[805,7183,7054],{"class":2526},[805,7185,7186],{"class":814},"Description of the changes",[805,7188,6694],{"class":2526},[805,7190,7191,7193],{"class":807,"line":667},[805,7192,6997],{"class":811},[805,7194,7195],{"class":814}," push\n",[11,7197,7198],{},"And that's it! You now have your Vue 3 project with Vite uploaded to GitHub.",[11,7200,7201],{},"It's common to always use a version control system like Git to manage your code, even in personal projects. It will help you keep a history of changes, collaborate with others, and protect your work.",[11,7203,7204,7205,7210],{},"I also recommend exploring ",[88,7206,7209],{"href":7207,"target":280,"rel":7208},"https:\u002F\u002Fwww.conventionalcommits.org\u002Fen\u002Fv1.0.0\u002F",[282,283],"Conventional Commits"," to maintain consistent and meaningful commit messages. It's something many companies, teams, and open source projects use.",[11,7212,7213],{},"Some resources to learn more about Git and GitHub:",[49,7215,7216,7223,7230],{},[52,7217,7218],{},[88,7219,7222],{"href":7220,"target":280,"rel":7221},"https:\u002F\u002Fgithub.com\u002Fdjayepro3\u002FGuide-Git-GitHub-VSCode",[282,283],"Git & GitHub with VS Code: A Beginner's Guide",[52,7224,7225],{},[88,7226,7229],{"href":7227,"target":280,"rel":7228},"https:\u002F\u002Fdocs.github.com\u002Fen\u002Fget-started\u002Fstart-your-journey",[282,283],"GitHub Quickstart Guide",[52,7231,7232],{},[88,7233,7236],{"href":7234,"target":280,"rel":7235},"https:\u002F\u002Fgit-scm.com\u002Fdoc",[282,283],"Git Documentation",[11,7238,7239],{},"VS Code Extensions:",[49,7241,7242,7249],{},[52,7243,7244],{},[88,7245,7248],{"href":7246,"target":280,"rel":7247},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=eamodio.gitlens",[282,283],"Gitlens",[52,7250,7251],{},[88,7252,7255],{"href":7253,"target":280,"rel":7254},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=mhutchie.git-graph",[282,283],"Git Graph",[18,7257],{},[21,7259,7261],{"id":7260},"extra-other-useful-extensions-for-vs-code","EXTRA: Other Useful Extensions for VS Code",[11,7263,7264],{},"Some extensions that I always recommend for VS Code (regardless of the type of project):",[49,7266,7267,7275,7283,7291],{},[52,7268,7269,7274],{},[88,7270,7273],{"href":7271,"target":280,"rel":7272},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=usernamehw.errorlens",[282,283],"Error Lens",": Highlights errors and warnings directly in the code, making them easier to identify and fix.",[52,7276,7277,7282],{},[88,7278,7281],{"href":7279,"target":280,"rel":7280},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=naumovs.color-highlight",[282,283],"Color Highlight",": Highlights colors defined in your CSS code, making them easier to visualize.",[52,7284,7285,7290],{},[88,7286,7289],{"href":7287,"target":280,"rel":7288},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=aaron-bond.better-comments",[282,283],"Better Comments",": Improves the readability of comments in the code through colors and styles.",[52,7292,7293,7298,7299,7304],{},[88,7294,7297],{"href":7295,"target":280,"rel":7296},"https:\u002F\u002Fmarketplace.visualstudio.com\u002Fitems?itemName=antfu.iconify",[282,283],"Iconify Intellisense",": If you want to work with icons, this extension allows you to search and insert icons from multiple libraries directly into your code. Additionally, I recommend exploring its library ",[88,7300,7303],{"href":7301,"target":280,"rel":7302},"https:\u002F\u002Ficon-sets.iconify.design\u002F",[282,283],"Iconify"," where you can find a wide variety of free icon sets.",[18,7306],{},[11,7308,7309],{},"Well, that's it for this introductory guide to Vue 3 with Vite. I hope you found it useful for starting your journey with this framework and its ecosystem.",[18,7311],{},[3463,7313,7314],{},"html pre.shiki code .s_xSY, html code.shiki .s_xSY{--shiki-default:#59873A;--shiki-dark:#80A665}html pre.shiki code .spP0B, html code.shiki .spP0B{--shiki-default:#B56959;--shiki-dark:#C98A7D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: 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.sqbOQ{--shiki-default:#2F798A;--shiki-dark:#4C9A91}",{"title":666,"searchDepth":667,"depth":667,"links":7316},[7317,7318,7319,7320,7324,7325,7326,7331,7332,7333,7334],{"id":710,"depth":670,"text":711},{"id":771,"depth":670,"text":772},{"id":791,"depth":670,"text":792},{"id":1137,"depth":670,"text":1138,"children":7321},[7322,7323],{"id":1444,"depth":667,"text":1445},{"id":1526,"depth":667,"text":1527},{"id":2499,"depth":670,"text":2500},{"id":3849,"depth":670,"text":3850},{"id":4498,"depth":670,"text":4499,"children":7327},[7328,7329,7330],{"id":5380,"depth":667,"text":5381},{"id":5414,"depth":667,"text":5415},{"id":5495,"depth":667,"text":5496},{"id":5552,"depth":670,"text":5553},{"id":5698,"depth":670,"text":5699},{"id":6978,"depth":670,"text":6979},{"id":7260,"depth":670,"text":7261},"2025-12-04","\u002Fblog\u002Fgetting-started-vue-vite\u002Fshared\u002Fvite+vue.webp","2025-12-22",{},"\u002Fblog\u002Fblog\u002Fgetting-started-vue-vite",{"title":698,"description":703},{"loc":7342,"priority":686,"lastmod":7337},"\u002Fblog\u002Fgetting-started-vue-vite","getting-started-vue-vite","blog\u002Fblog\u002Fgetting-started-vue-vite","A complete guide to configure your first Vue 3 project with Vite, including best practices and modern tools.",[7347,7348,7349],"Vue.js","Vite","Tailwind CSS","b6SxZbQcC3iRayzSkSVPR9EKbnU-AnUSqrWp6kaVGa8",{"id":7352,"title":7353,"author":6,"body":7354,"date":24700,"description":7358,"extension":678,"image":24701,"lastmod":24700,"meta":24702,"navigation":681,"order":667,"path":24703,"seo":24704,"sitemap":24705,"slug":24707,"stem":24708,"summary":24709,"tags":24710,"__hash__":24711},"content_en\u002Fblog\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations.md","Machine learning paradigms and mathematical foundations",{"type":8,"value":7355,"toc":24684},[7356,7359,7366,7368,7370,7374,7377,7380,7383,7977,8066,8069,8081,8083,8086,8295,8308,8310,8313,8316,8338,8341,8343,8347,8357,8360,8363,8366,8847,8850,9472,9534,9644,9743,9752,9755,9826,9866,10133,10136,10474,10477,10912,10914,11193,11196,11496,11499,11875,11917,11922,12100,12688,12929,12931,13241,13244,13282,13289,13474,13476,13715,13857,14432,14669,14671,14979,14982,15588,15946,16136,16261,16363,16366,16524,16531,16933,16935,17154,17159,17748,17753,17996,18415,18420,18719,18755,18763,18765,18768,18771,18774,18794,18799,18802,18924,19049,19056,19065,19515,19524,19718,19723,20349,20351,20892,20895,20900,20903,21092,21094,21649,21702,21705,21848,21855,21997,22298,22310,22313,22633,22636,23264,23266,23543,23546,23557,23560,23568,23571,23576,23851,23856,24151,24154,24158,24162,24168,24188,24192,24218,24223,24227,24234,24237,24344,24346,24464,24467,24565,24568,24577,24591,24594,24621,24625,24628],[11,7357,7358],{},"We continue learning about Machine Learning, this time we will delve into the different paradigms of machine learning and the mathematical foundations that support these models.",[11,7360,7361,7362],{},"Previous article: ",[88,7363,5],{"href":7364,"rel":7365},"https:\u002F\u002Fderas.dev\u002Fblog\u002Fmachine-learning-fundamentals",[92],[15,7367],{},[18,7369],{},[21,7371,7373],{"id":7372},"machine-learning-paradigms","Machine Learning Paradigms",[11,7375,7376],{},"In the previous article, we already mentioned the types of machine learning, in this section we will review them and then move on to the mathematical foundations.",[1442,7378,186],{"id":7379},"supervised-learning",[11,7381,7382],{},"Supervised learning requires a dataset where each input example X is associated with a label or output Y. The objective of the model is to learn a function that maps inputs to correct outputs.",[11,7384,7385,7386,7830,7831,7904,7905,7976],{},"Based on this, the training dataset is represented as a set of pairs 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is the input and ",[805,7906,7908,7926],{"className":7907},[7389],[805,7909,7911],{"className":7910},[7393],[7395,7912,7913],{"xmlns":7397},[7399,7914,7915,7923],{},[7402,7916,7917],{},[7411,7918,7919,7921],{},[7414,7920,828],{},[7414,7922,7849],{},[7487,7924,7925],{"encoding":7489},"y_i",[805,7927,7929],{"className":7928,"ariaHidden":7423},[7494],[805,7930,7932,7936],{"className":7931},[7498],[805,7933],{"className":7934,"style":7935},[7502],"height:0.625em;vertical-align:-0.1944em;",[805,7937,7939,7942],{"className":7938},[7512],[805,7940,828],{"className":7941,"style":7582},[7512,7516],[805,7943,7945],{"className":7944},[7520],[805,7946,7948,7968],{"className":7947},[7524,7525],[805,7949,7951,7965],{"className":7950},[7529],[805,7952,7954],{"className":7953,"style":7881},[7533],[805,7955,7956,7959],{"style":7597},[805,7957],{"className":7958,"style":7542},[7541],[805,7960,7962],{"className":7961},[7546,7547,7548,7549],[805,7963,7849],{"className":7964},[7512,7516,7549],[805,7966,7557],{"className":7967},[7556],[805,7969,7971],{"className":7970},[7529],[805,7972,7974],{"className":7973,"style":7564},[7533],[805,7975],{}," is the corresponding label. The supervised learning model attempts to find a function f that can predict the output Y from the input X:",[805,7978,7981],{"className":7979},[7980],"katex-display",[805,7982,7984,8011],{"className":7983},[7389],[805,7985,7987],{"className":7986},[7393],[7395,7988,7990],{"xmlns":7397,"display":7989},"block",[7399,7991,7992,8008],{},[7402,7993,7994,7997,7999,8002,8005],{},[7414,7995,7996],{},"f",[7405,7998,3573],{},[7414,8000,8001],{},"X",[7405,8003,8004],{},"→",[7414,8006,8007],{},"Y",[7487,8009,8010],{"encoding":7489},"f: X \\rightarrow Y",[805,8012,8014,8036,8056],{"className":8013,"ariaHidden":7423},[7494],[805,8015,8017,8021,8025,8029,8033],{"className":8016},[7498],[805,8018],{"className":8019,"style":8020},[7502],"height:0.8889em;vertical-align:-0.1944em;",[805,8022,7996],{"className":8023,"style":8024},[7512,7516],"margin-right:0.1076em;",[805,8026],{"className":8027,"style":8028},[7574],"margin-right:0.2778em;",[805,8030,3573],{"className":8031},[8032],"mrel",[805,8034],{"className":8035,"style":8028},[7574],[805,8037,8039,8043,8047,8050,8053],{"className":8038},[7498],[805,8040],{"className":8041,"style":8042},[7502],"height:0.6833em;",[805,8044,8001],{"className":8045,"style":8046},[7512,7516],"margin-right:0.0785em;",[805,8048],{"className":8049,"style":8028},[7574],[805,8051,8004],{"className":8052},[8032],[805,8054],{"className":8055,"style":8028},[7574],[805,8057,8059,8062],{"className":8058},[7498],[805,8060],{"className":8061,"style":8042},[7502],[805,8063,8007],{"className":8064,"style":8065},[7512,7516],"margin-right:0.2222em;",[11,8067,8068],{},"Depending on the type of output, supervised learning can be divided into:",[49,8070,8071,8076],{},[52,8072,8073,8075],{},[29,8074,266],{},": When the output Y is a discrete category (that is, a categorical value). For example, classifying whether an image contains a cat or a dog, whether a number is even or odd, or whether an email is spam or not spam.",[52,8077,8078,8080],{},[29,8079,272],{},": When the output Y is a continuous value (that is, a real number). For example, making predictions about future sales based on historical data, predicting the price of a house based on its features, or estimating the temperature of a city based on climatic factors.",[18,8082],{},[1442,8084,311],{"id":8085},"unsupervised-learning",[11,8087,8088,8089,8294],{},"In unsupervised learning, the model receives only the inputs X without associated labels: ",[805,8090,8092,8138],{"className":8091},[7389],[805,8093,8095],{"className":8094},[7393],[7395,8096,8097],{"xmlns":7397},[7399,8098,8099,8135],{},[7402,8100,8101,8103,8109,8111,8117,8119,8121,8123,8125,8127,8133],{},[7405,8102,7407],{"stretchy":1435},[7411,8104,8105,8107],{},[7414,8106,7416],{},[7418,8108,7420],{},[7405,8110,3096],{"separator":7423},[7411,8112,8113,8115],{},[7414,8114,7416],{},[7418,8116,7442],{},[7405,8118,3096],{"separator":7423},[7414,8120,44],{"mathvariant":7457},[7414,8122,44],{"mathvariant":7457},[7414,8124,44],{"mathvariant":7457},[7405,8126,3096],{"separator":7423},[7411,8128,8129,8131],{},[7414,8130,7416],{},[7414,8132,7472],{},[7405,8134,7485],{"stretchy":1435},[7487,8136,8137],{"encoding":7489},"\\{x_1, x_2, ..., x_n\\}",[805,8139,8141],{"className":8140,"ariaHidden":7423},[7494],[805,8142,8144,8147,8150,8190,8193,8196,8236,8239,8242,8245,8248,8251,8291],{"className":8143},[7498],[805,8145],{"className":8146,"style":7503},[7502],[805,8148,7407],{"className":8149},[7507],[805,8151,8153,8156],{"className":8152},[7512],[805,8154,7416],{"className":8155},[7512,7516],[805,8157,8159],{"className":8158},[7520],[805,8160,8162,8182],{"className":8161},[7524,7525],[805,8163,8165,8179],{"className":8164},[7529],[805,8166,8168],{"className":8167,"style":7534},[7533],[805,8169,8170,8173],{"style":7537},[805,8171],{"className":8172,"style":7542},[7541],[805,8174,8176],{"className":8175},[7546,7547,7548,7549],[805,8177,7420],{"className":8178},[7512,7549],[805,8180,7557],{"className":8181},[7556],[805,8183,8185],{"className":8184},[7529],[805,8186,8188],{"className":8187,"style":7564},[7533],[805,8189],{},[805,8191,3096],{"className":8192},[7570],[805,8194],{"className":8195,"style":7575},[7574],[805,8197,8199,8202],{"className":8198},[7512],[805,8200,7416],{"className":8201},[7512,7516],[805,8203,8205],{"className":8204},[7520],[805,8206,8208,8228],{"className":8207},[7524,7525],[805,8209,8211,8225],{"className":8210},[7529],[805,8212,8214],{"className":8213,"style":7534},[7533],[805,8215,8216,8219],{"style":7537},[805,8217],{"className":8218,"style":7542},[7541],[805,8220,8222],{"className":8221},[7546,7547,7548,7549],[805,8223,7442],{"className":8224},[7512,7549],[805,8226,7557],{"className":8227},[7556],[805,8229,8231],{"className":8230},[7529],[805,8232,8234],{"className":8233,"style":7564},[7533],[805,8235],{},[805,8237,3096],{"className":8238},[7570],[805,8240],{"className":8241,"style":7575},[7574],[805,8243,7729],{"className":8244},[7512],[805,8246,3096],{"className":8247},[7570],[805,8249],{"className":8250,"style":7575},[7574],[805,8252,8254,8257],{"className":8253},[7512],[805,8255,7416],{"className":8256},[7512,7516],[805,8258,8260],{"className":8259},[7520],[805,8261,8263,8283],{"className":8262},[7524,7525],[805,8264,8266,8280],{"className":8265},[7529],[805,8267,8269],{"className":8268,"style":7757},[7533],[805,8270,8271,8274],{"style":7537},[805,8272],{"className":8273,"style":7542},[7541],[805,8275,8277],{"className":8276},[7546,7547,7548,7549],[805,8278,7472],{"className":8279},[7512,7516,7549],[805,8281,7557],{"className":8282},[7556],[805,8284,8286],{"className":8285},[7529],[805,8287,8289],{"className":8288,"style":7564},[7533],[805,8290],{},[805,8292,7485],{"className":8293},[7621],". The objective is to find underlying patterns or structures in the data.\nSome examples of unsupervised learning techniques include:",[49,8296,8297,8303],{},[52,8298,8299,8302],{},[29,8300,8301],{},"Clustering",": Grouping similar data points into clusters. For example, segmenting customers into groups based on their purchasing behaviors.",[52,8304,8305,8307],{},[29,8306,340],{},": Reducing the number of variables in a dataset while preserving as much information as possible. For example, using PCA (Principal Component Analysis) to visualize data in 2D or 3D.",[18,8309],{},[1442,8311,419],{"id":8312},"reinforcement-learning",[11,8314,8315],{},"Reinforcement learning is based on the idea that an agent learns to make decisions by interacting with an environment. The agent receives rewards or penalties based on the actions it takes, and its objective is to maximize the accumulated reward over time.\nIn this paradigm, the agent learns an action policy that allows it to make optimal decisions based on the current state of the environment. A classic example of reinforcement learning is the game of chess, where the agent learns to play better as it plays more games and receives feedback about its moves.",[317,8317,8318,8323,8328,8333],{},[52,8319,8320,8322],{},[29,8321,435],{},": The system that makes decisions.",[52,8324,8325,8327],{},[29,8326,441],{},": The world with which the agent interacts.",[52,8329,8330,8332],{},[29,8331,447],{},": The feedback that the agent receives after taking an action.",[52,8334,8335,8337],{},[29,8336,453],{},": The strategy that the agent follows to make decisions.",[11,8339,8340],{},"Its applications include games, robotics, and recommendation systems.",[18,8342],{},[21,8344,8346],{"id":8345},"mathematical-foundations","Mathematical Foundations",[11,8348,8349,8350,8353,8354,44],{},"It is important to understand that machine learning is based on concepts from linear algebra, calculus, probability, and statistics. These fundamentals are essential for understanding how ",[29,8351,8352],{},"models work"," and how they are ",[29,8355,8356],{},"optimized",[1442,8358,272],{"id":8359},"regression",[11,8361,8362],{},"What is a regression problem? It is a type of problem where the objective is to predict a continuous value by finding the best line (or plane) that fits the data. We have already discussed application examples such as price calculation, sales forecasting, etc.",[11,8364,8365],{},"The simplest form of regression is linear regression, which can be mathematically expressed as:",[805,8367,8369],{"className":8368},[7980],[805,8370,8372,8452],{"className":8371},[7389],[805,8373,8375],{"className":8374},[7393],[7395,8376,8377],{"xmlns":7397,"display":7989},[7399,8378,8379,8449],{},[7402,8380,8381,8383,8385,8393,8396,8402,8408,8410,8416,8422,8424,8426,8428,8430,8432,8438,8444,8446],{},[7414,8382,828],{},[7405,8384,2823],{},[7411,8386,8387,8390],{},[7414,8388,8389],{},"β",[7418,8391,8392],{},"0",[7405,8394,8395],{},"+",[7411,8397,8398,8400],{},[7414,8399,8389],{},[7418,8401,7420],{},[7411,8403,8404,8406],{},[7414,8405,7416],{},[7418,8407,7420],{},[7405,8409,8395],{},[7411,8411,8412,8414],{},[7414,8413,8389],{},[7418,8415,7442],{},[7411,8417,8418,8420],{},[7414,8419,7416],{},[7418,8421,7442],{},[7405,8423,8395],{},[7414,8425,44],{"mathvariant":7457},[7414,8427,44],{"mathvariant":7457},[7414,8429,44],{"mathvariant":7457},[7405,8431,8395],{},[7411,8433,8434,8436],{},[7414,8435,8389],{},[7414,8437,11],{},[7411,8439,8440,8442],{},[7414,8441,7416],{},[7414,8443,11],{},[7405,8445,8395],{},[7414,8447,8448],{},"ϵ",[7487,8450,8451],{"encoding":7489},"y = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + ... + \\beta_p x_p + \\epsilon",[805,8453,8455,8473,8531,8626,8721,8740,8837],{"className":8454,"ariaHidden":7423},[7494],[805,8456,8458,8461,8464,8467,8470],{"className":8457},[7498],[805,8459],{"className":8460,"style":7935},[7502],[805,8462,828],{"className":8463,"style":7582},[7512,7516],[805,8465],{"className":8466,"style":8028},[7574],[805,8468,2823],{"className":8469},[8032],[805,8471],{"className":8472,"style":8028},[7574],[805,8474,8476,8479,8521,8524,8528],{"className":8475},[7498],[805,8477],{"className":8478,"style":8020},[7502],[805,8480,8482,8486],{"className":8481},[7512],[805,8483,8389],{"className":8484,"style":8485},[7512,7516],"margin-right:0.0528em;",[805,8487,8489],{"className":8488},[7520],[805,8490,8492,8513],{"className":8491},[7524,7525],[805,8493,8495,8510],{"className":8494},[7529],[805,8496,8498],{"className":8497,"style":7534},[7533],[805,8499,8501,8504],{"style":8500},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[805,8502],{"className":8503,"style":7542},[7541],[805,8505,8507],{"className":8506},[7546,7547,7548,7549],[805,8508,8392],{"className":8509},[7512,7549],[805,8511,7557],{"className":8512},[7556],[805,8514,8516],{"className":8515},[7529],[805,8517,8519],{"className":8518,"style":7564},[7533],[805,8520],{},[805,8522],{"className":8523,"style":8065},[7574],[805,8525,8395],{"className":8526},[8527],"mbin",[805,8529],{"className":8530,"style":8065},[7574],[805,8532,8534,8537,8577,8617,8620,8623],{"className":8533},[7498],[805,8535],{"className":8536,"style":8020},[7502],[805,8538,8540,8543],{"className":8539},[7512],[805,8541,8389],{"className":8542,"style":8485},[7512,7516],[805,8544,8546],{"className":8545},[7520],[805,8547,8549,8569],{"className":8548},[7524,7525],[805,8550,8552,8566],{"className":8551},[7529],[805,8553,8555],{"className":8554,"style":7534},[7533],[805,8556,8557,8560],{"style":8500},[805,8558],{"className":8559,"style":7542},[7541],[805,8561,8563],{"className":8562},[7546,7547,7548,7549],[805,8564,7420],{"className":8565},[7512,7549],[805,8567,7557],{"className":8568},[7556],[805,8570,8572],{"className":8571},[7529],[805,8573,8575],{"className":8574,"style":7564},[7533],[805,8576],{},[805,8578,8580,8583],{"className":8579},[7512],[805,8581,7416],{"className":8582},[7512,7516],[805,8584,8586],{"className":8585},[7520],[805,8587,8589,8609],{"className":8588},[7524,7525],[805,8590,8592,8606],{"className":8591},[7529],[805,8593,8595],{"className":8594,"style":7534},[7533],[805,8596,8597,8600],{"style":7537},[805,8598],{"className":8599,"style":7542},[7541],[805,8601,8603],{"className":8602},[7546,7547,7548,7549],[805,8604,7420],{"className":8605},[7512,7549],[805,8607,7557],{"className":8608},[7556],[805,8610,8612],{"className":8611},[7529],[805,8613,8615],{"className":8614,"style":7564},[7533],[805,8616],{},[805,8618],{"className":8619,"style":8065},[7574],[805,8621,8395],{"className":8622},[8527],[805,8624],{"className":8625,"style":8065},[7574],[805,8627,8629,8632,8672,8712,8715,8718],{"className":8628},[7498],[805,8630],{"className":8631,"style":8020},[7502],[805,8633,8635,8638],{"className":8634},[7512],[805,8636,8389],{"className":8637,"style":8485},[7512,7516],[805,8639,8641],{"className":8640},[7520],[805,8642,8644,8664],{"className":8643},[7524,7525],[805,8645,8647,8661],{"className":8646},[7529],[805,8648,8650],{"className":8649,"style":7534},[7533],[805,8651,8652,8655],{"style":8500},[805,8653],{"className":8654,"style":7542},[7541],[805,8656,8658],{"className":8657},[7546,7547,7548,7549],[805,8659,7442],{"className":8660},[7512,7549],[805,8662,7557],{"className":8663},[7556],[805,8665,8667],{"className":8666},[7529],[805,8668,8670],{"className":8669,"style":7564},[7533],[805,8671],{},[805,8673,8675,8678],{"className":8674},[7512],[805,8676,7416],{"className":8677},[7512,7516],[805,8679,8681],{"className":8680},[7520],[805,8682,8684,8704],{"className":8683},[7524,7525],[805,8685,8687,8701],{"className":8686},[7529],[805,8688,8690],{"className":8689,"style":7534},[7533],[805,8691,8692,8695],{"style":7537},[805,8693],{"className":8694,"style":7542},[7541],[805,8696,8698],{"className":8697},[7546,7547,7548,7549],[805,8699,7442],{"className":8700},[7512,7549],[805,8702,7557],{"className":8703},[7556],[805,8705,8707],{"className":8706},[7529],[805,8708,8710],{"className":8709,"style":7564},[7533],[805,8711],{},[805,8713],{"className":8714,"style":8065},[7574],[805,8716,8395],{"className":8717},[8527],[805,8719],{"className":8720,"style":8065},[7574],[805,8722,8724,8728,8731,8734,8737],{"className":8723},[7498],[805,8725],{"className":8726,"style":8727},[7502],"height:0.6667em;vertical-align:-0.0833em;",[805,8729,7729],{"className":8730},[7512],[805,8732],{"className":8733,"style":8065},[7574],[805,8735,8395],{"className":8736},[8527],[805,8738],{"className":8739,"style":8065},[7574],[805,8741,8743,8747,8788,8828,8831,8834],{"className":8742},[7498],[805,8744],{"className":8745,"style":8746},[7502],"height:0.9805em;vertical-align:-0.2861em;",[805,8748,8750,8753],{"className":8749},[7512],[805,8751,8389],{"className":8752,"style":8485},[7512,7516],[805,8754,8756],{"className":8755},[7520],[805,8757,8759,8779],{"className":8758},[7524,7525],[805,8760,8762,8776],{"className":8761},[7529],[805,8763,8765],{"className":8764,"style":7757},[7533],[805,8766,8767,8770],{"style":8500},[805,8768],{"className":8769,"style":7542},[7541],[805,8771,8773],{"className":8772},[7546,7547,7548,7549],[805,8774,11],{"className":8775},[7512,7516,7549],[805,8777,7557],{"className":8778},[7556],[805,8780,8782],{"className":8781},[7529],[805,8783,8786],{"className":8784,"style":8785},[7533],"height:0.2861em;",[805,8787],{},[805,8789,8791,8794],{"className":8790},[7512],[805,8792,7416],{"className":8793},[7512,7516],[805,8795,8797],{"className":8796},[7520],[805,8798,8800,8820],{"className":8799},[7524,7525],[805,8801,8803,8817],{"className":8802},[7529],[805,8804,8806],{"className":8805,"style":7757},[7533],[805,8807,8808,8811],{"style":7537},[805,8809],{"className":8810,"style":7542},[7541],[805,8812,8814],{"className":8813},[7546,7547,7548,7549],[805,8815,11],{"className":8816},[7512,7516,7549],[805,8818,7557],{"className":8819},[7556],[805,8821,8823],{"className":8822},[7529],[805,8824,8826],{"className":8825,"style":8785},[7533],[805,8827],{},[805,8829],{"className":8830,"style":8065},[7574],[805,8832,8395],{"className":8833},[8527],[805,8835],{"className":8836,"style":8065},[7574],[805,8838,8840,8844],{"className":8839},[7498],[805,8841],{"className":8842,"style":8843},[7502],"height:0.4306em;",[805,8845,8448],{"className":8846},[7512,7516],[11,8848,8849],{},"Where:",[49,8851,8852,8883,9082,9213,9411],{},[52,8853,8854,8882],{},[805,8855,8857,8870],{"className":8856},[7389],[805,8858,8860],{"className":8859},[7393],[7395,8861,8862],{"xmlns":7397},[7399,8863,8864,8868],{},[7402,8865,8866],{},[7414,8867,828],{},[7487,8869,828],{"encoding":7489},[805,8871,8873],{"className":8872,"ariaHidden":7423},[7494],[805,8874,8876,8879],{"className":8875},[7498],[805,8877],{"className":8878,"style":7935},[7502],[805,8880,828],{"className":8881,"style":7582},[7512,7516]," is the dependent variable (what we want to predict).",[52,8884,8885,9081],{},[805,8886,8888,8930],{"className":8887},[7389],[805,8889,8891],{"className":8890},[7393],[7395,8892,8893],{"xmlns":7397},[7399,8894,8895,8927],{},[7402,8896,8897,8903,8905,8911,8913,8915,8917,8919,8921],{},[7411,8898,8899,8901],{},[7414,8900,7416],{},[7418,8902,7420],{},[7405,8904,3096],{"separator":7423},[7411,8906,8907,8909],{},[7414,8908,7416],{},[7418,8910,7442],{},[7405,8912,3096],{"separator":7423},[7414,8914,44],{"mathvariant":7457},[7414,8916,44],{"mathvariant":7457},[7414,8918,44],{"mathvariant":7457},[7405,8920,3096],{"separator":7423},[7411,8922,8923,8925],{},[7414,8924,7416],{},[7414,8926,11],{},[7487,8928,8929],{"encoding":7489},"x_1, x_2, ..., x_p",[805,8931,8933],{"className":8932,"ariaHidden":7423},[7494],[805,8934,8936,8940,8980,8983,8986,9026,9029,9032,9035,9038,9041],{"className":8935},[7498],[805,8937],{"className":8938,"style":8939},[7502],"height:0.7167em;vertical-align:-0.2861em;",[805,8941,8943,8946],{"className":8942},[7512],[805,8944,7416],{"className":8945},[7512,7516],[805,8947,8949],{"className":8948},[7520],[805,8950,8952,8972],{"className":8951},[7524,7525],[805,8953,8955,8969],{"className":8954},[7529],[805,8956,8958],{"className":8957,"style":7534},[7533],[805,8959,8960,8963],{"style":7537},[805,8961],{"className":8962,"style":7542},[7541],[805,8964,8966],{"className":8965},[7546,7547,7548,7549],[805,8967,7420],{"className":8968},[7512,7549],[805,8970,7557],{"className":8971},[7556],[805,8973,8975],{"className":8974},[7529],[805,8976,8978],{"className":8977,"style":7564},[7533],[805,8979],{},[805,8981,3096],{"className":8982},[7570],[805,8984],{"className":8985,"style":7575},[7574],[805,8987,8989,8992],{"className":8988},[7512],[805,8990,7416],{"className":8991},[7512,7516],[805,8993,8995],{"className":8994},[7520],[805,8996,8998,9018],{"className":8997},[7524,7525],[805,8999,9001,9015],{"className":9000},[7529],[805,9002,9004],{"className":9003,"style":7534},[7533],[805,9005,9006,9009],{"style":7537},[805,9007],{"className":9008,"style":7542},[7541],[805,9010,9012],{"className":9011},[7546,7547,7548,7549],[805,9013,7442],{"className":9014},[7512,7549],[805,9016,7557],{"className":9017},[7556],[805,9019,9021],{"className":9020},[7529],[805,9022,9024],{"className":9023,"style":7564},[7533],[805,9025],{},[805,9027,3096],{"className":9028},[7570],[805,9030],{"className":9031,"style":7575},[7574],[805,9033,7729],{"className":9034},[7512],[805,9036,3096],{"className":9037},[7570],[805,9039],{"className":9040,"style":7575},[7574],[805,9042,9044,9047],{"className":9043},[7512],[805,9045,7416],{"className":9046},[7512,7516],[805,9048,9050],{"className":9049},[7520],[805,9051,9053,9073],{"className":9052},[7524,7525],[805,9054,9056,9070],{"className":9055},[7529],[805,9057,9059],{"className":9058,"style":7757},[7533],[805,9060,9061,9064],{"style":7537},[805,9062],{"className":9063,"style":7542},[7541],[805,9065,9067],{"className":9066},[7546,7547,7548,7549],[805,9068,11],{"className":9069},[7512,7516,7549],[805,9071,7557],{"className":9072},[7556],[805,9074,9076],{"className":9075},[7529],[805,9077,9079],{"className":9078,"style":8785},[7533],[805,9080],{}," are the independent variables (the features).",[52,9083,9084,9154,9155,9183,9184,9212],{},[805,9085,9087,9105],{"className":9086},[7389],[805,9088,9090],{"className":9089},[7393],[7395,9091,9092],{"xmlns":7397},[7399,9093,9094,9102],{},[7402,9095,9096],{},[7411,9097,9098,9100],{},[7414,9099,8389],{},[7418,9101,8392],{},[7487,9103,9104],{"encoding":7489},"\\beta_0",[805,9106,9108],{"className":9107,"ariaHidden":7423},[7494],[805,9109,9111,9114],{"className":9110},[7498],[805,9112],{"className":9113,"style":8020},[7502],[805,9115,9117,9120],{"className":9116},[7512],[805,9118,8389],{"className":9119,"style":8485},[7512,7516],[805,9121,9123],{"className":9122},[7520],[805,9124,9126,9146],{"className":9125},[7524,7525],[805,9127,9129,9143],{"className":9128},[7529],[805,9130,9132],{"className":9131,"style":7534},[7533],[805,9133,9134,9137],{"style":8500},[805,9135],{"className":9136,"style":7542},[7541],[805,9138,9140],{"className":9139},[7546,7547,7548,7549],[805,9141,8392],{"className":9142},[7512,7549],[805,9144,7557],{"className":9145},[7556],[805,9147,9149],{"className":9148},[7529],[805,9150,9152],{"className":9151,"style":7564},[7533],[805,9153],{}," is the intercept (the value of ",[805,9156,9158,9171],{"className":9157},[7389],[805,9159,9161],{"className":9160},[7393],[7395,9162,9163],{"xmlns":7397},[7399,9164,9165,9169],{},[7402,9166,9167],{},[7414,9168,828],{},[7487,9170,828],{"encoding":7489},[805,9172,9174],{"className":9173,"ariaHidden":7423},[7494],[805,9175,9177,9180],{"className":9176},[7498],[805,9178],{"className":9179,"style":7935},[7502],[805,9181,828],{"className":9182,"style":7582},[7512,7516]," when all ",[805,9185,9187,9200],{"className":9186},[7389],[805,9188,9190],{"className":9189},[7393],[7395,9191,9192],{"xmlns":7397},[7399,9193,9194,9198],{},[7402,9195,9196],{},[7414,9197,7416],{},[7487,9199,7416],{"encoding":7489},[805,9201,9203],{"className":9202,"ariaHidden":7423},[7494],[805,9204,9206,9209],{"className":9205},[7498],[805,9207],{"className":9208,"style":8843},[7502],[805,9210,7416],{"className":9211},[7512,7516]," are 0).",[52,9214,9215,9410],{},[805,9216,9218,9260],{"className":9217},[7389],[805,9219,9221],{"className":9220},[7393],[7395,9222,9223],{"xmlns":7397},[7399,9224,9225,9257],{},[7402,9226,9227,9233,9235,9241,9243,9245,9247,9249,9251],{},[7411,9228,9229,9231],{},[7414,9230,8389],{},[7418,9232,7420],{},[7405,9234,3096],{"separator":7423},[7411,9236,9237,9239],{},[7414,9238,8389],{},[7418,9240,7442],{},[7405,9242,3096],{"separator":7423},[7414,9244,44],{"mathvariant":7457},[7414,9246,44],{"mathvariant":7457},[7414,9248,44],{"mathvariant":7457},[7405,9250,3096],{"separator":7423},[7411,9252,9253,9255],{},[7414,9254,8389],{},[7414,9256,11],{},[7487,9258,9259],{"encoding":7489},"\\beta_1, \\beta_2, ..., \\beta_p",[805,9261,9263],{"className":9262,"ariaHidden":7423},[7494],[805,9264,9266,9269,9309,9312,9315,9355,9358,9361,9364,9367,9370],{"className":9265},[7498],[805,9267],{"className":9268,"style":8746},[7502],[805,9270,9272,9275],{"className":9271},[7512],[805,9273,8389],{"className":9274,"style":8485},[7512,7516],[805,9276,9278],{"className":9277},[7520],[805,9279,9281,9301],{"className":9280},[7524,7525],[805,9282,9284,9298],{"className":9283},[7529],[805,9285,9287],{"className":9286,"style":7534},[7533],[805,9288,9289,9292],{"style":8500},[805,9290],{"className":9291,"style":7542},[7541],[805,9293,9295],{"className":9294},[7546,7547,7548,7549],[805,9296,7420],{"className":9297},[7512,7549],[805,9299,7557],{"className":9300},[7556],[805,9302,9304],{"className":9303},[7529],[805,9305,9307],{"className":9306,"style":7564},[7533],[805,9308],{},[805,9310,3096],{"className":9311},[7570],[805,9313],{"className":9314,"style":7575},[7574],[805,9316,9318,9321],{"className":9317},[7512],[805,9319,8389],{"className":9320,"style":8485},[7512,7516],[805,9322,9324],{"className":9323},[7520],[805,9325,9327,9347],{"className":9326},[7524,7525],[805,9328,9330,9344],{"className":9329},[7529],[805,9331,9333],{"className":9332,"style":7534},[7533],[805,9334,9335,9338],{"style":8500},[805,9336],{"className":9337,"style":7542},[7541],[805,9339,9341],{"className":9340},[7546,7547,7548,7549],[805,9342,7442],{"className":9343},[7512,7549],[805,9345,7557],{"className":9346},[7556],[805,9348,9350],{"className":9349},[7529],[805,9351,9353],{"className":9352,"style":7564},[7533],[805,9354],{},[805,9356,3096],{"className":9357},[7570],[805,9359],{"className":9360,"style":7575},[7574],[805,9362,7729],{"className":9363},[7512],[805,9365,3096],{"className":9366},[7570],[805,9368],{"className":9369,"style":7575},[7574],[805,9371,9373,9376],{"className":9372},[7512],[805,9374,8389],{"className":9375,"style":8485},[7512,7516],[805,9377,9379],{"className":9378},[7520],[805,9380,9382,9402],{"className":9381},[7524,7525],[805,9383,9385,9399],{"className":9384},[7529],[805,9386,9388],{"className":9387,"style":7757},[7533],[805,9389,9390,9393],{"style":8500},[805,9391],{"className":9392,"style":7542},[7541],[805,9394,9396],{"className":9395},[7546,7547,7548,7549],[805,9397,11],{"className":9398},[7512,7516,7549],[805,9400,7557],{"className":9401},[7556],[805,9403,9405],{"className":9404},[7529],[805,9406,9408],{"className":9407,"style":8785},[7533],[805,9409],{}," are the coefficients that represent the influence of each feature on the dependent variable.",[52,9412,9413,9442,9443,9471],{},[805,9414,9416,9430],{"className":9415},[7389],[805,9417,9419],{"className":9418},[7393],[7395,9420,9421],{"xmlns":7397},[7399,9422,9423,9427],{},[7402,9424,9425],{},[7414,9426,8448],{},[7487,9428,9429],{"encoding":7489},"\\epsilon",[805,9431,9433],{"className":9432,"ariaHidden":7423},[7494],[805,9434,9436,9439],{"className":9435},[7498],[805,9437],{"className":9438,"style":8843},[7502],[805,9440,8448],{"className":9441},[7512,7516]," is the error or noise value, which represents the variability not explained by the model, that is, what affects ",[805,9444,9446,9459],{"className":9445},[7389],[805,9447,9449],{"className":9448},[7393],[7395,9450,9451],{"xmlns":7397},[7399,9452,9453,9457],{},[7402,9454,9455],{},[7414,9456,828],{},[7487,9458,828],{"encoding":7489},[805,9460,9462],{"className":9461,"ariaHidden":7423},[7494],[805,9463,9465,9468],{"className":9464},[7498],[805,9466],{"className":9467,"style":7935},[7502],[805,9469,828],{"className":9470,"style":7582},[7512,7516]," but is not included in the independent variables.",[11,9473,9474,9475,9504,9505,9533],{},"The objective of the regression model is to find the values of the coefficients ",[805,9476,9478,9492],{"className":9477},[7389],[805,9479,9481],{"className":9480},[7393],[7395,9482,9483],{"xmlns":7397},[7399,9484,9485,9489],{},[7402,9486,9487],{},[7414,9488,8389],{},[7487,9490,9491],{"encoding":7489},"\\beta",[805,9493,9495],{"className":9494,"ariaHidden":7423},[7494],[805,9496,9498,9501],{"className":9497},[7498],[805,9499],{"className":9500,"style":8020},[7502],[805,9502,8389],{"className":9503,"style":8485},[7512,7516]," that minimize the difference between the model's predictions and the real values of ",[805,9506,9508,9521],{"className":9507},[7389],[805,9509,9511],{"className":9510},[7393],[7395,9512,9513],{"xmlns":7397},[7399,9514,9515,9519],{},[7402,9516,9517],{},[7414,9518,828],{},[7487,9520,828],{"encoding":7489},[805,9522,9524],{"className":9523,"ariaHidden":7423},[7494],[805,9525,9527,9530],{"className":9526},[7498],[805,9528],{"className":9529,"style":7935},[7502],[805,9531,828],{"className":9532,"style":7582},[7512,7516],". This can be achieved using techniques such as the least squares method.",[11,9535,2472,9536,9539,9540,9578,9579,9612,9613,9643],{},[29,9537,9538],{},"intercept"," is represented as the first element of the coefficient vector ",[805,9541,9543,9558],{"className":9542},[7389],[805,9544,9546],{"className":9545},[7393],[7395,9547,9548],{"xmlns":7397},[7399,9549,9550,9555],{},[7402,9551,9552],{},[7414,9553,8389],{"mathvariant":9554},"bold-italic",[7487,9556,9557],{"encoding":7489},"\\boldsymbol{\\beta}",[805,9559,9561],{"className":9560,"ariaHidden":7423},[7494],[805,9562,9564,9567],{"className":9563},[7498],[805,9565],{"className":9566,"style":8020},[7502],[805,9568,9570],{"className":9569},[7512],[805,9571,9573],{"className":9572},[7512],[805,9574,8389],{"className":9575,"style":9577},[7512,9576],"boldsymbol","margin-right:0.034em;",", and it is the value of ",[805,9580,9582,9597],{"className":9581},[7389],[805,9583,9585],{"className":9584},[7393],[7395,9586,9587],{"xmlns":7397},[7399,9588,9589,9594],{},[7402,9590,9591],{},[7414,9592,828],{"mathvariant":9593},"bold",[7487,9595,9596],{"encoding":7489},"\\mathbf{y}",[805,9598,9600],{"className":9599,"ariaHidden":7423},[7494],[805,9601,9603,9607],{"className":9602},[7498],[805,9604],{"className":9605,"style":9606},[7502],"height:0.6389em;vertical-align:-0.1944em;",[805,9608,828],{"className":9609,"style":9611},[7512,9610],"mathbf","margin-right:0.016em;"," when all the independent variables in ",[805,9614,9616,9630],{"className":9615},[7389],[805,9617,9619],{"className":9618},[7393],[7395,9620,9621],{"xmlns":7397},[7399,9622,9623,9627],{},[7402,9624,9625],{},[7414,9626,8001],{"mathvariant":9593},[7487,9628,9629],{"encoding":7489},"\\mathbf{X}",[805,9631,9633],{"className":9632,"ariaHidden":7423},[7494],[805,9634,9636,9640],{"className":9635},[7498],[805,9637],{"className":9638,"style":9639},[7502],"height:0.6861em;",[805,9641,8001],{"className":9642},[7512,9610]," are zero, that is, it represents the point where the regression line crosses the Y-axis.",[11,9645,2472,9646,9649,9650,9684,9685,9713,9714,9742],{},[29,9647,9648],{},"slope"," is represented by the remaining coefficients in the vector ",[805,9651,9653,9666],{"className":9652},[7389],[805,9654,9656],{"className":9655},[7393],[7395,9657,9658],{"xmlns":7397},[7399,9659,9660,9664],{},[7402,9661,9662],{},[7414,9663,8389],{"mathvariant":9554},[7487,9665,9557],{"encoding":7489},[805,9667,9669],{"className":9668,"ariaHidden":7423},[7494],[805,9670,9672,9675],{"className":9671},[7498],[805,9673],{"className":9674,"style":8020},[7502],[805,9676,9678],{"className":9677},[7512],[805,9679,9681],{"className":9680},[7512],[805,9682,8389],{"className":9683,"style":9577},[7512,9576],", and each coefficient indicates the amount of change in the dependent variable ",[805,9686,9688,9701],{"className":9687},[7389],[805,9689,9691],{"className":9690},[7393],[7395,9692,9693],{"xmlns":7397},[7399,9694,9695,9699],{},[7402,9696,9697],{},[7414,9698,828],{"mathvariant":9593},[7487,9700,9596],{"encoding":7489},[805,9702,9704],{"className":9703,"ariaHidden":7423},[7494],[805,9705,9707,9710],{"className":9706},[7498],[805,9708],{"className":9709,"style":9606},[7502],[805,9711,828],{"className":9712,"style":9611},[7512,9610]," for each unit change in the corresponding independent variable in ",[805,9715,9717,9730],{"className":9716},[7389],[805,9718,9720],{"className":9719},[7393],[7395,9721,9722],{"xmlns":7397},[7399,9723,9724,9728],{},[7402,9725,9726],{},[7414,9727,8001],{"mathvariant":9593},[7487,9729,9629],{"encoding":7489},[805,9731,9733],{"className":9732,"ariaHidden":7423},[7494],[805,9734,9736,9739],{"className":9735},[7498],[805,9737],{"className":9738,"style":9639},[7502],[805,9740,8001],{"className":9741},[7512,9610],", keeping all other independent variables constant.\nGraphically we can see it as follows:",[11,9744,9745,9749],{},[833,9746],{"alt":9747,"src":9748},"Linear regression graph","\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations\u002Fshared\u002Fregression-model.webp",[838,9750,9751],{},"Linear Regression Graph",[11,9753,9754],{},"Each black point represents a training data point, and the red line represents the model.",[11,9756,9757,9758,9761,9762,9796,9797,9825],{},"The key point here is to understand that ",[29,9759,9760],{},"the main objective"," is to find the values of the coefficients ",[805,9763,9765,9778],{"className":9764},[7389],[805,9766,9768],{"className":9767},[7393],[7395,9769,9770],{"xmlns":7397},[7399,9771,9772,9776],{},[7402,9773,9774],{},[7414,9775,8389],{"mathvariant":9554},[7487,9777,9557],{"encoding":7489},[805,9779,9781],{"className":9780,"ariaHidden":7423},[7494],[805,9782,9784,9787],{"className":9783},[7498],[805,9785],{"className":9786,"style":8020},[7502],[805,9788,9790],{"className":9789},[7512],[805,9791,9793],{"className":9792},[7512],[805,9794,8389],{"className":9795,"style":9577},[7512,9576]," that make the predictions ",[805,9798,9800,9813],{"className":9799},[7389],[805,9801,9803],{"className":9802},[7393],[7395,9804,9805],{"xmlns":7397},[7399,9806,9807,9811],{},[7402,9808,9809],{},[7414,9810,828],{"mathvariant":9593},[7487,9812,9596],{"encoding":7489},[805,9814,9816],{"className":9815,"ariaHidden":7423},[7494],[805,9817,9819,9822],{"className":9818},[7498],[805,9820],{"className":9821,"style":9606},[7502],[805,9823,828],{"className":9824,"style":9611},[7512,9610]," (the red line) as close as possible to the actual values.",[11,9827,9828,9829,9865],{},"That is, we do not seek to predict with perfect accuracy, as (in reality) there will always be some error or difference between what the model predicts and what actually happens. The term ",[805,9830,9832,9846],{"className":9831},[7389],[805,9833,9835],{"className":9834},[7393],[7395,9836,9837],{"xmlns":7397},[7399,9838,9839,9843],{},[7402,9840,9841],{},[7414,9842,8448],{"mathvariant":9554},[7487,9844,9845],{"encoding":7489},"\\boldsymbol{\\epsilon}",[805,9847,9849],{"className":9848,"ariaHidden":7423},[7494],[805,9850,9852,9856],{"className":9851},[7498],[805,9853],{"className":9854,"style":9855},[7502],"height:0.4444em;",[805,9857,9859],{"className":9858},[7512],[805,9860,9862],{"className":9861},[7512],[805,9863,8448],{"className":9864},[7512,9576]," is the difference between the values predicted by the model and the actual values:",[805,9867,9869],{"className":9868},[7980],[805,9870,9872,9915],{"className":9871},[7389],[805,9873,9875],{"className":9874},[7393],[7395,9876,9877],{"xmlns":7397,"display":7989},[7399,9878,9879,9912],{},[7402,9880,9881,9889,9891,9897,9900],{},[7414,9882,9883],{},[7411,9884,9885,9887],{},[7414,9886,8448],{"mathvariant":9554},[7414,9888,7849],{"mathvariant":9554},[7405,9890,2823],{},[7411,9892,9893,9895],{},[7414,9894,828],{"mathvariant":9593},[7414,9896,7849],{"mathvariant":9593},[7405,9898,9899],{},"−",[9901,9902,9903,9909],"mover",{"accent":7423},[7411,9904,9905,9907],{},[7414,9906,828],{"mathvariant":9593},[7414,9908,7849],{"mathvariant":9593},[7405,9910,9911],{},"^",[7487,9913,9914],{"encoding":7489},"\\boldsymbol{\\epsilon _i} = \\mathbf{y_i} - \\hat{\\mathbf{y_i}}",[805,9916,9918,9981,10039],{"className":9917,"ariaHidden":7423},[7494],[805,9919,9921,9925,9972,9975,9978],{"className":9920},[7498],[805,9922],{"className":9923,"style":9924},[7502],"height:0.5944em;vertical-align:-0.15em;",[805,9926,9928],{"className":9927},[7512],[805,9929,9931],{"className":9930},[7512],[805,9932,9934,9937],{"className":9933},[7512],[805,9935,8448],{"className":9936},[7512,9576],[805,9938,9940],{"className":9939},[7520],[805,9941,9943,9964],{"className":9942},[7524,7525],[805,9944,9946,9961],{"className":9945},[7529],[805,9947,9950],{"className":9948,"style":9949},[7533],"height:0.3353em;",[805,9951,9952,9955],{"style":7537},[805,9953],{"className":9954,"style":7542},[7541],[805,9956,9958],{"className":9957},[7546,7547,7548,7549],[805,9959,7849],{"className":9960},[7512,9576,7549],[805,9962,7557],{"className":9963},[7556],[805,9965,9967],{"className":9966},[7529],[805,9968,9970],{"className":9969,"style":7564},[7533],[805,9971],{},[805,9973],{"className":9974,"style":8028},[7574],[805,9976,2823],{"className":9977},[8032],[805,9979],{"className":9980,"style":8028},[7574],[805,9982,9984,9988,10030,10033,10036],{"className":9983},[7498],[805,9985],{"className":9986,"style":9987},[7502],"height:0.7778em;vertical-align:-0.1944em;",[805,9989,9991,9994],{"className":9990},[7512],[805,9992,828],{"className":9993,"style":9611},[7512,9610],[805,9995,9997],{"className":9996},[7520],[805,9998,10000,10022],{"className":9999},[7524,7525],[805,10001,10003,10019],{"className":10002},[7529],[805,10004,10007],{"className":10005,"style":10006},[7533],"height:0.3361em;",[805,10008,10010,10013],{"style":10009},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[805,10011],{"className":10012,"style":7542},[7541],[805,10014,10016],{"className":10015},[7546,7547,7548,7549],[805,10017,7849],{"className":10018},[7512,9610,7549],[805,10020,7557],{"className":10021},[7556],[805,10023,10025],{"className":10024},[7529],[805,10026,10028],{"className":10027,"style":7564},[7533],[805,10029],{},[805,10031],{"className":10032,"style":8065},[7574],[805,10034,9899],{"className":10035},[8527],[805,10037],{"className":10038,"style":8065},[7574],[805,10040,10042,10046],{"className":10041},[7498],[805,10043],{"className":10044,"style":10045},[7502],"height:0.9023em;vertical-align:-0.1944em;",[805,10047,10050],{"className":10048},[7512,10049],"accent",[805,10051,10053,10124],{"className":10052},[7524,7525],[805,10054,10056,10121],{"className":10055},[7529],[805,10057,10060,10107],{"className":10058,"style":10059},[7533],"height:0.7079em;",[805,10061,10063,10067],{"style":10062},"top:-3em;",[805,10064],{"className":10065,"style":10066},[7541],"height:3em;",[805,10068,10070,10073],{"className":10069},[7512],[805,10071,828],{"className":10072,"style":9611},[7512,9610],[805,10074,10076],{"className":10075},[7520],[805,10077,10079,10099],{"className":10078},[7524,7525],[805,10080,10082,10096],{"className":10081},[7529],[805,10083,10085],{"className":10084,"style":10006},[7533],[805,10086,10087,10090],{"style":10009},[805,10088],{"className":10089,"style":7542},[7541],[805,10091,10093],{"className":10092},[7546,7547,7548,7549],[805,10094,7849],{"className":10095},[7512,9610,7549],[805,10097,7557],{"className":10098},[7556],[805,10100,10102],{"className":10101},[7529],[805,10103,10105],{"className":10104,"style":7564},[7533],[805,10106],{},[805,10108,10110,10113],{"style":10109},"top:-3.0134em;",[805,10111],{"className":10112,"style":10066},[7541],[805,10114,10118],{"className":10115,"style":10117},[10116],"accent-body","left:-0.25em;",[805,10119,9911],{"className":10120},[7512],[805,10122,7557],{"className":10123},[7556],[805,10125,10127],{"className":10126},[7529],[805,10128,10131],{"className":10129,"style":10130},[7533],"height:0.1944em;",[805,10132],{},[11,10134,10135],{},"Which can also be expressed as:",[805,10137,10139],{"className":10138},[7980],[805,10140,10142,10198],{"className":10141},[7389],[805,10143,10145],{"className":10144},[7393],[7395,10146,10147],{"xmlns":7397,"display":7989},[7399,10148,10149,10195],{},[7402,10150,10151,10159,10161,10167,10169,10171,10179,10181,10187,10193],{},[7414,10152,10153],{},[7411,10154,10155,10157],{},[7414,10156,8448],{"mathvariant":9554},[7414,10158,7849],{"mathvariant":9554},[7405,10160,2823],{},[7411,10162,10163,10165],{},[7414,10164,828],{"mathvariant":9593},[7414,10166,7849],{"mathvariant":9593},[7405,10168,9899],{},[7405,10170,2651],{"stretchy":1435},[7414,10172,10173],{},[7411,10174,10175,10177],{},[7414,10176,8389],{"mathvariant":9554},[7418,10178,8392],{"mathvariant":9593},[7405,10180,8395],{},[7411,10182,10183,10185],{},[7414,10184,8389],{},[7418,10186,7420],{},[7411,10188,10189,10191],{},[7414,10190,7416],{},[7414,10192,7849],{},[7405,10194,3993],{"stretchy":1435},[7487,10196,10197],{"encoding":7489},"\\boldsymbol{\\epsilon _i} = \\mathbf{y_i} - (\\boldsymbol{\\beta_0} + \\beta_1 x_{i})",[805,10199,10201,10262,10317,10382],{"className":10200,"ariaHidden":7423},[7494],[805,10202,10204,10207,10253,10256,10259],{"className":10203},[7498],[805,10205],{"className":10206,"style":9924},[7502],[805,10208,10210],{"className":10209},[7512],[805,10211,10213],{"className":10212},[7512],[805,10214,10216,10219],{"className":10215},[7512],[805,10217,8448],{"className":10218},[7512,9576],[805,10220,10222],{"className":10221},[7520],[805,10223,10225,10245],{"className":10224},[7524,7525],[805,10226,10228,10242],{"className":10227},[7529],[805,10229,10231],{"className":10230,"style":9949},[7533],[805,10232,10233,10236],{"style":7537},[805,10234],{"className":10235,"style":7542},[7541],[805,10237,10239],{"className":10238},[7546,7547,7548,7549],[805,10240,7849],{"className":10241},[7512,9576,7549],[805,10243,7557],{"className":10244},[7556],[805,10246,10248],{"className":10247},[7529],[805,10249,10251],{"className":10250,"style":7564},[7533],[805,10252],{},[805,10254],{"className":10255,"style":8028},[7574],[805,10257,2823],{"className":10258},[8032],[805,10260],{"className":10261,"style":8028},[7574],[805,10263,10265,10268,10308,10311,10314],{"className":10264},[7498],[805,10266],{"className":10267,"style":9987},[7502],[805,10269,10271,10274],{"className":10270},[7512],[805,10272,828],{"className":10273,"style":9611},[7512,9610],[805,10275,10277],{"className":10276},[7520],[805,10278,10280,10300],{"className":10279},[7524,7525],[805,10281,10283,10297],{"className":10282},[7529],[805,10284,10286],{"className":10285,"style":10006},[7533],[805,10287,10288,10291],{"style":10009},[805,10289],{"className":10290,"style":7542},[7541],[805,10292,10294],{"className":10293},[7546,7547,7548,7549],[805,10295,7849],{"className":10296},[7512,9610,7549],[805,10298,7557],{"className":10299},[7556],[805,10301,10303],{"className":10302},[7529],[805,10304,10306],{"className":10305,"style":7564},[7533],[805,10307],{},[805,10309],{"className":10310,"style":8065},[7574],[805,10312,9899],{"className":10313},[8527],[805,10315],{"className":10316,"style":8065},[7574],[805,10318,10320,10323,10326,10373,10376,10379],{"className":10319},[7498],[805,10321],{"className":10322,"style":7503},[7502],[805,10324,2651],{"className":10325},[7507],[805,10327,10329],{"className":10328},[7512],[805,10330,10332],{"className":10331},[7512],[805,10333,10335,10338],{"className":10334},[7512],[805,10336,8389],{"className":10337,"style":9577},[7512,9576],[805,10339,10341],{"className":10340},[7520],[805,10342,10344,10365],{"className":10343},[7524,7525],[805,10345,10347,10362],{"className":10346},[7529],[805,10348,10350],{"className":10349,"style":7534},[7533],[805,10351,10353,10356],{"style":10352},"top:-2.55em;margin-left:-0.034em;margin-right:0.05em;",[805,10354],{"className":10355,"style":7542},[7541],[805,10357,10359],{"className":10358},[7546,7547,7548,7549],[805,10360,8392],{"className":10361},[7512,9610,7549],[805,10363,7557],{"className":10364},[7556],[805,10366,10368],{"className":10367},[7529],[805,10369,10371],{"className":10370,"style":7564},[7533],[805,10372],{},[805,10374],{"className":10375,"style":8065},[7574],[805,10377,8395],{"className":10378},[8527],[805,10380],{"className":10381,"style":8065},[7574],[805,10383,10385,10388,10428,10471],{"className":10384},[7498],[805,10386],{"className":10387,"style":7503},[7502],[805,10389,10391,10394],{"className":10390},[7512],[805,10392,8389],{"className":10393,"style":8485},[7512,7516],[805,10395,10397],{"className":10396},[7520],[805,10398,10400,10420],{"className":10399},[7524,7525],[805,10401,10403,10417],{"className":10402},[7529],[805,10404,10406],{"className":10405,"style":7534},[7533],[805,10407,10408,10411],{"style":8500},[805,10409],{"className":10410,"style":7542},[7541],[805,10412,10414],{"className":10413},[7546,7547,7548,7549],[805,10415,7420],{"className":10416},[7512,7549],[805,10418,7557],{"className":10419},[7556],[805,10421,10423],{"className":10422},[7529],[805,10424,10426],{"className":10425,"style":7564},[7533],[805,10427],{},[805,10429,10431,10434],{"className":10430},[7512],[805,10432,7416],{"className":10433},[7512,7516],[805,10435,10437],{"className":10436},[7520],[805,10438,10440,10463],{"className":10439},[7524,7525],[805,10441,10443,10460],{"className":10442},[7529],[805,10444,10446],{"className":10445,"style":7881},[7533],[805,10447,10448,10451],{"style":7537},[805,10449],{"className":10450,"style":7542},[7541],[805,10452,10454],{"className":10453},[7546,7547,7548,7549],[805,10455,10457],{"className":10456},[7512,7549],[805,10458,7849],{"className":10459},[7512,7516,7549],[805,10461,7557],{"className":10462},[7556],[805,10464,10466],{"className":10465},[7529],[805,10467,10469],{"className":10468,"style":7564},[7533],[805,10470],{},[805,10472,3993],{"className":10473},[7621],[11,10475,10476],{},"To measure the error we can use a loss function. In the case of linear regression, a commonly used loss function is the mean squared error (MSE, for its acronym in English), which is defined as:",[805,10478,10480],{"className":10479},[7980],[805,10481,10483,10556],{"className":10482},[7389],[805,10484,10486],{"className":10485},[7393],[7395,10487,10488],{"xmlns":7397,"display":7989},[7399,10489,10490,10553],{},[7402,10491,10492,10495,10498,10501,10503,10510,10526,10528,10534,10536,10546],{},[7414,10493,10494],{},"M",[7414,10496,10497],{},"S",[7414,10499,10500],{},"E",[7405,10502,2823],{},[10504,10505,10506,10508],"mfrac",{},[7418,10507,7420],{},[7414,10509,7472],{},[10511,10512,10513,10516,10524],"munderover",{},[7405,10514,10515],{},"∑",[7402,10517,10518,10520,10522],{},[7414,10519,7849],{},[7405,10521,2823],{},[7418,10523,7420],{},[7414,10525,7472],{},[7405,10527,2651],{"stretchy":1435},[7411,10529,10530,10532],{},[7414,10531,828],{},[7414,10533,7849],{},[7405,10535,9899],{},[9901,10537,10538,10544],{"accent":7423},[7411,10539,10540,10542],{},[7414,10541,828],{},[7414,10543,7849],{},[7405,10545,9911],{},[10547,10548,10549,10551],"msup",{},[7405,10550,3993],{"stretchy":1435},[7418,10552,7442],{},[7487,10554,10555],{"encoding":7489},"MSE = \\frac{1}{n} \\sum_{i=1}^{n} (y_i - \\hat{y_i})^2",[805,10557,10559,10585,10794],{"className":10558,"ariaHidden":7423},[7494],[805,10560,10562,10565,10569,10573,10576,10579,10582],{"className":10561},[7498],[805,10563],{"className":10564,"style":8042},[7502],[805,10566,10494],{"className":10567,"style":10568},[7512,7516],"margin-right:0.109em;",[805,10570,10497],{"className":10571,"style":10572},[7512,7516],"margin-right:0.0576em;",[805,10574,10500],{"className":10575,"style":10572},[7512,7516],[805,10577],{"className":10578,"style":8028},[7574],[805,10580,2823],{"className":10581},[8032],[805,10583],{"className":10584,"style":8028},[7574],[805,10586,10588,10592,10662,10665,10742,10745,10785,10788,10791],{"className":10587},[7498],[805,10589],{"className":10590,"style":10591},[7502],"height:2.9291em;vertical-align:-1.2777em;",[805,10593,10595,10599,10659],{"className":10594},[7512],[805,10596],{"className":10597},[7507,10598],"nulldelimiter",[805,10600,10602],{"className":10601},[10504],[805,10603,10605,10650],{"className":10604},[7524,7525],[805,10606,10608,10647],{"className":10607},[7529],[805,10609,10612,10624,10635],{"className":10610,"style":10611},[7533],"height:1.3214em;",[805,10613,10615,10618],{"style":10614},"top:-2.314em;",[805,10616],{"className":10617,"style":10066},[7541],[805,10619,10621],{"className":10620},[7512],[805,10622,7472],{"className":10623},[7512,7516],[805,10625,10627,10630],{"style":10626},"top:-3.23em;",[805,10628],{"className":10629,"style":10066},[7541],[805,10631],{"className":10632,"style":10634},[10633],"frac-line","border-bottom-width:0.04em;",[805,10636,10638,10641],{"style":10637},"top:-3.677em;",[805,10639],{"className":10640,"style":10066},[7541],[805,10642,10644],{"className":10643},[7512],[805,10645,7420],{"className":10646},[7512],[805,10648,7557],{"className":10649},[7556],[805,10651,10653],{"className":10652},[7529],[805,10654,10657],{"className":10655,"style":10656},[7533],"height:0.686em;",[805,10658],{},[805,10660],{"className":10661},[7621,10598],[805,10663],{"className":10664,"style":7575},[7574],[805,10666,10670],{"className":10667},[10668,10669],"mop","op-limits",[805,10671,10673,10733],{"className":10672},[7524,7525],[805,10674,10676,10730],{"className":10675},[7529],[805,10677,10680,10702,10715],{"className":10678,"style":10679},[7533],"height:1.6514em;",[805,10681,10683,10687],{"style":10682},"top:-1.8723em;margin-left:0em;",[805,10684],{"className":10685,"style":10686},[7541],"height:3.05em;",[805,10688,10690],{"className":10689},[7546,7547,7548,7549],[805,10691,10693,10696,10699],{"className":10692},[7512,7549],[805,10694,7849],{"className":10695},[7512,7516,7549],[805,10697,2823],{"className":10698},[8032,7549],[805,10700,7420],{"className":10701},[7512,7549],[805,10703,10705,10708],{"style":10704},"top:-3.05em;",[805,10706],{"className":10707,"style":10686},[7541],[805,10709,10710],{},[805,10711,10515],{"className":10712},[10668,10713,10714],"op-symbol","large-op",[805,10716,10718,10721],{"style":10717},"top:-4.3em;margin-left:0em;",[805,10719],{"className":10720,"style":10686},[7541],[805,10722,10724],{"className":10723},[7546,7547,7548,7549],[805,10725,10727],{"className":10726},[7512,7549],[805,10728,7472],{"className":10729},[7512,7516,7549],[805,10731,7557],{"className":10732},[7556],[805,10734,10736],{"className":10735},[7529],[805,10737,10740],{"className":10738,"style":10739},[7533],"height:1.2777em;",[805,10741],{},[805,10743,2651],{"className":10744},[7507],[805,10746,10748,10751],{"className":10747},[7512],[805,10749,828],{"className":10750,"style":7582},[7512,7516],[805,10752,10754],{"className":10753},[7520],[805,10755,10757,10777],{"className":10756},[7524,7525],[805,10758,10760,10774],{"className":10759},[7529],[805,10761,10763],{"className":10762,"style":7881},[7533],[805,10764,10765,10768],{"style":7597},[805,10766],{"className":10767,"style":7542},[7541],[805,10769,10771],{"className":10770},[7546,7547,7548,7549],[805,10772,7849],{"className":10773},[7512,7516,7549],[805,10775,7557],{"className":10776},[7556],[805,10778,10780],{"className":10779},[7529],[805,10781,10783],{"className":10782,"style":7564},[7533],[805,10784],{},[805,10786],{"className":10787,"style":8065},[7574],[805,10789,9899],{"className":10790},[8527],[805,10792],{"className":10793,"style":8065},[7574],[805,10795,10797,10801,10881],{"className":10796},[7498],[805,10798],{"className":10799,"style":10800},[7502],"height:1.1141em;vertical-align:-0.25em;",[805,10802,10804],{"className":10803},[7512,10049],[805,10805,10807,10873],{"className":10806},[7524,7525],[805,10808,10810,10870],{"className":10809},[7529],[805,10811,10814,10859],{"className":10812,"style":10813},[7533],"height:0.6944em;",[805,10815,10816,10819],{"style":10062},[805,10817],{"className":10818,"style":10066},[7541],[805,10820,10822,10825],{"className":10821},[7512],[805,10823,828],{"className":10824,"style":7582},[7512,7516],[805,10826,10828],{"className":10827},[7520],[805,10829,10831,10851],{"className":10830},[7524,7525],[805,10832,10834,10848],{"className":10833},[7529],[805,10835,10837],{"className":10836,"style":7881},[7533],[805,10838,10839,10842],{"style":7597},[805,10840],{"className":10841,"style":7542},[7541],[805,10843,10845],{"className":10844},[7546,7547,7548,7549],[805,10846,7849],{"className":10847},[7512,7516,7549],[805,10849,7557],{"className":10850},[7556],[805,10852,10854],{"className":10853},[7529],[805,10855,10857],{"className":10856,"style":7564},[7533],[805,10858],{},[805,10860,10861,10864],{"style":10062},[805,10862],{"className":10863,"style":10066},[7541],[805,10865,10867],{"className":10866,"style":10117},[10116],[805,10868,9911],{"className":10869},[7512],[805,10871,7557],{"className":10872},[7556],[805,10874,10876],{"className":10875},[7529],[805,10877,10879],{"className":10878,"style":10130},[7533],[805,10880],{},[805,10882,10884,10887],{"className":10883},[7621],[805,10885,3993],{"className":10886},[7621],[805,10888,10890],{"className":10889},[7520],[805,10891,10893],{"className":10892},[7524],[805,10894,10896],{"className":10895},[7529],[805,10897,10900],{"className":10898,"style":10899},[7533],"height:0.8641em;",[805,10901,10903,10906],{"style":10902},"top:-3.113em;margin-right:0.05em;",[805,10904],{"className":10905,"style":7542},[7541],[805,10907,10909],{"className":10908},[7546,7547,7548,7549],[805,10910,7442],{"className":10911},[7512,7549],[11,10913,8849],{},[49,10915,10916,10947,11048],{},[52,10917,10918,10946],{},[805,10919,10921,10934],{"className":10920},[7389],[805,10922,10924],{"className":10923},[7393],[7395,10925,10926],{"xmlns":7397},[7399,10927,10928,10932],{},[7402,10929,10930],{},[7414,10931,7472],{},[7487,10933,7472],{"encoding":7489},[805,10935,10937],{"className":10936,"ariaHidden":7423},[7494],[805,10938,10940,10943],{"className":10939},[7498],[805,10941],{"className":10942,"style":8843},[7502],[805,10944,7472],{"className":10945},[7512,7516]," is the number of examples in the dataset.",[52,10948,10949,11018,11019,44],{},[805,10950,10952,10969],{"className":10951},[7389],[805,10953,10955],{"className":10954},[7393],[7395,10956,10957],{"xmlns":7397},[7399,10958,10959,10967],{},[7402,10960,10961],{},[7411,10962,10963,10965],{},[7414,10964,828],{},[7414,10966,7849],{},[7487,10968,7925],{"encoding":7489},[805,10970,10972],{"className":10971,"ariaHidden":7423},[7494],[805,10973,10975,10978],{"className":10974},[7498],[805,10976],{"className":10977,"style":7935},[7502],[805,10979,10981,10984],{"className":10980},[7512],[805,10982,828],{"className":10983,"style":7582},[7512,7516],[805,10985,10987],{"className":10986},[7520],[805,10988,10990,11010],{"className":10989},[7524,7525],[805,10991,10993,11007],{"className":10992},[7529],[805,10994,10996],{"className":10995,"style":7881},[7533],[805,10997,10998,11001],{"style":7597},[805,10999],{"className":11000,"style":7542},[7541],[805,11002,11004],{"className":11003},[7546,7547,7548,7549],[805,11005,7849],{"className":11006},[7512,7516,7549],[805,11008,7557],{"className":11009},[7556],[805,11011,11013],{"className":11012},[7529],[805,11014,11016],{"className":11015,"style":7564},[7533],[805,11017],{}," is the actual value of the dependent variable for example ",[805,11020,11022,11035],{"className":11021},[7389],[805,11023,11025],{"className":11024},[7393],[7395,11026,11027],{"xmlns":7397},[7399,11028,11029,11033],{},[7402,11030,11031],{},[7414,11032,7849],{},[7487,11034,7849],{"encoding":7489},[805,11036,11038],{"className":11037,"ariaHidden":7423},[7494],[805,11039,11041,11045],{"className":11040},[7498],[805,11042],{"className":11043,"style":11044},[7502],"height:0.6595em;",[805,11046,7849],{"className":11047},[7512,7516],[52,11049,11050,11164,11165,44],{},[805,11051,11053,11075],{"className":11052},[7389],[805,11054,11056],{"className":11055},[7393],[7395,11057,11058],{"xmlns":7397},[7399,11059,11060,11072],{},[7402,11061,11062],{},[7411,11063,11064,11070],{},[9901,11065,11066,11068],{"accent":7423},[7414,11067,828],{},[7405,11069,9911],{},[7414,11071,7849],{},[7487,11073,11074],{"encoding":7489},"\\hat{y}_i",[805,11076,11078],{"className":11077,"ariaHidden":7423},[7494],[805,11079,11081,11084],{"className":11080},[7498],[805,11082],{"className":11083,"style":8020},[7502],[805,11085,11087,11130],{"className":11086},[7512],[805,11088,11090],{"className":11089},[7512,10049],[805,11091,11093,11122],{"className":11092},[7524,7525],[805,11094,11096,11119],{"className":11095},[7529],[805,11097,11099,11107],{"className":11098,"style":10813},[7533],[805,11100,11101,11104],{"style":10062},[805,11102],{"className":11103,"style":10066},[7541],[805,11105,828],{"className":11106,"style":7582},[7512,7516],[805,11108,11109,11112],{"style":10062},[805,11110],{"className":11111,"style":10066},[7541],[805,11113,11116],{"className":11114,"style":11115},[10116],"left:-0.1944em;",[805,11117,9911],{"className":11118},[7512],[805,11120,7557],{"className":11121},[7556],[805,11123,11125],{"className":11124},[7529],[805,11126,11128],{"className":11127,"style":10130},[7533],[805,11129],{},[805,11131,11133],{"className":11132},[7520],[805,11134,11136,11156],{"className":11135},[7524,7525],[805,11137,11139,11153],{"className":11138},[7529],[805,11140,11142],{"className":11141,"style":7881},[7533],[805,11143,11144,11147],{"style":7597},[805,11145],{"className":11146,"style":7542},[7541],[805,11148,11150],{"className":11149},[7546,7547,7548,7549],[805,11151,7849],{"className":11152},[7512,7516,7549],[805,11154,7557],{"className":11155},[7556],[805,11157,11159],{"className":11158},[7529],[805,11160,11162],{"className":11161,"style":7564},[7533],[805,11163],{}," is the model's prediction for example ",[805,11166,11168,11181],{"className":11167},[7389],[805,11169,11171],{"className":11170},[7393],[7395,11172,11173],{"xmlns":7397},[7399,11174,11175,11179],{},[7402,11176,11177],{},[7414,11178,7849],{},[7487,11180,7849],{"encoding":7489},[805,11182,11184],{"className":11183,"ariaHidden":7423},[7494],[805,11185,11187,11190],{"className":11186},[7498],[805,11188],{"className":11189,"style":11044},[7502],[805,11191,7849],{"className":11192},[7512,7516],[11,11194,11195],{},"Minimizing the MSE is equivalent to minimizing the sum of squared errors:",[805,11197,11199],{"className":11198},[7980],[805,11200,11202,11254],{"className":11201},[7389],[805,11203,11205],{"className":11204},[7393],[7395,11206,11207],{"xmlns":7397,"display":7989},[7399,11208,11209,11251],{},[7402,11210,11211,11225,11227,11233,11235,11245],{},[10511,11212,11213,11215,11223],{},[7405,11214,10515],{},[7402,11216,11217,11219,11221],{},[7414,11218,7849],{},[7405,11220,2823],{},[7418,11222,7420],{},[7414,11224,7472],{},[7405,11226,2651],{"stretchy":1435},[7411,11228,11229,11231],{},[7414,11230,828],{},[7414,11232,7849],{},[7405,11234,9899],{},[9901,11236,11237,11243],{"accent":7423},[7411,11238,11239,11241],{},[7414,11240,828],{},[7414,11242,7849],{},[7405,11244,9911],{},[10547,11246,11247,11249],{},[7405,11248,3993],{"stretchy":1435},[7418,11250,7442],{},[7487,11252,11253],{"encoding":7489},"\\sum_{i=1}^{n} (y_i - \\hat{y_i})^2",[805,11255,11257,11382],{"className":11256,"ariaHidden":7423},[7494],[805,11258,11260,11263,11330,11333,11373,11376,11379],{"className":11259},[7498],[805,11261],{"className":11262,"style":10591},[7502],[805,11264,11266],{"className":11265},[10668,10669],[805,11267,11269,11322],{"className":11268},[7524,7525],[805,11270,11272,11319],{"className":11271},[7529],[805,11273,11275,11295,11305],{"className":11274,"style":10679},[7533],[805,11276,11277,11280],{"style":10682},[805,11278],{"className":11279,"style":10686},[7541],[805,11281,11283],{"className":11282},[7546,7547,7548,7549],[805,11284,11286,11289,11292],{"className":11285},[7512,7549],[805,11287,7849],{"className":11288},[7512,7516,7549],[805,11290,2823],{"className":11291},[8032,7549],[805,11293,7420],{"className":11294},[7512,7549],[805,11296,11297,11300],{"style":10704},[805,11298],{"className":11299,"style":10686},[7541],[805,11301,11302],{},[805,11303,10515],{"className":11304},[10668,10713,10714],[805,11306,11307,11310],{"style":10717},[805,11308],{"className":11309,"style":10686},[7541],[805,11311,11313],{"className":11312},[7546,7547,7548,7549],[805,11314,11316],{"className":11315},[7512,7549],[805,11317,7472],{"className":11318},[7512,7516,7549],[805,11320,7557],{"className":11321},[7556],[805,11323,11325],{"className":11324},[7529],[805,11326,11328],{"className":11327,"style":10739},[7533],[805,11329],{},[805,11331,2651],{"className":11332},[7507],[805,11334,11336,11339],{"className":11335},[7512],[805,11337,828],{"className":11338,"style":7582},[7512,7516],[805,11340,11342],{"className":11341},[7520],[805,11343,11345,11365],{"className":11344},[7524,7525],[805,11346,11348,11362],{"className":11347},[7529],[805,11349,11351],{"className":11350,"style":7881},[7533],[805,11352,11353,11356],{"style":7597},[805,11354],{"className":11355,"style":7542},[7541],[805,11357,11359],{"className":11358},[7546,7547,7548,7549],[805,11360,7849],{"className":11361},[7512,7516,7549],[805,11363,7557],{"className":11364},[7556],[805,11366,11368],{"className":11367},[7529],[805,11369,11371],{"className":11370,"style":7564},[7533],[805,11372],{},[805,11374],{"className":11375,"style":8065},[7574],[805,11377,9899],{"className":11378},[8527],[805,11380],{"className":11381,"style":8065},[7574],[805,11383,11385,11388,11467],{"className":11384},[7498],[805,11386],{"className":11387,"style":10800},[7502],[805,11389,11391],{"className":11390},[7512,10049],[805,11392,11394,11459],{"className":11393},[7524,7525],[805,11395,11397,11456],{"className":11396},[7529],[805,11398,11400,11445],{"className":11399,"style":10813},[7533],[805,11401,11402,11405],{"style":10062},[805,11403],{"className":11404,"style":10066},[7541],[805,11406,11408,11411],{"className":11407},[7512],[805,11409,828],{"className":11410,"style":7582},[7512,7516],[805,11412,11414],{"className":11413},[7520],[805,11415,11417,11437],{"className":11416},[7524,7525],[805,11418,11420,11434],{"className":11419},[7529],[805,11421,11423],{"className":11422,"style":7881},[7533],[805,11424,11425,11428],{"style":7597},[805,11426],{"className":11427,"style":7542},[7541],[805,11429,11431],{"className":11430},[7546,7547,7548,7549],[805,11432,7849],{"className":11433},[7512,7516,7549],[805,11435,7557],{"className":11436},[7556],[805,11438,11440],{"className":11439},[7529],[805,11441,11443],{"className":11442,"style":7564},[7533],[805,11444],{},[805,11446,11447,11450],{"style":10062},[805,11448],{"className":11449,"style":10066},[7541],[805,11451,11453],{"className":11452,"style":10117},[10116],[805,11454,9911],{"className":11455},[7512],[805,11457,7557],{"className":11458},[7556],[805,11460,11462],{"className":11461},[7529],[805,11463,11465],{"className":11464,"style":10130},[7533],[805,11466],{},[805,11468,11470,11473],{"className":11469},[7621],[805,11471,3993],{"className":11472},[7621],[805,11474,11476],{"className":11475},[7520],[805,11477,11479],{"className":11478},[7524],[805,11480,11482],{"className":11481},[7529],[805,11483,11485],{"className":11484,"style":10899},[7533],[805,11486,11487,11490],{"style":10902},[805,11488],{"className":11489,"style":7542},[7541],[805,11491,11493],{"className":11492},[7546,7547,7548,7549],[805,11494,7442],{"className":11495},[7512,7549],[11,11497,11498],{},"Or also:",[805,11500,11502],{"className":11501},[7980],[805,11503,11505,11571],{"className":11504},[7389],[805,11506,11508],{"className":11507},[7393],[7395,11509,11510],{"xmlns":7397,"display":7989},[7399,11511,11512,11568],{},[7402,11513,11514,11528,11530,11536,11538,11540,11546,11548,11554,11560,11562],{},[10511,11515,11516,11518,11526],{},[7405,11517,10515],{},[7402,11519,11520,11522,11524],{},[7414,11521,7849],{},[7405,11523,2823],{},[7418,11525,7420],{},[7414,11527,7472],{},[7405,11529,2651],{"stretchy":1435},[7411,11531,11532,11534],{},[7414,11533,828],{},[7414,11535,7849],{},[7405,11537,9899],{},[7405,11539,2651],{"stretchy":1435},[7411,11541,11542,11544],{},[7414,11543,8389],{},[7418,11545,8392],{},[7405,11547,8395],{},[7411,11549,11550,11552],{},[7414,11551,8389],{},[7418,11553,7420],{},[7411,11555,11556,11558],{},[7414,11557,7416],{},[7414,11559,7849],{},[7405,11561,3993],{"stretchy":1435},[10547,11563,11564,11566],{},[7405,11565,3993],{"stretchy":1435},[7418,11567,7442],{},[7487,11569,11570],{"encoding":7489},"\\sum_{i=1}^{n} (y_i - (\\beta_0 + \\beta_1 x_i))^2",[805,11572,11574,11699,11757],{"className":11573,"ariaHidden":7423},[7494],[805,11575,11577,11580,11647,11650,11690,11693,11696],{"className":11576},[7498],[805,11578],{"className":11579,"style":10591},[7502],[805,11581,11583],{"className":11582},[10668,10669],[805,11584,11586,11639],{"className":11585},[7524,7525],[805,11587,11589,11636],{"className":11588},[7529],[805,11590,11592,11612,11622],{"className":11591,"style":10679},[7533],[805,11593,11594,11597],{"style":10682},[805,11595],{"className":11596,"style":10686},[7541],[805,11598,11600],{"className":11599},[7546,7547,7548,7549],[805,11601,11603,11606,11609],{"className":11602},[7512,7549],[805,11604,7849],{"className":11605},[7512,7516,7549],[805,11607,2823],{"className":11608},[8032,7549],[805,11610,7420],{"className":11611},[7512,7549],[805,11613,11614,11617],{"style":10704},[805,11615],{"className":11616,"style":10686},[7541],[805,11618,11619],{},[805,11620,10515],{"className":11621},[10668,10713,10714],[805,11623,11624,11627],{"style":10717},[805,11625],{"className":11626,"style":10686},[7541],[805,11628,11630],{"className":11629},[7546,7547,7548,7549],[805,11631,11633],{"className":11632},[7512,7549],[805,11634,7472],{"className":11635},[7512,7516,7549],[805,11637,7557],{"className":11638},[7556],[805,11640,11642],{"className":11641},[7529],[805,11643,11645],{"className":11644,"style":10739},[7533],[805,11646],{},[805,11648,2651],{"className":11649},[7507],[805,11651,11653,11656],{"className":11652},[7512],[805,11654,828],{"className":11655,"style":7582},[7512,7516],[805,11657,11659],{"className":11658},[7520],[805,11660,11662,11682],{"className":11661},[7524,7525],[805,11663,11665,11679],{"className":11664},[7529],[805,11666,11668],{"className":11667,"style":7881},[7533],[805,11669,11670,11673],{"style":7597},[805,11671],{"className":11672,"style":7542},[7541],[805,11674,11676],{"className":11675},[7546,7547,7548,7549],[805,11677,7849],{"className":11678},[7512,7516,7549],[805,11680,7557],{"className":11681},[7556],[805,11683,11685],{"className":11684},[7529],[805,11686,11688],{"className":11687,"style":7564},[7533],[805,11689],{},[805,11691],{"className":11692,"style":8065},[7574],[805,11694,9899],{"className":11695},[8527],[805,11697],{"className":11698,"style":8065},[7574],[805,11700,11702,11705,11708,11748,11751,11754],{"className":11701},[7498],[805,11703],{"className":11704,"style":7503},[7502],[805,11706,2651],{"className":11707},[7507],[805,11709,11711,11714],{"className":11710},[7512],[805,11712,8389],{"className":11713,"style":8485},[7512,7516],[805,11715,11717],{"className":11716},[7520],[805,11718,11720,11740],{"className":11719},[7524,7525],[805,11721,11723,11737],{"className":11722},[7529],[805,11724,11726],{"className":11725,"style":7534},[7533],[805,11727,11728,11731],{"style":8500},[805,11729],{"className":11730,"style":7542},[7541],[805,11732,11734],{"className":11733},[7546,7547,7548,7549],[805,11735,8392],{"className":11736},[7512,7549],[805,11738,7557],{"className":11739},[7556],[805,11741,11743],{"className":11742},[7529],[805,11744,11746],{"className":11745,"style":7564},[7533],[805,11747],{},[805,11749],{"className":11750,"style":8065},[7574],[805,11752,8395],{"className":11753},[8527],[805,11755],{"className":11756,"style":8065},[7574],[805,11758,11760,11763,11803,11843,11846],{"className":11759},[7498],[805,11761],{"className":11762,"style":10800},[7502],[805,11764,11766,11769],{"className":11765},[7512],[805,11767,8389],{"className":11768,"style":8485},[7512,7516],[805,11770,11772],{"className":11771},[7520],[805,11773,11775,11795],{"className":11774},[7524,7525],[805,11776,11778,11792],{"className":11777},[7529],[805,11779,11781],{"className":11780,"style":7534},[7533],[805,11782,11783,11786],{"style":8500},[805,11784],{"className":11785,"style":7542},[7541],[805,11787,11789],{"className":11788},[7546,7547,7548,7549],[805,11790,7420],{"className":11791},[7512,7549],[805,11793,7557],{"className":11794},[7556],[805,11796,11798],{"className":11797},[7529],[805,11799,11801],{"className":11800,"style":7564},[7533],[805,11802],{},[805,11804,11806,11809],{"className":11805},[7512],[805,11807,7416],{"className":11808},[7512,7516],[805,11810,11812],{"className":11811},[7520],[805,11813,11815,11835],{"className":11814},[7524,7525],[805,11816,11818,11832],{"className":11817},[7529],[805,11819,11821],{"className":11820,"style":7881},[7533],[805,11822,11823,11826],{"style":7537},[805,11824],{"className":11825,"style":7542},[7541],[805,11827,11829],{"className":11828},[7546,7547,7548,7549],[805,11830,7849],{"className":11831},[7512,7516,7549],[805,11833,7557],{"className":11834},[7556],[805,11836,11838],{"className":11837},[7529],[805,11839,11841],{"className":11840,"style":7564},[7533],[805,11842],{},[805,11844,3993],{"className":11845},[7621],[805,11847,11849,11852],{"className":11848},[7621],[805,11850,3993],{"className":11851},[7621],[805,11853,11855],{"className":11854},[7520],[805,11856,11858],{"className":11857},[7524],[805,11859,11861],{"className":11860},[7529],[805,11862,11864],{"className":11863,"style":10899},[7533],[805,11865,11866,11869],{"style":10902},[805,11867],{"className":11868,"style":7542},[7541],[805,11870,11872],{"className":11871},[7546,7547,7548,7549],[805,11873,7442],{"className":11874},[7512,7549],[11,11876,11877,11878,11881,11882,11916],{},"This is known as the ",[29,11879,11880],{},"least squares method",", and it is a commonly used technique to find the coefficients ",[805,11883,11885,11898],{"className":11884},[7389],[805,11886,11888],{"className":11887},[7393],[7395,11889,11890],{"xmlns":7397},[7399,11891,11892,11896],{},[7402,11893,11894],{},[7414,11895,8389],{"mathvariant":9554},[7487,11897,9557],{"encoding":7489},[805,11899,11901],{"className":11900,"ariaHidden":7423},[7494],[805,11902,11904,11907],{"className":11903},[7498],[805,11905],{"className":11906,"style":8020},[7502],[805,11908,11910],{"className":11909},[7512],[805,11911,11913],{"className":11912},[7512],[805,11914,8389],{"className":11915,"style":9577},[7512,9576]," that best fit the data.",[78,11918,11919],{},[11,11920,11921],{},"Why are squares used? Because by squaring the differences, larger errors are penalized more, which helps to find a better fitting line for the data.",[11,11923,11924,11925,11959,11960,36,12029,12099],{},"Let's now see how we obtain those coefficients ",[805,11926,11928,11941],{"className":11927},[7389],[805,11929,11931],{"className":11930},[7393],[7395,11932,11933],{"xmlns":7397},[7399,11934,11935,11939],{},[7402,11936,11937],{},[7414,11938,8389],{"mathvariant":9554},[7487,11940,9557],{"encoding":7489},[805,11942,11944],{"className":11943,"ariaHidden":7423},[7494],[805,11945,11947,11950],{"className":11946},[7498],[805,11948],{"className":11949,"style":8020},[7502],[805,11951,11953],{"className":11952},[7512],[805,11954,11956],{"className":11955},[7512],[805,11957,8389],{"className":11958,"style":9577},[7512,9576]," using the least squares method. To find the values of ",[805,11961,11963,11980],{"className":11962},[7389],[805,11964,11966],{"className":11965},[7393],[7395,11967,11968],{"xmlns":7397},[7399,11969,11970,11978],{},[7402,11971,11972],{},[7411,11973,11974,11976],{},[7414,11975,8389],{},[7418,11977,8392],{},[7487,11979,9104],{"encoding":7489},[805,11981,11983],{"className":11982,"ariaHidden":7423},[7494],[805,11984,11986,11989],{"className":11985},[7498],[805,11987],{"className":11988,"style":8020},[7502],[805,11990,11992,11995],{"className":11991},[7512],[805,11993,8389],{"className":11994,"style":8485},[7512,7516],[805,11996,11998],{"className":11997},[7520],[805,11999,12001,12021],{"className":12000},[7524,7525],[805,12002,12004,12018],{"className":12003},[7529],[805,12005,12007],{"className":12006,"style":7534},[7533],[805,12008,12009,12012],{"style":8500},[805,12010],{"className":12011,"style":7542},[7541],[805,12013,12015],{"className":12014},[7546,7547,7548,7549],[805,12016,8392],{"className":12017},[7512,7549],[805,12019,7557],{"className":12020},[7556],[805,12022,12024],{"className":12023},[7529],[805,12025,12027],{"className":12026,"style":7564},[7533],[805,12028],{},[805,12030,12032,12050],{"className":12031},[7389],[805,12033,12035],{"className":12034},[7393],[7395,12036,12037],{"xmlns":7397},[7399,12038,12039,12047],{},[7402,12040,12041],{},[7411,12042,12043,12045],{},[7414,12044,8389],{},[7418,12046,7420],{},[7487,12048,12049],{"encoding":7489},"\\beta_1",[805,12051,12053],{"className":12052,"ariaHidden":7423},[7494],[805,12054,12056,12059],{"className":12055},[7498],[805,12057],{"className":12058,"style":8020},[7502],[805,12060,12062,12065],{"className":12061},[7512],[805,12063,8389],{"className":12064,"style":8485},[7512,7516],[805,12066,12068],{"className":12067},[7520],[805,12069,12071,12091],{"className":12070},[7524,7525],[805,12072,12074,12088],{"className":12073},[7529],[805,12075,12077],{"className":12076,"style":7534},[7533],[805,12078,12079,12082],{"style":8500},[805,12080],{"className":12081,"style":7542},[7541],[805,12083,12085],{"className":12084},[7546,7547,7548,7549],[805,12086,7420],{"className":12087},[7512,7549],[805,12089,7557],{"className":12090},[7556],[805,12092,12094],{"className":12093},[7529],[805,12095,12097],{"className":12096,"style":7564},[7533],[805,12098],{},", we can use the following formulas, which are obtained by taking the derivative of the MSE loss function with respect to the coefficients and setting the derivatives equal to zero:",[805,12101,12103],{"className":12102},[7980],[805,12104,12106,12205],{"className":12105},[7389],[805,12107,12109],{"className":12108},[7393],[7395,12110,12111],{"xmlns":7397,"display":7989},[7399,12112,12113,12202],{},[7402,12114,12115,12121,12123],{},[7411,12116,12117,12119],{},[7414,12118,8389],{},[7418,12120,7420],{},[7405,12122,2823],{},[10504,12124,12125,12165],{},[7402,12126,12127,12129,12131,12133,12139,12145,12147,12149,12151,12157,12159],{},[7414,12128,7472],{},[7405,12130,10515],{},[7405,12132,2651],{"stretchy":1435},[7411,12134,12135,12137],{},[7414,12136,7416],{},[7414,12138,7849],{},[7411,12140,12141,12143],{},[7414,12142,828],{},[7414,12144,7849],{},[7405,12146,3993],{"stretchy":1435},[7405,12148,9899],{},[7405,12150,10515],{},[7411,12152,12153,12155],{},[7414,12154,7416],{},[7414,12156,7849],{},[7405,12158,10515],{},[7411,12160,12161,12163],{},[7414,12162,828],{},[7414,12164,7849],{},[7402,12166,12167,12169,12171,12173,12182,12184,12186,12188,12190,12196],{},[7414,12168,7472],{},[7405,12170,10515],{},[7405,12172,2651],{"stretchy":1435},[12174,12175,12176,12178,12180],"msubsup",{},[7414,12177,7416],{},[7414,12179,7849],{},[7418,12181,7442],{},[7405,12183,3993],{"stretchy":1435},[7405,12185,9899],{},[7405,12187,2651],{"stretchy":1435},[7405,12189,10515],{},[7411,12191,12192,12194],{},[7414,12193,7416],{},[7414,12195,7849],{},[10547,12197,12198,12200],{},[7405,12199,3993],{"stretchy":1435},[7418,12201,7442],{},[7487,12203,12204],{"encoding":7489},"\\beta_1 = \\frac{n \\sum (x_i y_i) - \\sum x_i \\sum y_i}{n \\sum (x_i^2) - (\\sum x_i)^2}",[805,12206,12208,12263],{"className":12207,"ariaHidden":7423},[7494],[805,12209,12211,12214,12254,12257,12260],{"className":12210},[7498],[805,12212],{"className":12213,"style":8020},[7502],[805,12215,12217,12220],{"className":12216},[7512],[805,12218,8389],{"className":12219,"style":8485},[7512,7516],[805,12221,12223],{"className":12222},[7520],[805,12224,12226,12246],{"className":12225},[7524,7525],[805,12227,12229,12243],{"className":12228},[7529],[805,12230,12232],{"className":12231,"style":7534},[7533],[805,12233,12234,12237],{"style":8500},[805,12235],{"className":12236,"style":7542},[7541],[805,12238,12240],{"className":12239},[7546,7547,7548,7549],[805,12241,7420],{"className":12242},[7512,7549],[805,12244,7557],{"className":12245},[7556],[805,12247,12249],{"className":12248},[7529],[805,12250,12252],{"className":12251,"style":7564},[7533],[805,12253],{},[805,12255],{"className":12256,"style":8028},[7574],[805,12258,2823],{"className":12259},[8032],[805,12261],{"className":12262,"style":8028},[7574],[805,12264,12266,12270],{"className":12265},[7498],[805,12267],{"className":12268,"style":12269},[7502],"height:2.3899em;vertical-align:-0.9629em;",[805,12271,12273,12276,12685],{"className":12272},[7512],[805,12274],{"className":12275},[7507,10598],[805,12277,12279],{"className":12278},[10504],[805,12280,12282,12676],{"className":12281},[7524,7525],[805,12283,12285,12673],{"className":12284},[7529],[805,12286,12289,12458,12466],{"className":12287,"style":12288},[7533],"height:1.427em;",[805,12290,12291,12294],{"style":10614},[805,12292],{"className":12293,"style":10066},[7541],[805,12295,12297,12300,12303,12308,12311,12366,12369,12372,12375,12378,12381,12384,12387,12427],{"className":12296},[7512],[805,12298,7472],{"className":12299},[7512,7516],[805,12301],{"className":12302,"style":7575},[7574],[805,12304,10515],{"className":12305,"style":12307},[10668,10713,12306],"small-op","position:relative;top:0em;",[805,12309,2651],{"className":12310},[7507],[805,12312,12314,12317],{"className":12313},[7512],[805,12315,7416],{"className":12316},[7512,7516],[805,12318,12320],{"className":12319},[7520],[805,12321,12323,12357],{"className":12322},[7524,7525],[805,12324,12326,12354],{"className":12325},[7529],[805,12327,12330,12342],{"className":12328,"style":12329},[7533],"height:0.7959em;",[805,12331,12333,12336],{"style":12332},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[805,12334],{"className":12335,"style":7542},[7541],[805,12337,12339],{"className":12338},[7546,7547,7548,7549],[805,12340,7849],{"className":12341},[7512,7516,7549],[805,12343,12345,12348],{"style":12344},"top:-3.0448em;margin-right:0.05em;",[805,12346],{"className":12347,"style":7542},[7541],[805,12349,12351],{"className":12350},[7546,7547,7548,7549],[805,12352,7442],{"className":12353},[7512,7549],[805,12355,7557],{"className":12356},[7556],[805,12358,12360],{"className":12359},[7529],[805,12361,12364],{"className":12362,"style":12363},[7533],"height:0.2769em;",[805,12365],{},[805,12367,3993],{"className":12368},[7621],[805,12370],{"className":12371,"style":8065},[7574],[805,12373,9899],{"className":12374},[8527],[805,12376],{"className":12377,"style":8065},[7574],[805,12379,2651],{"className":12380},[7507],[805,12382,10515],{"className":12383,"style":12307},[10668,10713,12306],[805,12385],{"className":12386,"style":7575},[7574],[805,12388,12390,12393],{"className":12389},[7512],[805,12391,7416],{"className":12392},[7512,7516],[805,12394,12396],{"className":12395},[7520],[805,12397,12399,12419],{"className":12398},[7524,7525],[805,12400,12402,12416],{"className":12401},[7529],[805,12403,12405],{"className":12404,"style":7881},[7533],[805,12406,12407,12410],{"style":7537},[805,12408],{"className":12409,"style":7542},[7541],[805,12411,12413],{"className":12412},[7546,7547,7548,7549],[805,12414,7849],{"className":12415},[7512,7516,7549],[805,12417,7557],{"className":12418},[7556],[805,12420,12422],{"className":12421},[7529],[805,12423,12425],{"className":12424,"style":7564},[7533],[805,12426],{},[805,12428,12430,12433],{"className":12429},[7621],[805,12431,3993],{"className":12432},[7621],[805,12434,12436],{"className":12435},[7520],[805,12437,12439],{"className":12438},[7524],[805,12440,12442],{"className":12441},[7529],[805,12443,12446],{"className":12444,"style":12445},[7533],"height:0.7401em;",[805,12447,12449,12452],{"style":12448},"top:-2.989em;margin-right:0.05em;",[805,12450],{"className":12451,"style":7542},[7541],[805,12453,12455],{"className":12454},[7546,7547,7548,7549],[805,12456,7442],{"className":12457},[7512,7549],[805,12459,12460,12463],{"style":10626},[805,12461],{"className":12462,"style":10066},[7541],[805,12464],{"className":12465,"style":10634},[10633],[805,12467,12468,12471],{"style":10637},[805,12469],{"className":12470,"style":10066},[7541],[805,12472,12474,12477,12480,12483,12486,12526,12566,12569,12572,12575,12578,12581,12584,12624,12627,12630,12633],{"className":12473},[7512],[805,12475,7472],{"className":12476},[7512,7516],[805,12478],{"className":12479,"style":7575},[7574],[805,12481,10515],{"className":12482,"style":12307},[10668,10713,12306],[805,12484,2651],{"className":12485},[7507],[805,12487,12489,12492],{"className":12488},[7512],[805,12490,7416],{"className":12491},[7512,7516],[805,12493,12495],{"className":12494},[7520],[805,12496,12498,12518],{"className":12497},[7524,7525],[805,12499,12501,12515],{"className":12500},[7529],[805,12502,12504],{"className":12503,"style":7881},[7533],[805,12505,12506,12509],{"style":7537},[805,12507],{"className":12508,"style":7542},[7541],[805,12510,12512],{"className":12511},[7546,7547,7548,7549],[805,12513,7849],{"className":12514},[7512,7516,7549],[805,12516,7557],{"className":12517},[7556],[805,12519,12521],{"className":12520},[7529],[805,12522,12524],{"className":12523,"style":7564},[7533],[805,12525],{},[805,12527,12529,12532],{"className":12528},[7512],[805,12530,828],{"className":12531,"style":7582},[7512,7516],[805,12533,12535],{"className":12534},[7520],[805,12536,12538,12558],{"className":12537},[7524,7525],[805,12539,12541,12555],{"className":12540},[7529],[805,12542,12544],{"className":12543,"style":7881},[7533],[805,12545,12546,12549],{"style":7597},[805,12547],{"className":12548,"style":7542},[7541],[805,12550,12552],{"className":12551},[7546,7547,7548,7549],[805,12553,7849],{"className":12554},[7512,7516,7549],[805,12556,7557],{"className":12557},[7556],[805,12559,12561],{"className":12560},[7529],[805,12562,12564],{"className":12563,"style":7564},[7533],[805,12565],{},[805,12567,3993],{"className":12568},[7621],[805,12570],{"className":12571,"style":8065},[7574],[805,12573,9899],{"className":12574},[8527],[805,12576],{"className":12577,"style":8065},[7574],[805,12579,10515],{"className":12580,"style":12307},[10668,10713,12306],[805,12582],{"className":12583,"style":7575},[7574],[805,12585,12587,12590],{"className":12586},[7512],[805,12588,7416],{"className":12589},[7512,7516],[805,12591,12593],{"className":12592},[7520],[805,12594,12596,12616],{"className":12595},[7524,7525],[805,12597,12599,12613],{"className":12598},[7529],[805,12600,12602],{"className":12601,"style":7881},[7533],[805,12603,12604,12607],{"style":7537},[805,12605],{"className":12606,"style":7542},[7541],[805,12608,12610],{"className":12609},[7546,7547,7548,7549],[805,12611,7849],{"className":12612},[7512,7516,7549],[805,12614,7557],{"className":12615},[7556],[805,12617,12619],{"className":12618},[7529],[805,12620,12622],{"className":12621,"style":7564},[7533],[805,12623],{},[805,12625],{"className":12626,"style":7575},[7574],[805,12628,10515],{"className":12629,"style":12307},[10668,10713,12306],[805,12631],{"className":12632,"style":7575},[7574],[805,12634,12636,12639],{"className":12635},[7512],[805,12637,828],{"className":12638,"style":7582},[7512,7516],[805,12640,12642],{"className":12641},[7520],[805,12643,12645,12665],{"className":12644},[7524,7525],[805,12646,12648,12662],{"className":12647},[7529],[805,12649,12651],{"className":12650,"style":7881},[7533],[805,12652,12653,12656],{"style":7597},[805,12654],{"className":12655,"style":7542},[7541],[805,12657,12659],{"className":12658},[7546,7547,7548,7549],[805,12660,7849],{"className":12661},[7512,7516,7549],[805,12663,7557],{"className":12664},[7556],[805,12666,12668],{"className":12667},[7529],[805,12669,12671],{"className":12670,"style":7564},[7533],[805,12672],{},[805,12674,7557],{"className":12675},[7556],[805,12677,12679],{"className":12678},[7529],[805,12680,12683],{"className":12681,"style":12682},[7533],"height:0.9629em;",[805,12684],{},[805,12686],{"className":12687},[7621,10598],[805,12689,12691],{"className":12690},[7980],[805,12692,12694,12735],{"className":12693},[7389],[805,12695,12697],{"className":12696},[7393],[7395,12698,12699],{"xmlns":7397,"display":7989},[7399,12700,12701,12732],{},[7402,12702,12703,12709,12711,12718,12720,12726],{},[7411,12704,12705,12707],{},[7414,12706,8389],{},[7418,12708,8392],{},[7405,12710,2823],{},[9901,12712,12713,12715],{"accent":7423},[7414,12714,828],{},[7405,12716,12717],{},"ˉ",[7405,12719,9899],{},[7411,12721,12722,12724],{},[7414,12723,8389],{},[7418,12725,7420],{},[9901,12727,12728,12730],{"accent":7423},[7414,12729,7416],{},[7405,12731,12717],{},[7487,12733,12734],{"encoding":7489},"\\beta_0 = \\bar{y} - \\beta_1 \\bar{x}",[805,12736,12738,12793,12851],{"className":12737,"ariaHidden":7423},[7494],[805,12739,12741,12744,12784,12787,12790],{"className":12740},[7498],[805,12742],{"className":12743,"style":8020},[7502],[805,12745,12747,12750],{"className":12746},[7512],[805,12748,8389],{"className":12749,"style":8485},[7512,7516],[805,12751,12753],{"className":12752},[7520],[805,12754,12756,12776],{"className":12755},[7524,7525],[805,12757,12759,12773],{"className":12758},[7529],[805,12760,12762],{"className":12761,"style":7534},[7533],[805,12763,12764,12767],{"style":8500},[805,12765],{"className":12766,"style":7542},[7541],[805,12768,12770],{"className":12769},[7546,7547,7548,7549],[805,12771,8392],{"className":12772},[7512,7549],[805,12774,7557],{"className":12775},[7556],[805,12777,12779],{"className":12778},[7529],[805,12780,12782],{"className":12781,"style":7564},[7533],[805,12783],{},[805,12785],{"className":12786,"style":8028},[7574],[805,12788,2823],{"className":12789},[8032],[805,12791],{"className":12792,"style":8028},[7574],[805,12794,12796,12799,12842,12845,12848],{"className":12795},[7498],[805,12797],{"className":12798,"style":9987},[7502],[805,12800,12802],{"className":12801},[7512,10049],[805,12803,12805,12834],{"className":12804},[7524,7525],[805,12806,12808,12831],{"className":12807},[7529],[805,12809,12812,12820],{"className":12810,"style":12811},[7533],"height:0.5678em;",[805,12813,12814,12817],{"style":10062},[805,12815],{"className":12816,"style":10066},[7541],[805,12818,828],{"className":12819,"style":7582},[7512,7516],[805,12821,12822,12825],{"style":10062},[805,12823],{"className":12824,"style":10066},[7541],[805,12826,12828],{"className":12827,"style":11115},[10116],[805,12829,12717],{"className":12830},[7512],[805,12832,7557],{"className":12833},[7556],[805,12835,12837],{"className":12836},[7529],[805,12838,12840],{"className":12839,"style":10130},[7533],[805,12841],{},[805,12843],{"className":12844,"style":8065},[7574],[805,12846,9899],{"className":12847},[8527],[805,12849],{"className":12850,"style":8065},[7574],[805,12852,12854,12857,12897],{"className":12853},[7498],[805,12855],{"className":12856,"style":8020},[7502],[805,12858,12860,12863],{"className":12859},[7512],[805,12861,8389],{"className":12862,"style":8485},[7512,7516],[805,12864,12866],{"className":12865},[7520],[805,12867,12869,12889],{"className":12868},[7524,7525],[805,12870,12872,12886],{"className":12871},[7529],[805,12873,12875],{"className":12874,"style":7534},[7533],[805,12876,12877,12880],{"style":8500},[805,12878],{"className":12879,"style":7542},[7541],[805,12881,12883],{"className":12882},[7546,7547,7548,7549],[805,12884,7420],{"className":12885},[7512,7549],[805,12887,7557],{"className":12888},[7556],[805,12890,12892],{"className":12891},[7529],[805,12893,12895],{"className":12894,"style":7564},[7533],[805,12896],{},[805,12898,12900],{"className":12899},[7512,10049],[805,12901,12903],{"className":12902},[7524],[805,12904,12906],{"className":12905},[7529],[805,12907,12909,12917],{"className":12908,"style":12811},[7533],[805,12910,12911,12914],{"style":10062},[805,12912],{"className":12913,"style":10066},[7541],[805,12915,7416],{"className":12916},[7512,7516],[805,12918,12919,12922],{"style":10062},[805,12920],{"className":12921,"style":10066},[7541],[805,12923,12926],{"className":12924,"style":12925},[10116],"left:-0.2222em;",[805,12927,12717],{"className":12928},[7512],[11,12930,8849],{},[49,12932,12933,12963,13104],{},[52,12934,12935,10946],{},[805,12936,12938,12951],{"className":12937},[7389],[805,12939,12941],{"className":12940},[7393],[7395,12942,12943],{"xmlns":7397},[7399,12944,12945,12949],{},[7402,12946,12947],{},[7414,12948,7472],{},[7487,12950,7472],{"encoding":7489},[805,12952,12954],{"className":12953,"ariaHidden":7423},[7494],[805,12955,12957,12960],{"className":12956},[7498],[805,12958],{"className":12959,"style":8843},[7502],[805,12961,7472],{"className":12962},[7512,7516],[52,12964,12965,36,13034,13103],{},[805,12966,12968,12985],{"className":12967},[7389],[805,12969,12971],{"className":12970},[7393],[7395,12972,12973],{"xmlns":7397},[7399,12974,12975,12983],{},[7402,12976,12977],{},[7411,12978,12979,12981],{},[7414,12980,7416],{},[7414,12982,7849],{},[7487,12984,7852],{"encoding":7489},[805,12986,12988],{"className":12987,"ariaHidden":7423},[7494],[805,12989,12991,12994],{"className":12990},[7498],[805,12992],{"className":12993,"style":7862},[7502],[805,12995,12997,13000],{"className":12996},[7512],[805,12998,7416],{"className":12999},[7512,7516],[805,13001,13003],{"className":13002},[7520],[805,13004,13006,13026],{"className":13005},[7524,7525],[805,13007,13009,13023],{"className":13008},[7529],[805,13010,13012],{"className":13011,"style":7881},[7533],[805,13013,13014,13017],{"style":7537},[805,13015],{"className":13016,"style":7542},[7541],[805,13018,13020],{"className":13019},[7546,7547,7548,7549],[805,13021,7849],{"className":13022},[7512,7516,7549],[805,13024,7557],{"className":13025},[7556],[8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are the values of the independent and dependent variables for each example.",[52,13105,13106,36,13167,13240],{},[805,13107,13109,13127],{"className":13108},[7389],[805,13110,13112],{"className":13111},[7393],[7395,13113,13114],{"xmlns":7397},[7399,13115,13116,13124],{},[7402,13117,13118],{},[9901,13119,13120,13122],{"accent":7423},[7414,13121,7416],{},[7405,13123,12717],{},[7487,13125,13126],{"encoding":7489},"\\bar{x}",[805,13128,13130],{"className":13129,"ariaHidden":7423},[7494],[805,13131,13133,13136],{"className":13132},[7498],[805,13134],{"className":13135,"style":12811},[7502],[805,13137,13139],{"className":13138},[7512,10049],[805,13140,13142],{"className":13141},[7524],[805,13143,13145],{"className":13144},[7529],[805,13146,13148,13156],{"className":13147,"style":12811},[7533],[805,13149,13150,13153],{"style":10062},[805,13151],{"className":13152,"style":10066},[7541],[805,13154,7416],{"className":13155},[7512,7516],[805,13157,13158,13161],{"style":10062},[805,13159],{"className":13160,"style":10066},[7541],[805,13162,13164],{"className":13163,"style":12925},[10116],[805,13165,12717],{"className":13166},[7512],[805,13168,13170,13188],{"className":13169},[7389],[805,13171,13173],{"className":13172},[7393],[7395,13174,13175],{"xmlns":7397},[7399,13176,13177,13185],{},[7402,13178,13179],{},[9901,13180,13181,13183],{"accent":7423},[7414,13182,828],{},[7405,13184,12717],{},[7487,13186,13187],{"encoding":7489},"\\bar{y}",[805,13189,13191],{"className":13190,"ariaHidden":7423},[7494],[805,13192,13194,13198],{"className":13193},[7498],[805,13195],{"className":13196,"style":13197},[7502],"height:0.7622em;vertical-align:-0.1944em;",[805,13199,13201],{"className":13200},[7512,10049],[805,13202,13204,13232],{"className":13203},[7524,7525],[805,13205,13207,13229],{"className":13206},[7529],[805,13208,13210,13218],{"className":13209,"style":12811},[7533],[805,13211,13212,13215],{"style":10062},[805,13213],{"className":13214,"style":10066},[7541],[805,13216,828],{"className":13217,"style":7582},[7512,7516],[805,13219,13220,13223],{"style":10062},[805,13221],{"className":13222,"style":10066},[7541],[805,13224,13226],{"className":13225,"style":11115},[10116],[805,13227,12717],{"className":13228},[7512],[805,13230,7557],{"className":13231},[7556],[805,13233,13235],{"className":13234},[7529],[805,13236,13238],{"className":13237,"style":10130},[7533],[805,13239],{}," are the means (or averages) of the independent and dependent variables, respectively.",[11,13242,13243],{},"Let's do an example to understand how to apply it. Suppose we have a dataset with the following characteristics:",[201,13245,13246,13256],{},[204,13247,13248],{},[207,13249,13250,13253],{},[210,13251,13252],{},"Size (m²)",[210,13254,13255],{},"Price (USD)",[223,13257,13258,13266,13274],{},[207,13259,13260,13263],{},[228,13261,13262],{},"50",[228,13264,13265],{},"100,000",[207,13267,13268,13271],{},[228,13269,13270],{},"100",[228,13272,13273],{},"200,000",[207,13275,13276,13279],{},[228,13277,13278],{},"150",[228,13280,13281],{},"300,000",[11,13283,13284,13285,13288],{},"We want to build a linear regression model to ",[29,13286,13287],{},"predict the price"," of a house based on its size. In this case, the independent variable is the size (X) and the dependent variable is the price (Y). The linear regression model can be expressed as:",[805,13290,13292],{"className":13291},[7980],[805,13293,13295,13331],{"className":13294},[7389],[805,13296,13298],{"className":13297},[7393],[7395,13299,13300],{"xmlns":7397,"display":7989},[7399,13301,13302,13328],{},[7402,13303,13304,13306,13308,13314,13316,13322,13324,13326],{},[7414,13305,828],{},[7405,13307,2823],{},[7411,13309,13310,13312],{},[7414,13311,8389],{},[7418,13313,8392],{},[7405,13315,8395],{},[7411,13317,13318,13320],{},[7414,13319,8389],{},[7418,13321,7420],{},[7414,13323,7416],{},[7405,13325,8395],{},[7414,13327,8448],{},[7487,13329,13330],{"encoding":7489},"y = \\beta_0 + \\beta_1 x + \\epsilon",[805,13332,13334,13352,13407,13465],{"className":13333,"ariaHidden":7423},[7494],[805,13335,13337,13340,13343,13346,13349],{"className":13336},[7498],[805,13338],{"className":13339,"style":7935},[7502],[805,13341,828],{"className":13342,"style":7582},[7512,7516],[805,13344],{"className":13345,"style":8028},[7574],[805,13347,2823],{"className":13348},[8032],[805,13350],{"className":13351,"style":8028},[7574],[805,13353,13355,13358,13398,13401,13404],{"className":13354},[7498],[805,13356],{"className":13357,"style":8020},[7502],[805,13359,13361,13364],{"className":13360},[7512],[805,13362,8389],{"className":13363,"style":8485},[7512,7516],[805,13365,13367],{"className":13366},[7520],[805,13368,13370,13390],{"className":13369},[7524,7525],[805,13371,13373,13387],{"className":13372},[7529],[805,13374,13376],{"className":13375,"style":7534},[7533],[805,13377,13378,13381],{"style":8500},[805,13379],{"className":13380,"style":7542},[7541],[805,13382,13384],{"className":13383},[7546,7547,7548,7549],[805,13385,8392],{"className":13386},[7512,7549],[805,13388,7557],{"className":13389},[7556],[805,13391,13393],{"className":13392},[7529],[805,13394,13396],{"className":13395,"style":7564},[7533],[805,13397],{},[805,13399],{"className":13400,"style":8065},[7574],[805,13402,8395],{"className":13403},[8527],[805,13405],{"className":13406,"style":8065},[7574],[805,13408,13410,13413,13453,13456,13459,13462],{"className":13409},[7498],[805,13411],{"className":13412,"style":8020},[7502],[805,13414,13416,13419],{"className":13415},[7512],[805,13417,8389],{"className":13418,"style":8485},[7512,7516],[805,13420,13422],{"className":13421},[7520],[805,13423,13425,13445],{"className":13424},[7524,7525],[805,13426,13428,13442],{"className":13427},[7529],[805,13429,13431],{"className":13430,"style":7534},[7533],[805,13432,13433,13436],{"style":8500},[805,13434],{"className":13435,"style":7542},[7541],[805,13437,13439],{"className":13438},[7546,7547,7548,7549],[805,13440,7420],{"className":13441},[7512,7549],[805,13443,7557],{"className":13444},[7556],[805,13446,13448],{"className":13447},[7529],[805,13449,13451],{"className":13450,"style":7564},[7533],[805,13452],{},[805,13454,7416],{"className":13455},[7512,7516],[805,13457],{"className":13458,"style":8065},[7574],[805,13460,8395],{"className":13461},[8527],[805,13463],{"className":13464,"style":8065},[7574],[805,13466,13468,13471],{"className":13467},[7498],[805,13469],{"className":13470,"style":8843},[7502],[805,13472,8448],{"className":13473},[7512,7516],[11,13475,8849],{},[49,13477,13478,13509,13540,13612,13684],{},[52,13479,13480,13508],{},[805,13481,13483,13496],{"className":13482},[7389],[805,13484,13486],{"className":13485},[7393],[7395,13487,13488],{"xmlns":7397},[7399,13489,13490,13494],{},[7402,13491,13492],{},[7414,13493,828],{},[7487,13495,828],{"encoding":7489},[805,13497,13499],{"className":13498,"ariaHidden":7423},[7494],[805,13500,13502,13505],{"className":13501},[7498],[805,13503],{"className":13504,"style":7935},[7502],[805,13506,828],{"className":13507,"style":7582},[7512,7516]," is the price of the house.",[52,13510,13511,13539],{},[805,13512,13514,13527],{"className":13513},[7389],[805,13515,13517],{"className":13516},[7393],[7395,13518,13519],{"xmlns":7397},[7399,13520,13521,13525],{},[7402,13522,13523],{},[7414,13524,7416],{},[7487,13526,7416],{"encoding":7489},[805,13528,13530],{"className":13529,"ariaHidden":7423},[7494],[805,13531,13533,13536],{"className":13532},[7498],[805,13534],{"className":13535,"style":8843},[7502],[805,13537,7416],{"className":13538},[7512,7516]," is the size of the house.",[52,13541,13542,13611],{},[805,13543,13545,13562],{"className":13544},[7389],[805,13546,13548],{"className":13547},[7393],[7395,13549,13550],{"xmlns":7397},[7399,13551,13552,13560],{},[7402,13553,13554],{},[7411,13555,13556,13558],{},[7414,13557,8389],{},[7418,13559,8392],{},[7487,13561,9104],{"encoding":7489},[805,13563,13565],{"className":13564,"ariaHidden":7423},[7494],[805,13566,13568,13571],{"className":13567},[7498],[805,13569],{"className":13570,"style":8020},[7502],[805,13572,13574,13577],{"className":13573},[7512],[805,13575,8389],{"className":13576,"style":8485},[7512,7516],[805,13578,13580],{"className":13579},[7520],[805,13581,13583,13603],{"className":13582},[7524,7525],[805,13584,13586,13600],{"className":13585},[7529],[805,13587,13589],{"className":13588,"style":7534},[7533],[805,13590,13591,13594],{"style":8500},[805,13592],{"className":13593,"style":7542},[7541],[805,13595,13597],{"className":13596},[7546,7547,7548,7549],[805,13598,8392],{"className":13599},[7512,7549],[805,13601,7557],{"className":13602},[7556],[805,13604,13606],{"className":13605},[7529],[805,13607,13609],{"className":13608,"style":7564},[7533],[805,13610],{}," is the y-intercept.",[52,13613,13614,13683],{},[805,13615,13617,13634],{"className":13616},[7389],[805,13618,13620],{"className":13619},[7393],[7395,13621,13622],{"xmlns":7397},[7399,13623,13624,13632],{},[7402,13625,13626],{},[7411,13627,13628,13630],{},[7414,13629,8389],{},[7418,13631,7420],{},[7487,13633,12049],{"encoding":7489},[805,13635,13637],{"className":13636,"ariaHidden":7423},[7494],[805,13638,13640,13643],{"className":13639},[7498],[805,13641],{"className":13642,"style":8020},[7502],[805,13644,13646,13649],{"className":13645},[7512],[805,13647,8389],{"className":13648,"style":8485},[7512,7516],[805,13650,13652],{"className":13651},[7520],[805,13653,13655,13675],{"className":13654},[7524,7525],[805,13656,13658,13672],{"className":13657},[7529],[805,13659,13661],{"className":13660,"style":7534},[7533],[805,13662,13663,13666],{"style":8500},[805,13664],{"className":13665,"style":7542},[7541],[805,13667,13669],{"className":13668},[7546,7547,7548,7549],[805,13670,7420],{"className":13671},[7512,7549],[805,13673,7557],{"className":13674},[7556],[805,13676,13678],{"className":13677},[7529],[805,13679,13681],{"className":13680,"style":7564},[7533],[805,13682],{}," is the slope.",[52,13685,13686,13714],{},[805,13687,13689,13702],{"className":13688},[7389],[805,13690,13692],{"className":13691},[7393],[7395,13693,13694],{"xmlns":7397},[7399,13695,13696,13700],{},[7402,13697,13698],{},[7414,13699,8448],{},[7487,13701,9429],{"encoding":7489},[805,13703,13705],{"className":13704,"ariaHidden":7423},[7494],[805,13706,13708,13711],{"className":13707},[7498],[805,13709],{"className":13710,"style":8843},[7502],[805,13712,8448],{"className":13713},[7512,7516]," is the error or noise, which for this example we will assume is zero for simplicity.",[11,13716,13717,13718,36,13787,13856],{},"To find the values of ",[805,13719,13721,13738],{"className":13720},[7389],[805,13722,13724],{"className":13723},[7393],[7395,13725,13726],{"xmlns":7397},[7399,13727,13728,13736],{},[7402,13729,13730],{},[7411,13731,13732,13734],{},[7414,13733,8389],{},[7418,13735,8392],{},[7487,13737,9104],{"encoding":7489},[805,13739,13741],{"className":13740,"ariaHidden":7423},[7494],[805,13742,13744,13747],{"className":13743},[7498],[805,13745],{"className":13746,"style":8020},[7502],[805,13748,13750,13753],{"className":13749},[7512],[805,13751,8389],{"className":13752,"style":8485},[7512,7516],[805,13754,13756],{"className":13755},[7520],[805,13757,13759,13779],{"className":13758},[7524,7525],[805,13760,13762,13776],{"className":13761},[7529],[805,13763,13765],{"className":13764,"style":7534},[7533],[805,13766,13767,13770],{"style":8500},[805,13768],{"className":13769,"style":7542},[7541],[805,13771,13773],{"className":13772},[7546,7547,7548,7549],[805,13774,8392],{"className":13775},[7512,7549],[805,13777,7557],{"className":13778},[7556],[805,13780,13782],{"className":13781},[7529],[805,13783,13785],{"className":13784,"style":7564},[7533],[805,13786],{},[805,13788,13790,13807],{"className":13789},[7389],[805,13791,13793],{"className":13792},[7393],[7395,13794,13795],{"xmlns":7397},[7399,13796,13797,13805],{},[7402,13798,13799],{},[7411,13800,13801,13803],{},[7414,13802,8389],{},[7418,13804,7420],{},[7487,13806,12049],{"encoding":7489},[805,13808,13810],{"className":13809,"ariaHidden":7423},[7494],[805,13811,13813,13816],{"className":13812},[7498],[805,13814],{"className":13815,"style":8020},[7502],[805,13817,13819,13822],{"className":13818},[7512],[805,13820,8389],{"className":13821,"style":8485},[7512,7516],[805,13823,13825],{"className":13824},[7520],[805,13826,13828,13848],{"className":13827},[7524,7525],[805,13829,13831,13845],{"className":13830},[7529],[805,13832,13834],{"className":13833,"style":7534},[7533],[805,13835,13836,13839],{"style":8500},[805,13837],{"className":13838,"style":7542},[7541],[805,13840,13842],{"className":13841},[7546,7547,7548,7549],[805,13843,7420],{"className":13844},[7512,7549],[805,13846,7557],{"className":13847},[7556],[805,13849,13851],{"className":13850},[7529],[805,13852,13854],{"className":13853,"style":7564},[7533],[805,13855],{},", we can use the least squares method with the formula we mentioned earlier:",[805,13858,13860],{"className":13859},[7980],[805,13861,13863,13960],{"className":13862},[7389],[805,13864,13866],{"className":13865},[7393],[7395,13867,13868],{"xmlns":7397,"display":7989},[7399,13869,13870,13958],{},[7402,13871,13872,13878,13880],{},[7411,13873,13874,13876],{},[7414,13875,8389],{},[7418,13877,7420],{},[7405,13879,2823],{},[10504,13881,13882,13922],{},[7402,13883,13884,13886,13888,13890,13896,13902,13904,13906,13908,13914,13916],{},[7414,13885,7472],{},[7405,13887,10515],{},[7405,13889,2651],{"stretchy":1435},[7411,13891,13892,13894],{},[7414,13893,7416],{},[7414,13895,7849],{},[7411,13897,13898,13900],{},[7414,13899,828],{},[7414,13901,7849],{},[7405,13903,3993],{"stretchy":1435},[7405,13905,9899],{},[7405,13907,10515],{},[7411,13909,13910,13912],{},[7414,13911,7416],{},[7414,13913,7849],{},[7405,13915,10515],{},[7411,13917,13918,13920],{},[7414,13919,828],{},[7414,13921,7849],{},[7402,13923,13924,13926,13928,13930,13938,13940,13942,13944,13946,13952],{},[7414,13925,7472],{},[7405,13927,10515],{},[7405,13929,2651],{"stretchy":1435},[12174,13931,13932,13934,13936],{},[7414,13933,7416],{},[7414,13935,7849],{},[7418,13937,7442],{},[7405,13939,3993],{"stretchy":1435},[7405,13941,9899],{},[7405,13943,2651],{"stretchy":1435},[7405,13945,10515],{},[7411,13947,13948,13950],{},[7414,13949,7416],{},[7414,13951,7849],{},[10547,13953,13954,13956],{},[7405,13955,3993],{"stretchy":1435},[7418,13957,7442],{},[7487,13959,12204],{"encoding":7489},[805,13961,13963,14018],{"className":13962,"ariaHidden":7423},[7494],[805,13964,13966,13969,14009,14012,14015],{"className":13965},[7498],[805,13967],{"className":13968,"style":8020},[7502],[805,13970,13972,13975],{"className":13971},[7512],[805,13973,8389],{"className":13974,"style":8485},[7512,7516],[805,13976,13978],{"className":13977},[7520],[805,13979,13981,14001],{"className":13980},[7524,7525],[805,13982,13984,13998],{"className":13983},[7529],[805,13985,13987],{"className":13986,"style":7534},[7533],[805,13988,13989,13992],{"style":8500},[805,13990],{"className":13991,"style":7542},[7541],[805,13993,13995],{"className":13994},[7546,7547,7548,7549],[805,13996,7420],{"className":13997},[7512,7549],[805,13999,7557],{"className":14000},[7556],[805,14002,14004],{"className":14003},[7529],[805,14005,14007],{"className":14006,"style":7564},[7533],[805,14008],{},[805,14010],{"className":14011,"style":8028},[7574],[805,14013,2823],{"className":14014},[8032],[805,14016],{"className":14017,"style":8028},[7574],[805,14019,14021,14024],{"className":14020},[7498],[805,14022],{"className":14023,"style":12269},[7502],[805,14025,14027,14030,14429],{"className":14026},[7512],[805,14028],{"className":14029},[7507,10598],[805,14031,14033],{"className":14032},[10504],[805,14034,14036,14421],{"className":14035},[7524,7525],[805,14037,14039,14418],{"className":14038},[7529],[805,14040,14042,14203,14211],{"className":14041,"style":12288},[7533],[805,14043,14044,14047],{"style":10614},[805,14045],{"className":14046,"style":10066},[7541],[805,14048,14050,14053,14056,14059,14062,14113,14116,14119,14122,14125,14128,14131,14134,14174],{"className":14049},[7512],[805,14051,7472],{"className":14052},[7512,7516],[805,14054],{"className":14055,"style":7575},[7574],[805,14057,10515],{"className":14058,"style":12307},[10668,10713,12306],[805,14060,2651],{"className":14061},[7507],[805,14063,14065,14068],{"className":14064},[7512],[805,14066,7416],{"className":14067},[7512,7516],[805,14069,14071],{"className":14070},[7520],[805,14072,14074,14105],{"className":14073},[7524,7525],[805,14075,14077,14102],{"className":14076},[7529],[805,14078,14080,14091],{"className":14079,"style":12329},[7533],[805,14081,14082,14085],{"style":12332},[805,14083],{"className":14084,"style":7542},[7541],[805,14086,14088],{"className":14087},[7546,7547,7548,7549],[805,14089,7849],{"className":14090},[7512,7516,7549],[805,14092,14093,14096],{"style":12344},[805,14094],{"className":14095,"style":7542},[7541],[805,14097,14099],{"className":14098},[7546,7547,7548,7549],[805,14100,7442],{"className":14101},[7512,7549],[805,14103,7557],{"className":14104},[7556],[805,14106,14108],{"className":14107},[7529],[805,14109,14111],{"className":14110,"style":12363},[7533],[805,14112],{},[805,14114,3993],{"className":14115},[7621],[805,14117],{"className":14118,"style":8065},[7574],[805,14120,9899],{"className":14121},[8527],[805,14123],{"className":14124,"style":8065},[7574],[805,14126,2651],{"className":14127},[7507],[805,14129,10515],{"className":14130,"style":12307},[10668,10713,12306],[805,14132],{"className":14133,"style":7575},[7574],[805,14135,14137,14140],{"className":14136},[7512],[805,14138,7416],{"className":14139},[7512,7516],[805,14141,14143],{"className":14142},[7520],[805,14144,14146,14166],{"className":14145},[7524,7525],[805,14147,14149,14163],{"className":14148},[7529],[805,14150,14152],{"className":14151,"style":7881},[7533],[805,14153,14154,14157],{"style":7537},[805,14155],{"className":14156,"sty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is the number of examples (in this case, 3).",[52,14705,14706,36,14775,14844],{},[805,14707,14709,14726],{"className":14708},[7389],[805,14710,14712],{"className":14711},[7393],[7395,14713,14714],{"xmlns":7397},[7399,14715,14716,14724],{},[7402,14717,14718],{},[7411,14719,14720,14722],{},[7414,14721,7416],{},[7414,14723,7849],{},[7487,14725,7852],{"encoding":7489},[805,14727,14729],{"className":14728,"ariaHidden":7423},[7494],[805,14730,14732,14735],{"className":14731},[7498],[805,14733],{"className":14734,"style":7862},[7502],[805,14736,14738,14741],{"className":14737},[7512],[805,14739,7416],{"className":14740},[7512,7516],[805,14742,14744],{"className":14743},[7520],[805,14745,14747,14767],{"className":14746},[7524,7525],[805,14748,14750,14764],{"className":14749},[7529],[805,14751,14753],{"className":14752,"style":7881},[7533],[805,14754,14755,14758],{"style":7537},[805,14756],{"className":14757,"style":7542},[7541],[805,14759,14761],{"className":14760},[7546,7547,7548,7549],[805,14762,7849],{"className":14763},[7512,7516,7549],[805,14765,7557],{"className":14766},[7556],[805,14768,14770],{"className":14769},[7529],[805,14771,14773],{"className":14772,"style":7564},[7533],[805,14774],{},[805,14776,14778,14795],{"className":14777},[7389],[805,14779,14781],{"className":14780},[7393],[7395,14782,14783],{"xmlns":7397},[7399,14784,14785,14793],{},[7402,14786,14787],{},[7411,14788,14789,14791],{},[7414,14790,828],{},[7414,14792,7849],{},[7487,14794,7925],{"encoding":7489},[805,14796,14798],{"className":14797,"ariaHidden":7423},[7494],[805,14799,14801,14804],{"className":14800},[7498],[805,14802],{"className":14803,"style":7935},[7502],[805,14805,14807,14810],{"className":14806},[7512],[805,14808,828],{"className":14809,"style":7582},[7512,7516],[805,14811,14813],{"className":14812},[7520],[805,14814,14816,14836],{"className":14815},[7524,7525],[805,14817,14819,14833],{"className":14818},[7529],[805,14820,14822],{"className":14821,"style":7881},[7533],[805,14823,14824,14827],{"style":7597},[805,14825],{"className":14826,"style":7542},[7541],[805,14828,14830],{"className":14829},[7546,7547,7548,7549],[805,14831,7849],{"className":14832},[7512,7516,7549],[805,14834,7557],{"className":14835},[7556],[805,14837,14839],{"className":14838},[7529],[805,14840,14842],{"className":14841,"style":7564},[7533],[805,14843],{}," are the values of size and price for each example.",[52,14846,14847,36,14907,14978],{},[805,14848,14850,14867],{"className":14849},[7389],[805,14851,14853],{"className":14852},[7393],[7395,14854,14855],{"xmlns":7397},[7399,14856,14857,14865],{},[7402,14858,14859],{},[9901,14860,14861,14863],{"accent":7423},[7414,14862,7416],{},[7405,14864,12717],{},[7487,14866,13126],{"encoding":7489},[805,14868,14870],{"className":14869,"ariaHidden":7423},[7494],[805,14871,14873,14876],{"className":14872},[7498],[805,14874],{"className":14875,"style":12811},[7502],[805,14877,14879],{"className":14878},[7512,10049],[805,14880,14882],{"className":14881},[7524],[805,14883,14885],{"className":14884},[7529],[805,14886,14888,14896],{"className":14887,"style":12811},[7533],[805,14889,14890,14893],{"style":10062},[805,14891],{"className":14892,"style":10066},[7541],[805,14894,7416],{"className":14895},[7512,7516],[805,14897,14898,14901],{"style":10062},[805,14899],{"className":14900,"style":10066},[7541],[805,14902,14904],{"className":14903,"style":12925},[10116],[805,14905,12717],{"className":14906},[7512],[805,14908,14910,14927],{"className":14909},[7389],[805,14911,14913],{"className":14912},[7393],[7395,14914,14915],{"xmlns":7397},[7399,14916,14917,14925],{},[7402,14918,14919],{},[9901,14920,14921,14923],{"accent":7423},[7414,14922,828],{},[7405,14924,12717],{},[7487,14926,13187],{"encoding":7489},[805,14928,14930],{"className":14929,"ariaHidden":7423},[7494],[805,14931,14933,14936],{"className":14932},[7498],[805,14934],{"className":14935,"style":13197},[7502],[805,14937,14939],{"className":14938},[7512,10049],[805,14940,14942,14970],{"className":14941},[7524,7525],[805,14943,14945,14967],{"className":14944},[7529],[805,14946,14948,14956],{"className":14947,"style":12811},[7533],[805,14949,14950,14953],{"style":10062},[805,14951],{"className":14952,"style":10066},[7541],[805,14954,828],{"className":14955,"style":7582},[7512,7516],[805,14957,14958,14961],{"style":10062},[805,14959],{"className":14960,"style":10066},[7541],[805,14962,14964],{"className":14963,"style":11115},[10116],[805,14965,12717],{"className":14966},[7512],[805,14968,7557],{"className":14969},[7556],[805,14971,14973],{"className":14972},[7529],[805,14974,14976],{"className":14975,"style":10130},[7533],[805,14977],{}," are the means of the independent and dependent variables, respectively.",[11,14980,14981],{},"We have then for this case:",[805,14983,14985],{"className":14984},[7980],[805,14986,14988,15125],{"className":14987},[7389],[805,14989,14991],{"className":14990},[7393],[7395,14992,14993],{"xmlns":7397,"display":7989},[7399,14994,14995,15122],{},[7402,14996,14997,15003,15005],{},[7411,14998,14999,15001],{},[7414,15000,8389],{},[7418,15002,7420],{},[7405,15004,2823],{},[10504,15006,15007,15072],{},[7402,15008,15009,15012,15014,15016,15019,15022,15024,15026,15028,15031,15033,15035,15037,15040,15042,15044,15046,15048,15050,15052,15054,15056,15058,15060,15062,15064,15066,15068,15070],{},[7418,15010,15011],{},"3",[7405,15013,2651],{"stretchy":1435},[7418,15015,13262],{},[7405,15017,15018],{},"∗",[7418,15020,15021],{},"100000",[7405,15023,8395],{},[7418,15025,13270],{},[7405,15027,15018],{},[7418,15029,15030],{},"200000",[7405,15032,8395],{},[7418,15034,13278],{},[7405,15036,15018],{},[7418,15038,15039],{},"300000",[7405,15041,3993],{"stretchy":1435},[7405,15043,9899],{},[7405,15045,2651],{"stretchy":1435},[7418,15047,13262],{},[7405,15049,8395],{},[7418,15051,13270],{},[7405,15053,8395],{},[7418,15055,13278],{},[7405,15057,3993],{"stretchy":1435},[7405,15059,2651],{"stretchy":1435},[7418,15061,15021],{},[7405,15063,8395],{},[7418,15065,15030],{},[7405,15067,8395],{},[7418,15069,15039],{},[7405,15071,3993],{"stretchy":1435},[7402,15073,15074,15076,15078,15084,15086,15092,15094,15100,15102,15104,15106,15108,15110,15112,15114,15116],{},[7418,15075,15011],{},[7405,15077,2651],{"stretchy":1435},[10547,15079,15080,15082],{},[7418,15081,13262],{},[7418,15083,7442],{},[7405,15085,8395],{},[10547,15087,15088,15090],{},[7418,15089,13270],{},[7418,15091,7442],{},[7405,15093,8395],{},[10547,15095,15096,15098],{},[7418,15097,13278],{},[7418,15099,7442],{},[7405,15101,3993],{"stretchy":1435},[7405,15103,9899],{},[7405,15105,2651],{"stretchy":1435},[7418,15107,13262],{},[7405,15109,8395],{},[7418,15111,13270],{},[7405,15113,8395],{},[7418,15115,13278],{},[10547,15117,15118,15120],{},[7405,15119,3993],{"stretchy":1435},[7418,15121,7442],{},[7487,15123,15124],{"encoding":7489},"\\beta_1 = \\frac{3(50*100000 + 100*200000 + 150*300000) - (50 + 100 + 150)(100000 + 200000 + 300000)}{3(50^2 + 100^2 + 150^2) - (50 + 100 + 150)^2}",[805,15126,15128,15183],{"className":15127,"ariaHidden":7423},[7494],[805,15129,15131,15134,15174,15177,15180],{"className":15130},[7498],[805,15132],{"className":15133,"style":8020},[7502],[805,15135,15137,15140],{"className":15136},[7512],[805,15138,8389],{"className":15139,"style":8485},[7512,7516],[805,15141,15143],{"className":15142},[7520],[805,15144,15146,15166],{"className":15145},[7524,7525],[805,15147,15149,15163],{"className":15148},[7529],[805,15150,15152],{"className":15151,"style":7534},[7533],[805,15153,15154,15157],{"style":8500},[805,15155],{"className":15156,"style":7542},[7541],[805,15158,15160],{"className":15159},[7546,7547,7548,7549],[805,15161,7420],{"className":15162},[7512,7549],[805,15164,7557],{"className":15165},[7556],[805,15167,15169],{"className":15168},[7529],[805,15170,15172],{"className":15171,"style":7564},[7533],[805,15173],{},[805,15175],{"className":15176,"style":8028},[7574],[805,15178,2823],{"className":15179},[8032],[805,15181],{"className":15182,"style":8028},[7574],[805,15184,15186,15190],{"className":15185},[7498],[805,15187],{"className":15188,"style":15189},[7502],"height:2.363em;vertical-align:-0.936em;",[805,15191,15193,15196,15585],{"className":15192},[7512],[805,15194],{"className":15195},[7507,10598],[805,15197,15199],{"className":15198},[10504],[805,15200,15202,15576],{"className":15201},[7524,7525],[805,15203,15205,15573],{"className":15204},[7529],[805,15206,15208,15410,15418],{"className":15207,"style":12288},[7533],[805,15209,15210,15213],{"style":10614},[805,15211],{"className":15212,"style":10066},[7541],[805,15214,15216,15219,15222,15226,15255,15258,15261,15264,15268,15297,15300,15303,15306,15310,15339,15342,15345,15348,15351,15354,15357,15360,15363,15366,15369,15372,15375,15378,15381],{"className":15215},[7512],[805,15217,15011],{"className":15218},[7512],[805,15220,2651],{"className":15221},[7507],[805,15223,15225],{"className":15224},[7512],"5",[805,15227,15229,15232],{"className":15228},[7512],[805,15230,8392],{"className":15231},[7512],[805,15233,15235],{"className":15234},[7520],[805,15236,15238],{"className":15237},[7524],[805,15239,15241],{"className":15240},[7529],[805,15242,15244],{"className":15243,"style":12445},[7533],[805,15245,15246,15249],{"style":12448},[805,15247],{"className":15248,"style":7542},[7541],[805,15250,15252],{"className":15251},[7546,7547,7548,7549],[805,15253,7442],{"className":15254},[7512,7549],[805,15256],{"className":15257,"style":8065},[7574],[805,15259,8395],{"className":15260},[8527],[805,15262],{"className":15263,"style":8065},[7574],[805,15265,15267],{"className":15266},[7512],"10",[805,15269,15271,15274],{"className":15270},[7512],[805,15272,8392],{"className":15273},[7512],[805,15275,15277],{"className":15276},[7520],[805,15278,15280],{"className":15279},[7524],[805,15281,15283],{"className":15282},[7529],[805,15284,15286],{"className":15285,"style":12445},[7533],[805,15287,15288,15291],{"style":12448},[805,15289],{"className":15290,"style":7542},[7541],[805,15292,15294],{"className":15293},[7546,7547,7548,7549],[805,15295,7442],{"className":15296},[7512,7549],[805,15298],{"className":15299,"style":8065},[7574],[805,15301,8395],{"className":15302},[8527],[805,15304],{"className":15305,"style":8065},[7574],[805,15307,15309],{"className":15308},[7512],"15",[805,15311,15313,15316],{"className":15312},[7512],[805,15314,8392],{"className":15315},[7512],[805,15317,15319],{"className":15318},[7520],[805,15320,15322],{"className":15321},[7524],[805,15323,15325],{"className":15324},[7529],[805,15326,15328],{"className":15327,"style":12445},[7533],[805,15329,15330,15333],{"style":12448},[805,15331],{"className":15332,"style":7542},[7541],[805,15334,15336],{"className":15335},[7546,7547,7548,7549],[805,15337,7442],{"className":15338},[7512,7549],[805,15340,3993],{"className":15341},[7621],[805,15343],{"className":15344,"style":8065},[7574],[805,15346,9899],{"className":15347},[8527],[805,15349],{"className":15350,"style":8065},[7574],[805,15352,2651],{"className":15353},[7507],[805,15355,13262],{"className":15356},[7512],[805,15358],{"className":15359,"style":8065},[7574],[805,15361,8395],{"className":15362},[8527],[805,15364],{"className":15365,"style":8065},[7574],[805,15367,13270],{"className":15368},[7512],[805,15370],{"className":15371,"style":8065},[7574],[805,15373,8395],{"className":15374},[8527],[805,15376],{"className":15377,"style":8065},[7574],[805,15379,13278],{"className":15380},[7512],[805,15382,15384,15387],{"className":15383},[7621],[805,15385,3993],{"className":15386},[7621],[805,15388,15390],{"className":15389},[7520],[805,15391,15393],{"className":15392},[7524],[805,15394,15396],{"className":15395},[7529],[805,15397,15399],{"className":15398,"style":12445},[7533],[805,15400,15401,15404],{"style":12448},[805,15402],{"className":15403,"style":7542},[7541],[805,15405,15407],{"className":15406},[7546,7547,7548,7549],[805,15408,7442],{"className":15409},[7512,7549],[805,15411,15412,15415],{"style":10626},[805,15413],{"className":15414,"style":10066},[7541],[805,15416],{"className":15417,"style":10634},[10633],[805,15419,15420,15423],{"style":10637},[805,15421],{"className":15422,"style":10066},[7541],[805,15424,15426,15429,15432,15435,15438,15441,15444,15447,15450,15453,15456,15459,15462,15465,15468,15471,15474,15477,15480,15483,15486,15489,15492,15495,15498,15501,15504,15507,15510,15513,15516,15519,15522,15525,15528,15531,15534,15537,15540,15543,15546,15549,15552,15555,15558,15561,15564,15567,15570],{"className":15425},[7512],[805,15427,15011],{"className":15428},[7512],[805,15430,2651],{"className":15431},[7507],[805,15433,13262],{"className":15434},[7512],[805,15436],{"className":15437,"style":8065},[7574],[805,15439,15018],{"className":15440},[8527],[805,15442],{"className":15443,"style":8065},[7574],[805,15445,15021],{"className":15446},[7512],[805,15448],{"className":15449,"style":8065},[7574],[805,15451,8395],{"className":15452},[8527],[805,15454],{"className":15455,"style":8065},[7574],[805,15457,13270],{"className":15458},[7512],[805,15460],{"className":15461,"style":8065},[7574],[805,15463,15018],{"className":15464},[8527],[805,15466],{"className":15467,"style":8065},[7574],[805,15469,15030],{"className":15470},[7512],[805,15472],{"className":15473,"style":8065},[7574],[805,15475,8395],{"className":15476},[8527],[805,15478],{"className":15479,"style":8065},[7574],[805,15481,13278],{"className":15482},[7512],[805,15484],{"className":15485,"style":8065},[7574],[805,15487,15018],{"className":15488},[8527],[805,15490],{"className":15491,"style":8065},[7574],[805,15493,15039],{"className":15494},[7512],[805,15496,3993],{"className":15497},[7621],[805,15499],{"className":15500,"style":8065},[7574],[805,15502,9899],{"className":15503},[8527],[805,15505],{"className":15506,"style":8065},[7574],[805,15508,2651],{"className":15509},[7507],[805,15511,13262],{"className":15512},[7512],[805,15514],{"className":15515,"style":8065},[7574],[805,15517,8395],{"className":15518},[8527],[805,15520],{"className":15521,"style":8065},[7574],[805,15523,13270],{"className":15524},[7512],[805,15526],{"className":15527,"style":8065},[7574],[805,15529,8395],{"className":15530},[8527],[805,15532],{"className":15533,"style":8065},[7574],[805,15535,13278],{"className":15536},[7512],[805,15538,3993],{"className":15539},[7621],[805,15541,2651],{"className":15542},[7507],[805,15544,15021],{"className":15545},[7512],[805,15547],{"className":15548,"style":8065},[7574],[805,15550,8395],{"className":15551},[8527],[805,15553],{"className":15554,"style":8065},[7574],[805,15556,15030],{"className":15557},[7512],[805,15559],{"className":15560,"style":8065},[7574],[805,15562,8395],{"className":15563},[8527],[805,15565],{"className":15566,"style":8065},[7574],[805,15568,15039],{"className":15569},[7512],[805,15571,3993],{"className":15572},[7621],[805,15574,7557],{"className":15575},[7556],[805,15577,15579],{"className":15578},[7529],[805,15580,15583],{"className":15581,"style":15582},[7533],"height:0.936em;",[805,15584],{},[805,15586],{"className":15587},[7621,10598],[805,15589,15591],{"className":15590},[7980],[805,15592,15594,15654],{"className":15593},[7389],[805,15595,15597],{"className":15596},[7393],[7395,15598,15599],{"xmlns":7397,"display":7989},[7399,15600,15601,15651],{},[7402,15602,15603,15609,15611,15627,15629,15635],{},[7411,15604,15605,15607],{},[7414,15606,8389],{},[7418,15608,8392],{},[7405,15610,2823],{},[10504,15612,15613,15625],{},[7402,15614,15615,15617,15619,15621,15623],{},[7418,15616,15021],{},[7405,15618,8395],{},[7418,15620,15030],{},[7405,15622,8395],{},[7418,15624,15039],{},[7418,15626,15011],{},[7405,15628,9899],{},[7411,15630,15631,15633],{},[7414,15632,8389],{},[7418,15634,7420],{},[10504,15636,15637,15649],{},[7402,15638,15639,15641,15643,15645,15647],{},[7418,15640,13262],{},[7405,15642,8395],{},[7418,15644,13270],{},[7405,15646,8395],{},[7418,15648,13278],{},[7418,15650,15011],{},[7487,15652,15653],{"encoding":7489},"\\beta_0 = \\frac{100000 + 200000 + 300000}{3} - \\beta_1 \\frac{50 + 100 + 150}{3}",[805,15655,15657,15712,15814],{"className":15656,"ariaHidden":7423},[7494],[805,15658,15660,15663,15703,15706,15709],{"className":15659},[7498],[805,15661],{"className":15662,"style":8020},[7502],[805,15664,15666,15669],{"className":15665},[7512],[805,15667,8389],{"className":15668,"style":8485},[7512,7516],[805,15670,15672],{"className":15671},[7520],[805,15673,15675,15695],{"className":15674},[7524,7525],[805,15676,15678,15692],{"className":15677},[7529],[805,15679,15681],{"className":15680,"style":7534},[7533],[805,15682,15683,15686],{"style":8500},[805,15684],{"className":15685,"style":7542},[7541],[805,15687,15689],{"className":15688},[7546,7547,7548,7549],[805,15690,8392],{"className":15691},[7512,7549],[805,15693,7557],{"className":15694},[7556],[805,15696,15698],{"className":15697},[7529],[805,15699,15701],{"className":15700,"style":7564},[7533],[805,15702],{},[805,15704],{"className":15705,"style":8028},[7574],[805,15707,2823],{"className":15708},[8032],[805,15710],{"className":15711,"style":8028},[7574],[805,15713,15715,15719,15805,15808,15811],{"className":15714},[7498],[805,15716],{"className":15717,"style":15718},[7502],"height:2.0074em;vertical-align:-0.686em;",[805,15720,15722,15725,15802],{"className":15721},[7512],[805,15723],{"className":15724},[7507,10598],[805,15726,15728],{"className":15727},[10504],[805,15729,15731,15794],{"className":15730},[7524,7525],[805,15732,15734,15791],{"className":15733},[7529],[805,15735,15737,15748,15756],{"className":15736,"style":10611},[7533],[805,15738,15739,15742],{"style":10614},[805,15740],{"className":15741,"style":10066},[7541],[805,15743,15745],{"className":15744},[7512],[805,15746,15011],{"className":15747},[7512],[805,15749,15750,15753],{"style":10626},[805,15751],{"className":15752,"style":10066},[7541],[805,15754],{"className":15755,"style":10634},[10633],[805,15757,15758,15761],{"style":10637},[805,15759],{"className":15760,"style":10066},[7541],[805,15762,15764,15767,15770,15773,15776,15779,15782,15785,15788],{"className":15763},[7512],[805,15765,15021],{"className":15766},[7512],[805,15768],{"className":15769,"style":8065},[7574],[805,15771,8395],{"className":15772},[8527],[805,15774],{"className":15775,"style":8065},[7574],[805,15777,15030],{"className":15778},[7512],[805,15780],{"className":15781,"style":8065},[7574],[805,15783,8395],{"className":15784},[8527],[805,15786],{"className":15787,"style":8065},[7574],[805,15789,15039],{"className":15790},[7512],[805,15792,7557],{"className":15793},[7556],[805,15795,15797],{"className":15796},[7529],[805,15798,15800],{"className":15799,"style":10656},[7533],[805,15801],{},[805,15803],{"className":15804},[7621,10598],[805,15806],{"className":15807,"style":8065},[7574],[805,15809,9899],{"className":15810},[8527],[805,15812],{"className":15813,"style":8065},[7574],[805,15815,15817,15820,15860],{"className":15816},[7498],[805,15818],{"className":15819,"style":15718},[7502],[805,15821,15823,15826],{"className":15822},[7512],[805,15824,8389],{"className":15825,"style":8485},[7512,7516],[805,15827,15829],{"className":15828},[7520],[805,15830,15832,15852],{"className":15831},[7524,7525],[805,15833,15835,15849],{"className":15834},[7529],[805,15836,15838],{"className":15837,"style":7534},[7533],[805,15839,15840,15843],{"style":8500},[805,15841],{"className":15842,"style":7542},[7541],[805,15844,15846],{"className":15845},[7546,7547,7548,7549],[805,15847,7420],{"className":15848},[7512,7549],[805,15850,7557],{"className":15851},[7556],[805,15853,15855],{"className":15854},[7529],[805,15856,15858],{"className":15857,"style":7564},[7533],[805,15859],{},[805,15861,15863,15866,15943],{"className":15862},[7512],[805,15864],{"className":15865},[7507,10598],[805,15867,15869],{"className":15868},[10504],[805,15870,15872,15935],{"className":15871},[7524,7525],[805,15873,15875,15932],{"className":15874},[7529],[805,15876,15878,15889,15897],{"className":15877,"style":10611},[7533],[805,15879,15880,15883],{"style":10614},[805,15881],{"className":15882,"style":10066},[7541],[805,15884,15886],{"className":15885},[7512],[805,15887,15011],{"className":15888},[7512],[805,15890,15891,15894],{"style":10626},[805,15892],{"className":15893,"style":10066},[7541],[805,15895],{"className":15896,"style":10634},[10633],[805,15898,15899,15902],{"style":10637},[805,15900],{"className":15901,"style":10066},[7541],[805,15903,15905,15908,15911,15914,15917,15920,15923,15926,15929],{"className":15904},[7512],[805,15906,13262],{"className":15907},[7512],[805,15909],{"className":15910,"style":8065},[7574],[805,15912,8395],{"className":15913},[8527],[805,15915],{"className":15916,"style":8065},[7574],[805,15918,13270],{"className":15919},[7512],[805,15921],{"className":15922,"style":8065},[7574],[805,15924,8395],{"className":15925},[8527],[805,15927],{"className":15928,"style":8065},[7574],[805,15930,13278],{"className":15931},[7512],[805,15933,7557],{"className":15934},[7556],[805,15936,15938],{"className":15937},[7529],[805,15939,15941],{"className":15940,"style":10656},[7533],[805,15942],{},[805,15944],{"className":15945},[7621,10598],[11,15947,15948,15949,36,16042,16135],{},"Solving these formulas, we obtain the values ​​of ",[805,15950,15952,15974],{"className":15951},[7389],[805,15953,15955],{"className":15954},[7393],[7395,15956,15957],{"xmlns":7397},[7399,15958,15959,15971],{},[7402,15960,15961,15967,15969],{},[7411,15962,15963,15965],{},[7414,15964,8389],{},[7418,15966,8392],{},[7405,15968,2823],{},[7418,15970,8392],{},[7487,15972,15973],{"encoding":7489},"\\beta_0 = 0",[805,15975,15977,16032],{"className":15976,"ariaHidden":7423},[7494],[805,15978,15980,15983,16023,16026,16029],{"className":15979},[7498],[805,15981],{"className":15982,"style":8020},[7502],[805,15984,15986,15989],{"className":15985},[7512],[805,15987,8389],{"className":15988,"style":8485},[7512,7516],[805,15990,15992],{"className":15991},[7520],[805,15993,15995,16015],{"className":15994},[7524,7525],[805,15996,15998,16012],{"className":15997},[7529],[805,15999,16001],{"className":16000,"style":7534},[7533],[805,16002,16003,16006],{"style":8500},[805,16004],{"className":16005,"style":7542},[7541],[805,16007,16009],{"className":16008},[7546,7547,7548,7549],[805,16010,8392],{"className":16011},[7512,7549],[805,16013,7557],{"className":16014},[7556],[805,16016,16018],{"className":16017},[7529],[805,16019,16021],{"className":16020,"style":7564},[7533],[805,16022],{},[805,16024],{"className":16025,"style":8028},[7574],[805,16027,2823],{"className":16028},[8032],[805,16030],{"className":16031,"style":8028},[7574],[805,16033,16035,16039],{"className":16034},[7498],[805,16036],{"className":16037,"style":16038},[7502],"height:0.6444em;",[805,16040,8392],{"className":16041},[7512],[805,16043,16045,16068],{"className":16044},[7389],[805,16046,16048],{"className":16047},[7393],[7395,16049,16050],{"xmlns":7397},[7399,16051,16052,16065],{},[7402,16053,16054,16060,16062],{},[7411,16055,16056,16058],{},[7414,16057,8389],{},[7418,16059,7420],{},[7405,16061,2823],{},[7418,16063,16064],{},"2000",[7487,16066,16067],{"encoding":7489},"\\beta_1 = 2000",[805,16069,16071,16126],{"className":16070,"ariaHidden":7423},[7494],[805,16072,16074,16077,16117,16120,16123],{"className":16073},[7498],[805,16075],{"className":16076,"style":8020},[7502],[805,16078,16080,16083],{"className":16079},[7512],[805,16081,8389],{"className":16082,"style":8485},[7512,7516],[805,16084,16086],{"className":16085},[7520],[805,16087,16089,16109],{"className":16088},[7524,7525],[805,16090,16092,16106],{"className":16091},[7529],[805,16093,16095],{"className":16094,"style":7534},[7533],[805,16096,16097,16100],{"style":8500},[805,16098],{"className":16099,"style":7542},[7541],[805,16101,16103],{"className":16102},[7546,7547,7548,7549],[805,16104,7420],{"className":16105},[7512,7549],[805,16107,7557],{"className":16108},[7556],[805,16110,16112],{"className":16111},[7529],[805,16113,16115],{"className":16114,"style":7564},[7533],[805,16116],{},[805,16118],{"className":16119,"style":8028},[7574],[805,16121,2823],{"className":16122},[8032],[805,16124],{"className":16125,"style":8028},[7574],[805,16127,16129,16132],{"className":16128},[7498],[805,16130],{"className":16131,"style":16038},[7502],[805,16133,16064],{"className":16134},[7512],", which allow us to build the following model:",[805,16137,16139],{"className":16138},[7980],[805,16140,16142,16170],{"className":16141},[7389],[805,16143,16145],{"className":16144},[7393],[7395,16146,16147],{"xmlns":7397,"display":7989},[7399,16148,16149,16167],{},[7402,16150,16151,16157,16159,16161,16163,16165],{},[9901,16152,16153,16155],{"accent":7423},[7414,16154,828],{},[7405,16156,9911],{},[7405,16158,2823],{},[7418,16160,8392],{},[7405,16162,8395],{},[7418,16164,16064],{},[7414,16166,7416],{},[7487,16168,16169],{"encoding":7489},"\\hat{y} = 0 + 2000x",[805,16171,16173,16230,16249],{"className":16172,"ariaHidden":7423},[7494],[805,16174,16176,16179,16221,16224,16227],{"className":16175},[7498],[805,16177],{"className":16178,"style":8020},[7502],[805,16180,16182],{"className":16181},[7512,10049],[805,16183,16185,16213],{"className":16184},[7524,7525],[805,16186,16188,16210],{"className":16187},[7529],[805,16189,16191,16199],{"className":16190,"style":10813},[7533],[805,16192,16193,16196],{"style":10062},[805,16194],{"className":16195,"style":10066},[7541],[805,16197,828],{"className":16198,"style":7582},[7512,7516],[805,16200,16201,16204],{"style":10062},[805,16202],{"className":16203,"style":10066},[7541],[805,16205,16207],{"className":16206,"style":11115},[10116],[805,16208,9911],{"className":16209},[7512],[805,16211,7557],{"className":16212},[7556],[805,16214,16216],{"className":16215},[7529],[805,16217,16219],{"className":16218,"style":10130},[7533],[805,16220],{},[805,16222],{"className":16223,"style":8028},[7574],[805,16225,2823],{"className":16226},[8032],[805,16228],{"className":16229,"style":8028},[7574],[805,16231,16233,16237,16240,16243,16246],{"className":16232},[7498],[805,16234],{"className":16235,"style":16236},[7502],"height:0.7278em;vertical-align:-0.0833em;",[805,16238,8392],{"className":16239},[7512],[805,16241],{"className":16242,"style":8065},[7574],[805,16244,8395],{"className":16245},[8527],[805,16247],{"className":16248,"style":8065},[7574],[805,16250,16252,16255,16258],{"className":16251},[7498],[805,16253],{"className":16254,"style":16038},[7502],[805,16256,16064],{"className":16257},[7512],[805,16259,7416],{"className":16260},[7512,7516],[805,16262,16264],{"className":16263},[7980],[805,16265,16267,16291],{"className":16266},[7389],[805,16268,16270],{"className":16269},[7393],[7395,16271,16272],{"xmlns":7397,"display":7989},[7399,16273,16274,16288],{},[7402,16275,16276,16282,16284,16286],{},[9901,16277,16278,16280],{"accent":7423},[7414,16279,828],{},[7405,16281,9911],{},[7405,16283,2823],{},[7418,16285,16064],{},[7414,16287,7416],{},[7487,16289,16290],{"encoding":7489},"\\hat{y} = 2000x",[805,16292,16294,16351],{"className":16293,"ariaHidden":7423},[7494],[805,16295,16297,16300,16342,16345,16348],{"className":16296},[7498],[805,16298],{"className":16299,"style":8020},[7502],[805,16301,16303],{"className":16302},[7512,10049],[805,16304,16306,16334],{"className":16305},[7524,7525],[805,16307,16309,16331],{"className":16308},[7529],[805,16310,16312,16320],{"className":16311,"style":10813},[7533],[805,16313,16314,16317],{"style":10062},[805,16315],{"className":16316,"style":10066},[7541],[805,16318,828],{"className":16319,"style":7582},[7512,7516],[805,16321,16322,16325],{"style":10062},[805,16323],{"className":16324,"style":10066},[7541],[805,16326,16328],{"className":16327,"style":11115},[10116],[805,16329,9911],{"className":16330},[7512],[805,16332,7557],{"className":16333},[7556],[805,16335,16337],{"className":16336},[7529],[805,16338,16340],{"className":16339,"style":10130},[7533],[805,16341],{},[805,16343],{"className":16344,"style":8028},[7574],[805,16346,2823],{"className":16347},[8032],[805,16349],{"className":16350,"style":8028},[7574],[805,16352,16354,16357,16360],{"className":16353},[7498],[805,16355],{"className":16356,"style":16038},[7502],[805,16358,16064],{"className":16359},[7512],[805,16361,7416],{"className":16362},[7512,7516],[11,16364,16365],{},"We can now make predictions for new size values. For example, for a 120 m² house, the model predicts a price of:",[805,16367,16369],{"className":16368},[7980],[805,16370,16372,16409],{"className":16371},[7389],[805,16373,16375],{"className":16374},[7393],[7395,16376,16377],{"xmlns":7397,"display":7989},[7399,16378,16379,16406],{},[7402,16380,16381,16387,16389,16391,16393,16396,16398,16401,16403],{},[9901,16382,16383,16385],{"accent":7423},[7414,16384,828],{},[7405,16386,9911],{},[7405,16388,2823],{},[7418,16390,16064],{},[7405,16392,15018],{},[7418,16394,16395],{},"120",[7405,16397,2823],{},[7418,16399,16400],{},"240",[7405,16402,3096],{"separator":7423},[7418,16404,16405],{},"000",[7487,16407,16408],{"encoding":7489},"\\hat{y} = 2000 * 120 = 240,000",[805,16410,16412,16469,16487,16505],{"className":16411,"ariaHidden":7423},[7494],[805,16413,16415,16418,16460,16463,16466],{"className":16414},[7498],[805,16416],{"className":16417,"style":8020},[7502],[805,16419,16421],{"className":16420},[7512,10049],[805,16422,16424,16452],{"className":16423},[7524,7525],[805,16425,16427,16449],{"className":16426},[7529],[805,16428,16430,16438],{"className":16429,"style":10813},[7533],[805,16431,16432,16435],{"style":10062},[805,16433],{"className":16434,"style":10066},[7541],[805,16436,828],{"className":16437,"style":7582},[7512,7516],[805,16439,16440,16443],{"style":10062},[805,16441],{"className":16442,"style":10066},[7541],[805,16444,16446],{"className":16445,"style":11115},[10116],[805,16447,9911],{"className":16448},[7512],[805,16450,7557],{"className":16451},[7556],[805,16453,16455],{"className":16454},[7529],[805,16456,16458],{"className":16457,"style":10130},[7533],[805,16459],{},[805,16461],{"className":16462,"style":8028},[7574],[805,16464,2823],{"className":16465},[8032],[805,16467],{"className":16468,"style":8028},[7574],[805,16470,16472,16475,16478,16481,16484],{"className":16471},[7498],[805,16473],{"className":16474,"style":16038},[7502],[805,16476,16064],{"className":16477},[7512],[805,16479],{"className":16480,"style":8065},[7574],[805,16482,15018],{"className":16483},[8527],[805,16485],{"className":16486,"style":8065},[7574],[805,16488,16490,16493,16496,16499,16502],{"className":16489},[7498],[805,16491],{"className":16492,"style":16038},[7502],[805,16494,16395],{"className":16495},[7512],[805,16497],{"className":16498,"style":8028},[7574],[805,16500,2823],{"className":16501},[8032],[805,16503],{"className":16504,"style":8028},[7574],[805,16506,16508,16512,16515,16518,16521],{"className":16507},[7498],[805,16509],{"className":16510,"style":16511},[7502],"height:0.8389em;vertical-align:-0.1944em;",[805,16513,16400],{"className":16514},[7512],[805,16516,3096],{"className":16517},[7570],[805,16519],{"className":16520,"style":7575},[7574],[805,16522,16405],{"className":16523},[7512],[11,16525,16526,16527,16530],{},"Our calculations have yielded the best coefficients that ",[29,16528,16529],{},"minimize"," the mean squared error (but do not eliminate it completely). To measure the error of our model, we can calculate the MSE using the formula mentioned earlier:",[805,16532,16534],{"className":16533},[7980],[805,16535,16537,16602],{"className":16536},[7389],[805,16538,16540],{"className":16539},[7393],[7395,16541,16542],{"xmlns":7397,"display":7989},[7399,16543,16544,16600],{},[7402,16545,16546,16548,16550,16552,16554,16560,16574,16576,16582,16584,16594],{},[7414,16547,10494],{},[7414,16549,10497],{},[7414,16551,10500],{},[7405,16553,2823],{},[10504,16555,16556,16558],{},[7418,16557,7420],{},[7414,16559,7472],{},[10511,16561,16562,16564,16572],{},[7405,16563,10515],{},[7402,16565,16566,16568,16570],{},[7414,16567,7849],{},[7405,16569,2823],{},[7418,16571,7420],{},[7414,16573,7472],{},[7405,16575,2651],{"stretchy":1435},[7411,16577,16578,16580],{},[7414,16579,828],{},[7414,16581,7849],{},[7405,16583,9899],{},[9901,16585,16586,16592],{"accent":7423},[7411,16587,16588,16590],{},[7414,16589,828],{},[7414,16591,7849],{},[7405,16593,9911],{},[10547,16595,16596,16598],{},[7405,16597,3993],{"stretchy":1435},[7418,16599,7442],{},[7487,16601,10555],{"encoding":7489},[805,16603,16605,16629,16819],{"className":16604,"ariaHidden":7423},[7494],[805,16606,16608,16611,16614,16617,16620,16623,16626],{"className":16607},[7498],[805,16609],{"className":16610,"style":8042},[7502],[805,16612,10494],{"className":16613,"style":10568},[7512,7516],[805,16615,10497],{"className":16616,"style":10572},[7512,7516],[805,16618,10500],{"className":16619,"style":10572},[7512,7516],[805,16621],{"className":16622,"style":8028},[7574],[805,16624,2823],{"className":16625},[8032],[805,16627],{"className":16628,"style":8028},[7574],[805,16630,16632,16635,16697,16700,16767,16770,16810,16813,16816],{"className":16631},[7498],[805,16633],{"className":16634,"style":10591},[7502],[805,16636,16638,16641,16694],{"className":16637},[7512],[805,16639],{"className":16640},[7507,10598],[805,16642,16644],{"className":16643},[10504],[805,16645,16647,16686],{"className":16646},[7524,7525],[805,16648,16650,16683],{"className":16649},[7529],[805,16651,16653,16664,16672],{"className":16652,"style":10611},[7533],[805,16654,16655,16658],{"style":10614},[805,16656],{"className":16657,"style":10066},[7541],[805,16659,16661],{"className":16660},[7512],[805,16662,7472],{"className":16663},[7512,7516],[805,16665,16666,16669],{"style":10626},[805,16667],{"className":16668,"style":10066},[7541],[805,16670],{"className":16671,"style":10634},[10633],[805,16673,16674,16677],{"style":10637},[805,16675],{"className":16676,"style":10066},[7541],[805,16678,16680],{"className":16679},[7512],[805,16681,7420],{"className":16682},[7512],[805,16684,7557],{"className":16685},[7556],[805,16687,16689],{"className":16688},[7529],[805,16690,16692],{"className":16691,"style":10656},[7533],[805,16693],{},[805,16695],{"className":16696},[7621,10598],[805,16698],{"className":16699,"style":7575},[7574],[805,16701,16703],{"className":16702},[10668,10669],[805,16704,16706,16759],{"className":16705},[7524,7525],[805,16707,16709,16756],{"className":16708},[7529],[805,16710,16712,16732,16742],{"className":16711,"style":10679},[7533],[805,16713,16714,16717],{"style":10682},[805,16715],{"className":16716,"style":10686},[7541],[805,16718,16720],{"className":16719},[7546,7547,7548,7549],[805,16721,16723,16726,16729],{"className":16722},[7512,7549],[805,16724,7849],{"className":16725},[7512,7516,7549],[805,16727,2823],{"className":16728},[8032,7549],[805,16730,7420],{"className":16731},[7512,7549],[805,16733,16734,16737],{"style":10704},[805,16735],{"className":16736,"style":10686},[7541],[805,16738,16739],{},[805,16740,10515],{"className":16741},[10668,10713,10714],[805,16743,16744,16747],{"style":10717},[805,16745],{"className":16746,"style":10686},[7541],[805,16748,16750],{"className":16749},[7546,7547,7548,7549],[805,16751,16753],{"className":16752},[7512,7549],[805,16754,7472],{"className":16755},[7512,7516,7549],[805,16757,7557],{"className":16758},[7556],[805,16760,16762],{"className":16761},[7529],[805,16763,16765],{"className":16764,"style":10739},[7533],[805,16766],{},[805,16768,2651],{"className":16769},[7507],[805,16771,16773,16776],{"className":16772},[7512],[805,16774,828],{"className":16775,"style":7582},[7512,7516],[805,16777,16779],{"className":16778},[7520],[805,16780,16782,16802],{"className":16781},[7524,7525],[805,16783,16785,16799],{"className":16784},[7529],[805,16786,16788],{"className":16787,"style":7881},[7533],[805,16789,16790,16793],{"style":7597},[805,16791],{"className":16792,"style":7542},[7541],[805,16794,16796],{"className":16795},[7546,7547,7548,7549],[805,16797,7849],{"className":16798},[7512,7516,7549],[805,16800,7557],{"className":16801},[7556],[805,16803,16805],{"className":16804},[7529],[805,16806,16808],{"className":16807,"style":7564},[7533],[805,16809],{},[805,16811],{"className":16812,"style":8065},[7574],[805,16814,9899],{"className":16815},[8527],[805,16817],{"className":16818,"style":8065},[7574],[805,16820,16822,16825,16904],{"className":16821},[7498],[805,16823],{"className":16824,"style":10800},[7502],[805,16826,16828],{"className":16827},[7512,10049],[805,16829,16831,16896],{"className":16830},[7524,7525],[805,16832,16834,16893],{"className":16833},[7529],[805,16835,16837,16882],{"className":16836,"style":10813},[7533],[805,16838,16839,16842],{"style":10062},[805,16840],{"className":16841,"style":10066},[7541],[805,16843,16845,16848],{"className":16844},[7512],[805,16846,828],{"className":16847,"style":7582},[7512,7516],[805,16849,16851],{"className":16850},[7520],[805,16852,16854,16874],{"className":16853},[7524,7525],[805,16855,16857,16871],{"className":16856},[7529],[805,16858,16860],{"className":16859,"style":7881},[7533],[805,16861,16862,16865],{"style":7597},[805,16863],{"className":16864,"style":7542},[7541],[805,16866,16868],{"className":16867},[7546,7547,7548,7549],[805,16869,7849],{"className":16870},[7512,7516,7549],[805,16872,7557],{"className":16873},[7556],[805,16875,16877],{"className":16876},[7529],[805,16878,16880],{"className":16879,"style":7564},[7533],[805,16881],{},[805,16883,16884,16887],{"style":10062},[805,16885],{"className":16886,"style":10066},[7541],[805,16888,16890],{"className":16889,"style":10117},[10116],[805,16891,9911],{"className":16892},[7512],[805,16894,7557],{"className":16895},[7556],[805,16897,16899],{"className":16898},[7529],[805,16900,16902],{"className":16901,"style":10130},[7533],[805,16903],{},[805,16905,16907,16910],{"className":16906},[7621],[805,16908,3993],{"className":16909},[7621],[805,16911,16913],{"className":16912},[7520],[805,16914,16916],{"className":16915},[7524],[805,16917,16919],{"className":16918},[7529],[805,16920,16922],{"className":16921,"style":10899},[7533],[805,16923,16924,16927],{"style":10902},[805,16925],{"className":16926,"style":7542},[7541],[805,16928,16930],{"className":16929},[7546,7547,7548,7549],[805,16931,7442],{"className":16932},[7512,7549],[11,16934,8849],{},[49,16936,16937,16967,17039],{},[52,16938,16939,14703],{},[805,16940,16942,16955],{"className":16941},[7389],[805,16943,16945],{"className":16944},[7393],[7395,16946,16947],{"xmlns":7397},[7399,16948,16949,16953],{},[7402,16950,16951],{},[7414,16952,7472],{},[7487,16954,7472],{"encoding":7489},[805,16956,16958],{"className":16957,"ariaHidden":7423},[7494],[805,16959,16961,16964],{"className":16960},[7498],[805,16962],{"className":16963,"style":8843},[7502],[805,16965,7472],{"className":16966},[7512,7516],[52,16968,16969,17038],{},[805,16970,16972,16989],{"className":16971},[7389],[805,16973,16975],{"className":16974},[7393],[7395,16976,16977],{"xmlns":7397},[7399,16978,16979,16987],{},[7402,16980,16981],{},[7411,16982,16983,16985],{},[7414,16984,828],{},[7414,16986,7849],{},[7487,16988,7925],{"encoding":7489},[805,16990,16992],{"className":16991,"ariaHidden":7423},[7494],[805,16993,16995,16998],{"className":16994},[7498],[805,16996],{"className":16997,"style":7935},[7502],[805,16999,17001,17004],{"className":17000},[7512],[805,17002,828],{"className":17003,"style":7582},[7512,7516],[805,17005,17007],{"className":17006},[7520],[805,17008,17010,17030],{"className":17009},[7524,7525],[805,17011,17013,17027],{"className":17012},[7529],[805,17014,17016],{"className":17015,"style":7881},[7533],[805,17017,17018,17021],{"style":7597},[805,17019],{"className":17020,"style":7542},[7541],[805,17022,17024],{"className":17023},[7546,7547,7548,7549],[805,17025,7849],{"className":17026},[7512,7516,7549],[805,17028,7557],{"className":17029},[7556],[805,17031,17033],{"className":17032},[7529],[805,17034,17036],{"className":17035,"style":7564},[7533],[805,17037],{}," are the real values of price for each example.",[52,17040,17041,17153],{},[805,17042,17044,17065],{"className":17043},[7389],[805,17045,17047],{"className":17046},[7393],[7395,17048,17049],{"xmlns":7397},[7399,17050,17051,17063],{},[7402,17052,17053],{},[7411,17054,17055,17061],{},[9901,17056,17057,17059],{"accent":7423},[7414,17058,828],{},[7405,17060,9911],{},[7414,17062,7849],{},[7487,17064,11074],{"encoding":7489},[805,17066,17068],{"className":17067,"ariaHidden":7423},[7494],[805,17069,17071,17074],{"className":17070},[7498],[805,17072],{"className":17073,"style":8020},[7502],[805,17075,17077,17119],{"className":17076},[7512],[805,17078,17080],{"className":17079},[7512,10049],[805,17081,17083,17111],{"className":17082},[7524,7525],[805,17084,17086,17108],{"className":17085},[7529],[805,17087,17089,17097],{"className":17088,"style":10813},[7533],[805,17090,17091,17094],{"style":10062},[805,17092],{"className":17093,"style":10066},[7541],[805,17095,828],{"className":17096,"style":7582},[7512,7516],[805,17098,17099,17102],{"style":10062},[805,17100],{"className":17101,"style":10066},[7541],[805,17103,17105],{"className":17104,"style":11115},[10116],[805,17106,9911],{"className":17107},[7512],[805,17109,7557],{"className":17110},[7556],[805,17112,17114],{"className":17113},[7529],[805,17115,17117],{"className":17116,"style":10130},[7533],[805,17118],{},[805,17120,17122],{"className":17121},[7520],[805,17123,17125,17145],{"className":17124},[7524,7525],[805,17126,17128,17142],{"className":17127},[7529],[805,17129,17131],{"className":17130,"style":7881},[7533],[805,17132,17133,17136],{"style":7597},[805,17134],{"className":17135,"style":7542},[7541],[805,17137,17139],{"className":17138},[7546,7547,7548,7549],[805,17140,7849],{"className":17141},[7512,7516,7549],[805,17143,7557],{"className":17144},[7556],[805,17146,17148],{"className":17147},[7529],[805,17149,17151],{"className":17150,"style":7564},[7533],[805,17152],{}," are the model predictions for each example, which are calculated using the linear regression model.",[317,17155,17156],{},[52,17157,17158],{},"We calculate the predictions for each example:",[49,17160,17161,17356,17552],{},[52,17162,17163,17164],{},"For 50 m²: ",[805,17165,17167,17205],{"className":17166},[7389],[805,17168,17170],{"className":17169},[7393],[7395,17171,17172],{"xmlns":7397},[7399,17173,17174,17202],{},[7402,17175,17176,17186,17188,17190,17192,17194,17196,17198,17200],{},[9901,17177,17178,17184],{"accent":7423},[7411,17179,17180,17182],{},[7414,17181,828],{},[7418,17183,7420],{},[7405,17185,9911],{},[7405,17187,2823],{},[7418,17189,16064],{},[7405,17191,15018],{},[7418,17193,13262],{},[7405,17195,2823],{},[7418,17197,13270],{},[7405,17199,3096],{"separator":7423},[7418,17201,16405],{},[7487,17203,17204],{"encoding":7489},"\\hat{y_1} = 2000 * 50 = 100,000",[805,17206,17208,17302,17320,17338],{"className":17207,"ariaHidden":7423},[7494],[805,17209,17211,17214,17293,17296,17299],{"className":17210},[7498],[805,17212],{"className":17213,"style":8020},[7502],[805,17215,17217],{"className":17216},[7512,10049],[805,17218,17220,17285],{"className":17219},[7524,7525],[805,17221,17223,17282],{"className":17222},[7529],[805,17224,17226,17271],{"className":17225,"style":10813},[7533],[805,17227,17228,17231],{"style":10062},[805,17229],{"className":17230,"style":10066},[7541],[805,17232,17234,17237],{"className":17233},[7512],[805,17235,828],{"className":17236,"style":7582},[7512,7516],[805,17238,17240],{"className":17239},[7520],[805,17241,17243,17263],{"className":17242},[7524,7525],[805,17244,17246,17260],{"className":17245},[7529],[805,17247,17249],{"className":17248,"style":7534},[7533],[805,17250,17251,17254],{"style":7597},[805,17252],{"className":17253,"style":7542},[7541],[805,17255,17257],{"className":17256},[7546,7547,7548,7549],[805,17258,7420],{"className":17259},[7512,7549],[805,17261,7557],{"className":17262},[7556],[805,17264,17266],{"className":17265},[7529],[805,17267,17269],{"className":17268,"style":7564},[7533],[805,17270],{},[805,17272,17273,17276],{"style":10062},[805,17274],{"className":17275,"style":10066},[7541],[805,17277,17279],{"className":17278,"style":10117},[10116],[805,17280,9911],{"className":17281},[7512],[805,17283,7557],{"className":17284},[7556],[805,17286,17288],{"className":17287},[7529],[805,17289,17291],{"className":17290,"style":10130},[7533],[805,17292],{},[805,17294],{"className":17295,"style":8028},[7574],[805,17297,2823],{"className":17298},[8032],[805,17300],{"className":17301,"style":8028},[7574],[805,17303,17305,17308,17311,17314,17317],{"className":17304},[7498],[805,17306],{"className":17307,"style":16038},[7502],[805,17309,16064],{"className":17310},[7512],[805,17312],{"className":17313,"style":8065},[7574],[805,17315,15018],{"className":17316},[8527],[805,17318],{"className":17319,"style":8065},[7574],[805,17321,17323,17326,17329,17332,17335],{"className":17322},[7498],[805,17324],{"className":17325,"style":16038},[7502],[805,17327,13262],{"className":17328},[7512],[805,17330],{"className":17331,"style":8028},[7574],[805,17333,2823],{"className":17334},[8032],[805,17336],{"className":17337,"style":8028},[7574],[805,17339,17341,17344,17347,17350,17353],{"className":17340},[7498],[805,17342],{"className":17343,"style":16511},[7502],[805,17345,13270],{"className":17346},[7512],[805,17348,3096],{"className":17349},[7570],[805,17351],{"className":17352,"style":7575},[7574],[805,17354,16405],{"className":17355},[7512],[52,17357,17358,17359],{},"For 100 m²: ",[805,17360,17362,17401],{"className":17361},[7389],[805,17363,17365],{"className":17364},[7393],[7395,17366,17367],{"xmlns":7397},[7399,17368,17369,17398],{},[7402,17370,17371,17381,17383,17385,17387,17389,17391,17394,17396],{},[9901,17372,17373,17379],{"accent":7423},[7411,17374,17375,17377],{},[7414,17376,828],{},[7418,17378,7442],{},[7405,17380,9911],{},[7405,17382,2823],{},[7418,17384,16064],{},[7405,17386,15018],{},[7418,17388,13270],{},[7405,17390,2823],{},[7418,17392,17393],{},"200",[7405,17395,3096],{"separator":7423},[7418,17397,16405],{},[7487,17399,17400],{"encoding":7489},"\\hat{y_2} = 2000 * 100 = 200,000",[805,17402,17404,17498,17516,17534],{"className":17403,"ariaHidden":7423},[7494],[805,17405,17407,17410,17489,17492,17495],{"className":17406},[7498],[805,17408],{"className":17409,"style":8020},[7502],[805,17411,17413],{"className":17412},[7512,10049],[805,17414,17416,17481],{"className":17415},[7524,7525],[805,17417,17419,17478],{"className":17418},[7529],[805,17420,17422,17467],{"className":17421,"style":10813},[7533],[805,17423,17424,17427],{"style":10062},[805,17425],{"className":17426,"style":10066},[7541],[805,17428,17430,17433],{"className":17429},[7512],[805,17431,828],{"className":17432,"style":7582},[7512,7516],[805,17434,17436],{"className":17435},[7520],[805,17437,17439,17459],{"className":17438},[7524,7525],[805,17440,17442,17456],{"className":17441},[7529],[805,17443,17445],{"className":17444,"style":7534},[7533],[805,17446,17447,17450],{"style":7597},[805,17448],{"className":17449,"style":7542},[7541],[805,17451,17453],{"className":17452},[7546,7547,7548,7549],[805,17454,7442],{"className":17455},[7512,7549],[805,17457,7557],{"className":17458},[7556],[805,17460,17462],{"className":17461},[7529],[805,17463,17465],{"className":17464,"style":7564},[7533],[805,17466],{},[805,17468,17469,17472],{"style":10062},[805,17470],{"className":17471,"style":10066},[7541],[805,17473,17475],{"className":17474,"style":10117},[10116],[805,17476,9911],{"className":17477},[7512],[805,17479,7557],{"className":17480},[7556],[805,17482,17484],{"className":17483},[7529],[805,17485,17487],{"className":17486,"style":10130},[7533],[805,17488],{},[805,17490],{"className":17491,"style":8028},[7574],[805,17493,2823],{"className":17494},[8032],[805,17496],{"className":17497,"style":8028},[7574],[805,17499,17501,17504,17507,17510,17513],{"className":17500},[7498],[805,17502],{"className":17503,"style":16038},[7502],[805,17505,16064],{"className":17506},[7512],[805,17508],{"className":17509,"style":8065},[7574],[805,17511,15018],{"className":17512},[8527],[805,17514],{"className":17515,"style":8065},[7574],[805,17517,17519,17522,17525,17528,17531],{"className":17518},[7498],[805,17520],{"className":17521,"style":16038},[7502],[805,17523,13270],{"className":17524},[7512],[805,17526],{"className":17527,"style":8028},[7574],[805,17529,2823],{"className":17530},[8032],[805,17532],{"className":17533,"style":8028},[7574],[805,17535,17537,17540,17543,17546,17549],{"className":17536},[7498],[805,17538],{"className":17539,"style":16511},[7502],[805,17541,17393],{"className":17542},[7512],[805,17544,3096],{"className":17545},[7570],[805,17547],{"className":17548,"style":7575},[7574],[805,17550,16405],{"className":17551},[7512],[52,17553,17554,17555],{},"For 150 m²: ",[805,17556,17558,17597],{"className":17557},[7389],[805,17559,17561],{"className":17560},[7393],[7395,17562,17563],{"xmlns":7397},[7399,17564,17565,17594],{},[7402,17566,17567,17577,17579,17581,17583,17585,17587,17590,17592],{},[9901,17568,17569,17575],{"accent":7423},[7411,17570,17571,17573],{},[7414,17572,828],{},[7418,17574,15011],{},[7405,17576,9911],{},[7405,17578,2823],{},[7418,17580,16064],{},[7405,17582,15018],{},[7418,17584,13278],{},[7405,17586,2823],{},[7418,17588,17589],{},"300",[7405,17591,3096],{"separator":7423},[7418,17593,16405],{},[7487,17595,17596],{"encoding":7489},"\\hat{y_3} = 2000 * 150 = 300,000",[805,17598,17600,17694,17712,17730],{"className":17599,"ariaHidden":7423},[7494],[805,17601,17603,17606,17685,17688,17691],{"className":17602},[7498],[805,17604],{"className":17605,"style":8020},[7502],[805,17607,17609],{"className":17608},[7512,10049],[805,17610,17612,17677],{"className":17611},[7524,7525],[805,17613,17615,17674],{"className":17614},[7529],[805,17616,17618,17663],{"className":17617,"style":10813},[7533],[805,17619,17620,17623],{"style":10062},[805,17621],{"className":17622,"style":10066},[7541],[805,17624,17626,17629],{"className":17625},[7512],[805,17627,828],{"className":17628,"style":7582},[7512,7516],[805,17630,17632],{"className":17631},[7520],[805,17633,17635,17655],{"className":17634},[7524,7525],[805,17636,17638,17652],{"className":17637},[7529],[805,17639,17641],{"className":17640,"style":7534},[7533],[805,17642,17643,17646],{"style":7597},[805,17644],{"className":17645,"style":7542},[7541],[805,17647,17649],{"className":17648},[7546,7547,7548,7549],[805,17650,15011],{"className":17651},[7512,7549],[805,17653,7557],{"className":17654},[7556],[805,17656,17658],{"className":17657},[7529],[805,17659,17661],{"className":17660,"style":7564},[7533],[805,17662],{},[805,17664,17665,17668],{"style":10062},[805,17666],{"className":17667,"style":10066},[7541],[805,17669,17671],{"className":17670,"style":10117},[10116],[805,17672,9911],{"className":17673},[7512],[805,17675,7557],{"className":17676},[7556],[805,17678,17680],{"className":17679},[7529],[805,17681,17683],{"className":17682,"style":10130},[7533],[805,17684],{},[805,17686],{"className":17687,"style":8028},[7574],[805,17689,2823],{"className":17690},[8032],[805,17692],{"className":17693,"style":8028},[7574],[805,17695,17697,17700,17703,17706,17709],{"className":17696},[7498],[805,17698],{"className":17699,"style":16038},[7502],[805,17701,16064],{"className":17702},[7512],[805,17704],{"className":17705,"style":8065},[7574],[805,17707,15018],{"className":17708},[8527],[805,17710],{"className":17711,"style":8065},[7574],[805,17713,17715,17718,17721,17724,17727],{"className":17714},[7498],[805,17716],{"className":17717,"style":16038},[7502],[805,17719,13278],{"className":17720},[7512],[805,17722],{"className":17723,"style":8028},[7574],[805,17725,2823],{"className":17726},[8032],[805,17728],{"className":17729,"style":8028},[7574],[805,17731,17733,17736,17739,17742,17745],{"className":17732},[7498],[805,17734],{"className":17735,"style":16511},[7502],[805,17737,17589],{"className":17738},[7512],[805,17740,3096],{"className":17741},[7570],[805,17743],{"className":17744,"style":7575},[7574],[805,17746,16405],{"className":17747},[7512],[317,17749,17750],{"start":670},[52,17751,17752],{},"We calculate the errors for each example:",[805,17754,17756],{"className":17755},[7980],[805,17757,17759,17798],{"className":17758},[7389],[805,17760,17762],{"className":17761},[7393],[7395,17763,17764],{"xmlns":7397,"display":7989},[7399,17765,17766,17795],{},[7402,17767,17768,17775,17777,17783,17785],{},[7411,17769,17770,17773],{},[7414,17771,17772],{},"e",[7414,17774,7849],{},[7405,17776,2823],{},[7411,17778,17779,17781],{},[7414,17780,828],{},[7414,17782,7849],{},[7405,17784,9899],{},[9901,17786,17787,17793],{"accent":7423},[7411,17788,17789,17791],{},[7414,17790,828],{},[7414,17792,7849],{},[7405,17794,9911],{},[7487,17796,17797],{"encoding":7489},"e_i = y_i - \\hat{y_i}",[805,17799,17801,17856,17911],{"className":17800,"ariaHidden":7423},[7494],[805,17802,17804,17807,17847,17850,17853],{"className":17803},[7498],[805,17805],{"className":17806,"style":7862},[7502],[805,17808,17810,17813],{"className":17809},[7512],[805,17811,17772],{"className":17812},[7512,7516],[805,17814,17816],{"className":17815},[7520],[805,17817,17819,17839],{"className":17818},[7524,7525],[805,17820,17822,17836],{"className":17821},[7529],[805,17823,17825],{"className":17824,"style":7881},[7533],[805,17826,17827,17830],{"style":7537},[805,17828],{"className":17829,"style":7542},[7541],[805,17831,17833],{"className":17832},[7546,7547,7548,7549],[805,17834,7849],{"className":17835},[7512,7516,7549],[805,17837,7557],{"className":17838},[7556],[805,17840,17842],{"className":17841},[7529],[805,17843,17845],{"className":17844,"style":7564},[7533],[805,17846],{},[805,17848],{"className":17849,"style":8028},[7574],[805,17851,2823],{"className":17852},[8032],[805,17854],{"className":17855,"style":8028},[7574],[805,17857,17859,17862,17902,17905,17908],{"className":17858},[7498],[805,17860],{"className":17861,"style":9987},[7502],[805,17863,17865,17868],{"className":17864},[7512],[805,17866,828],{"className":17867,"style":7582},[7512,7516],[805,17869,17871],{"className":17870},[7520],[805,17872,17874,17894],{"className":17873},[7524,7525],[805,17875,17877,17891],{"className":17876},[7529],[805,17878,17880],{"className":17879,"style":7881},[7533],[805,17881,17882,17885],{"style":7597},[805,17883],{"className":17884,"style":7542},[7541],[805,17886,17888],{"className":17887},[7546,7547,7548,7549],[805,17889,7849],{"className":17890},[7512,7516,7549],[805,17892,7557],{"className":17893},[7556],[805,17895,17897],{"className":17896},[7529],[805,17898,17900],{"className":17899,"style":7564},[7533],[805,17901],{},[805,17903],{"className":17904,"style":8065},[7574],[805,17906,9899],{"className":17907},[8527],[805,17909],{"className":17910,"style":8065},[7574],[805,17912,17914,17917],{"className":17913},[7498],[805,17915],{"className":17916,"style":8020},[7502],[805,17918,17920],{"className":17919},[7512,10049],[805,17921,17923,17988],{"className":17922},[7524,7525],[805,17924,17926,17985],{"className":17925},[7529],[805,17927,17929,17974],{"className":17928,"style":10813},[7533],[805,17930,17931,17934],{"style":10062},[805,17932],{"className":17933,"style":10066},[7541],[805,17935,17937,17940],{"className":17936},[7512],[805,17938,828],{"className":17939,"style":7582},[7512,7516],[805,17941,17943],{"className":17942},[7520],[805,17944,17946,17966],{"className":17945},[7524,7525],[805,17947,17949,17963],{"className":17948},[7529],[805,17950,17952],{"className":17951,"style":7881},[7533],[805,17953,17954,17957],{"style":7597},[805,17955],{"className":17956,"style":7542},[7541],[805,17958,17960],{"className":17959},[7546,7547,7548,7549],[805,17961,7849],{"className":17962},[7512,7516,7549],[805,17964,7557],{"className":17965},[7556],[805,17967,17969],{"className":17968},[7529],[805,17970,17972],{"className":17971,"style":7564},[7533],[805,17973],{},[805,17975,17976,17979],{"style":10062},[805,17977],{"className":17978,"style":10066},[7541],[805,17980,17982],{"className":17981,"style":10117},[10116],[805,17983,9911],{"className":17984},[7512],[805,17986,7557],{"className":17987},[7556],[805,17989,17991],{"className":17990},[7529],[805,17992,17994],{"className":17993,"style":10130},[7533],[805,17995],{},[49,17997,17998,18137,18276],{},[52,17999,18000,18001],{},"Error for 50 m²: ",[805,18002,18004,18034],{"className":18003},[7389],[805,18005,18007],{"className":18006},[7393],[7395,18008,18009],{"xmlns":7397},[7399,18010,18011,18031],{},[7402,18012,18013,18019,18021,18023,18025,18027,18029],{},[7411,18014,18015,18017],{},[7414,18016,17772],{},[7418,18018,7420],{},[7405,18020,2823],{},[7418,18022,15021],{},[7405,18024,9899],{},[7418,18026,15021],{},[7405,18028,2823],{},[7418,18030,8392],{},[7487,18032,18033],{"encoding":7489},"e_1 = 100000 - 100000 = 0",[805,18035,18037,18092,18110,18128],{"className":18036,"ariaHidden":7423},[7494],[805,18038,18040,18043,18083,18086,18089],{"className":18039},[7498],[805,18041],{"className":18042,"style":7862},[7502],[805,18044,18046,18049],{"className":18045},[7512],[805,18047,17772],{"className":18048},[7512,7516],[805,18050,18052],{"className":18051},[7520],[805,18053,18055,18075],{"className":18054},[7524,7525],[805,18056,18058,18072],{"className":18057},[7529],[805,18059,18061],{"className":18060,"style":7534},[7533],[805,18062,18063,18066],{"style":7537},[805,18064],{"className":18065,"style":7542},[7541],[805,18067,18069],{"className":18068},[7546,7547,7548,7549],[805,18070,7420],{"className":18071},[7512,7549],[805,18073,7557],{"className":18074},[7556],[805,18076,18078],{"className":18077},[7529],[805,18079,18081],{"className":18080,"style":7564},[7533],[805,18082],{},[805,18084],{"className":18085,"style":8028},[7574],[805,18087,2823],{"className":18088},[8032],[805,18090],{"className":18091,"style":8028},[7574],[805,18093,18095,18098,18101,18104,18107],{"className":18094},[7498],[805,18096],{"className":18097,"style":16236},[7502],[805,18099,15021],{"className":18100},[7512],[805,18102],{"className":18103,"style":8065},[7574],[805,18105,9899],{"className":18106},[8527],[805,18108],{"className":18109,"style":8065},[7574],[805,18111,18113,18116,18119,18122,18125],{"className":18112},[7498],[805,18114],{"className":18115,"style":16038},[7502],[805,18117,15021],{"className":18118},[7512],[805,18120],{"className":18121,"style":8028},[7574],[805,18123,2823],{"className":18124},[8032],[805,18126],{"className":18127,"style":8028},[7574],[805,18129,18131,18134],{"className":18130},[7498],[805,18132],{"className":18133,"style":16038},[7502],[805,18135,8392],{"className":18136},[7512],[52,18138,18139,18140],{},"Error for 100 m²: ",[805,18141,18143,18173],{"className":18142},[7389],[805,18144,18146],{"className":18145},[7393],[7395,18147,18148],{"xmlns":7397},[7399,18149,18150,18170],{},[7402,18151,18152,18158,18160,18162,18164,18166,18168],{},[7411,18153,18154,18156],{},[7414,18155,17772],{},[7418,18157,7442],{},[7405,18159,2823],{},[7418,18161,15030],{},[7405,18163,9899],{},[7418,18165,15030],{},[7405,18167,2823],{},[7418,18169,8392],{},[7487,18171,18172],{"encoding":7489},"e_2 = 200000 - 200000 = 0",[805,18174,18176,18231,18249,18267],{"className":18175,"ariaHidden":7423},[7494],[805,18177,18179,18182,18222,18225,18228],{"className":18178},[7498],[805,18180],{"className":18181,"style":7862},[7502],[805,18183,18185,18188],{"className":18184},[7512],[805,18186,17772],{"className":18187},[7512,7516],[805,18189,18191],{"className":18190},[7520],[805,18192,18194,18214],{"className":18193},[7524,7525],[805,18195,18197,18211],{"className":18196},[7529],[805,18198,18200],{"className":18199,"style":7534},[7533],[805,18201,18202,18205],{"style":7537},[805,18203],{"className":18204,"style":7542},[7541],[805,18206,18208],{"className":18207},[7546,7547,7548,7549],[805,18209,7442],{"className":18210},[7512,7549],[805,18212,7557],{"className":18213},[7556],[805,18215,18217],{"className":18216},[7529],[805,18218,18220],{"className":18219,"style":7564},[7533],[805,18221],{},[805,18223],{"className":18224,"style":8028},[7574],[805,18226,2823],{"className":18227},[8032],[805,18229],{"className":18230,"style":8028},[7574],[805,18232,18234,18237,18240,18243,18246],{"className":18233},[7498],[805,18235],{"className":18236,"style":16236},[7502],[805,18238,15030],{"className":18239},[7512],[805,18241],{"className":18242,"style":8065},[7574],[805,18244,9899],{"className":18245},[8527],[805,18247],{"className":18248,"style":8065},[7574],[805,18250,18252,18255,18258,18261,18264],{"className":18251},[7498],[805,18253],{"className":18254,"style":16038},[7502],[805,18256,15030],{"className":18257},[7512],[805,18259],{"className":18260,"style":8028},[7574],[805,18262,2823],{"className":18263},[8032],[805,18265],{"className":18266,"style":8028},[7574],[805,18268,18270,18273],{"className":18269},[7498],[805,18271],{"className":18272,"style":16038},[7502],[805,18274,8392],{"className":18275},[7512],[52,18277,18278,18279],{},"Error for 150 m²: ",[805,18280,18282,18312],{"className":18281},[7389],[805,18283,18285],{"className":18284},[7393],[7395,18286,18287],{"xmlns":7397},[7399,18288,18289,18309],{},[7402,18290,18291,18297,18299,18301,18303,18305,18307],{},[7411,18292,18293,18295],{},[7414,18294,17772],{},[7418,18296,15011],{},[7405,18298,2823],{},[7418,18300,15039],{},[7405,18302,9899],{},[7418,18304,15039],{},[7405,18306,2823],{},[7418,18308,8392],{},[7487,18310,18311],{"encoding":7489},"e_3 = 300000 - 300000 = 0",[805,18313,18315,18370,18388,18406],{"className":18314,"ariaHidden":7423},[7494],[805,18316,18318,18321,18361,18364,18367],{"className":18317},[7498],[805,18319],{"className":18320,"style":7862},[7502],[805,18322,18324,18327],{"className":18323},[7512],[805,18325,17772],{"className":18326},[7512,7516],[805,18328,18330],{"className":18329},[7520],[805,18331,18333,18353],{"className":18332},[7524,7525],[805,18334,18336,18350],{"className":18335},[7529],[805,18337,18339],{"className":18338,"style":7534},[7533],[805,18340,18341,18344],{"style":7537},[805,18342],{"className":18343,"style":7542},[7541],[805,18345,18347],{"className":18346},[7546,7547,7548,7549],[805,18348,15011],{"className":18349},[7512,7549],[805,18351,7557],{"className":18352},[7556],[805,18354,18356],{"className":18355},[7529],[805,18357,18359],{"className":18358,"style":7564},[7533],[805,18360],{},[805,18362],{"className":18363,"style":8028},[7574],[805,18365,2823],{"className":18366},[8032],[805,18368],{"className":18369,"style":8028},[7574],[805,18371,18373,18376,18379,18382,18385],{"className":18372},[7498],[805,18374],{"className":18375,"style":16236},[7502],[805,18377,15039],{"className":18378},[7512],[805,18380],{"className":18381,"style":8065},[7574],[805,18383,9899],{"className":18384},[8527],[805,18386],{"className":18387,"style":8065},[7574],[805,18389,18391,18394,18397,18400,18403],{"className":18390},[7498],[805,18392],{"className":18393,"style":16038},[7502],[805,18395,15039],{"className":18396},[7512],[805,18398],{"className":18399,"style":8028},[7574],[805,18401,2823],{"className":18402},[8032],[805,18404],{"className":18405,"style":8028},[7574],[805,18407,18409,18412],{"className":18408},[7498],[805,18410],{"className":18411,"style":16038},[7502],[805,18413,8392],{"className":18414},[7512],[317,18416,18417],{"start":667},[52,18418,18419],{},"We square the errors and average them to obtain the MSE:",[805,18421,18423],{"className":18422},[7980],[805,18424,18426,18482],{"className":18425},[7389],[805,18427,18429],{"className":18428},[7393],[7395,18430,18431],{"xmlns":7397,"display":7989},[7399,18432,18433,18479],{},[7402,18434,18435,18437,18439,18441,18443,18449,18451,18457,18459,18465,18467,18473,18475,18477],{},[7414,18436,10494],{},[7414,18438,10497],{},[7414,18440,10500],{},[7405,18442,2823],{},[10504,18444,18445,18447],{},[7418,18446,7420],{},[7418,18448,15011],{},[7405,18450,2651],{"stretchy":1435},[10547,18452,18453,18455],{},[7418,18454,8392],{},[7418,18456,7442],{},[7405,18458,8395],{},[10547,18460,18461,18463],{},[7418,18462,8392],{},[7418,18464,7442],{},[7405,18466,8395],{},[10547,18468,18469,18471],{},[7418,18470,8392],{},[7418,18472,7442],{},[7405,18474,3993],{"stretchy":1435},[7405,18476,2823],{},[7418,18478,8392],{},[7487,18480,18481],{"encoding":7489},"MSE = \\frac{1}{3} (0^2 + 0^2 + 0^2) = 0",[805,18483,18485,18509,18618,18663,18710],{"className":18484,"ariaHidden":7423},[7494],[805,18486,18488,18491,18494,18497,18500,18503,18506],{"className":18487},[7498],[805,18489],{"className":18490,"style":8042},[7502],[805,18492,10494],{"className":18493,"style":10568},[7512,7516],[805,18495,10497],{"className":18496,"style":10572},[7512,7516],[805,18498,10500],{"className":18499,"style":10572},[7512,7516],[805,18501],{"className":18502,"style":8028},[7574],[805,18504,2823],{"className":18505},[8032],[805,18507],{"className":18508,"style":8028},[7574],[805,18510,18512,18515,18577,18580,18609,18612,18615],{"className":18511},[7498],[805,18513],{"className":18514,"style":15718},[7502],[805,18516,18518,18521,18574],{"className":18517},[7512],[805,18519],{"className":18520},[7507,10598],[805,18522,18524],{"className":18523},[10504],[805,18525,18527,18566],{"className":18526},[7524,7525],[805,18528,18530,18563],{"className":18529},[7529],[805,18531,18533,18544,18552],{"className":18532,"style":10611},[7533],[805,18534,18535,18538],{"style":10614},[805,18536],{"className":18537,"style":10066},[7541],[805,18539,18541],{"className":18540},[7512],[805,18542,15011],{"className":18543},[7512],[805,18545,18546,18549],{"style":10626},[805,18547],{"className":18548,"style":10066},[7541],[805,18550],{"className":18551,"style":10634},[10633],[805,18553,18554,18557],{"style":10637},[805,18555],{"className":18556,"style":10066},[7541],[805,18558,18560],{"className":18559},[7512],[805,18561,7420],{"className":18562},[7512],[805,18564,7557],{"className":18565},[7556],[805,18567,18569],{"className":18568},[7529],[805,18570,18572],{"className":18571,"style":10656},[7533],[805,18573],{},[805,18575],{"className":18576},[7621,10598],[805,18578,2651],{"className":18579},[7507],[805,18581,18583,18586],{"className":18582},[7512],[805,18584,8392],{"className":18585},[7512],[805,18587,18589],{"className":18588},[7520],[805,18590,18592],{"className":18591},[7524],[805,18593,18595],{"className":18594},[7529],[805,18596,18598],{"className":18597,"style":10899},[7533],[805,18599,18600,18603],{"style":10902},[805,18601],{"className":18602,"style":7542},[7541],[805,18604,18606],{"className":18605},[7546,7547,7548,7549],[805,18607,7442],{"className":18608},[7512,7549],[805,18610],{"className":18611,"style":8065},[7574],[805,18613,8395],{"className":18614},[8527],[805,18616],{"className":18617,"style":8065},[7574],[805,18619,18621,18625,18654,18657,18660],{"className":18620},[7498],[805,18622],{"className":18623,"style":18624},[7502],"height:0.9474em;vertical-align:-0.0833em;",[805,18626,18628,18631],{"className":18627},[7512],[805,18629,8392],{"className":18630},[7512],[805,18632,18634],{"className":18633},[7520],[805,18635,18637],{"className":18636},[7524],[805,18638,18640],{"className":18639},[7529],[805,18641,18643],{"className":18642,"style":10899},[7533],[805,18644,18645,18648],{"style":10902},[805,18646],{"className":18647,"style":7542},[7541],[805,18649,18651],{"className":18650},[7546,7547,7548,7549],[805,18652,7442],{"className":18653},[7512,7549],[805,18655],{"className":18656,"style":8065},[7574],[805,18658,8395],{"className":18659},[8527],[805,18661],{"className":18662,"style":8065},[7574],[805,18664,18666,18669,18698,18701,18704,18707],{"className":18665},[7498],[805,18667],{"className":18668,"style":10800},[7502],[805,18670,18672,18675],{"className":18671},[7512],[805,18673,8392],{"className":18674},[7512],[805,18676,18678],{"className":18677},[7520],[805,18679,18681],{"className":18680},[7524],[805,18682,18684],{"className":18683},[7529],[805,18685,18687],{"className":18686,"style":10899},[7533],[805,18688,18689,18692],{"style":10902},[805,18690],{"className":18691,"style":7542},[7541],[805,18693,18695],{"className":18694},[7546,7547,7548,7549],[805,18696,7442],{"className":18697},[7512,7549],[805,18699,3993],{"className":18700},[7621],[805,18702],{"className":18703,"style":8028},[7574],[805,18705,2823],{"className":18706},[8032],[805,18708],{"className":18709,"style":8028},[7574],[805,18711,18713,18716],{"className":18712},[7498],[805,18714],{"className":18715,"style":16038},[7502],[805,18717,8392],{"className":18718},[7512],[11,18720,18721,18722,18725,18726,18754],{},"For this case, an MSE of 0 indicates that the model predicts the real values perfectly for ",[29,18723,18724],{},"this training dataset",". Of course, in practice, real data will contain noise and will be even more variable, so the epsilon ",[805,18727,18729,18742],{"className":18728},[7389],[805,18730,18732],{"className":18731},[7393],[7395,18733,18734],{"xmlns":7397},[7399,18735,18736,18740],{},[7402,18737,18738],{},[7414,18739,8448],{},[7487,18741,9429],{"encoding":7489},[805,18743,18745],{"className":18744,"ariaHidden":7423},[7494],[805,18746,18748,18751],{"className":18747},[7498],[805,18749],{"className":18750,"style":8843},[7502],[805,18752,8448],{"className":18753},[7512,7516]," will not be zero.",[11,18756,18757,18758],{},"To explore more with regression, you can use this Google Colab that contains a complete example of linear regression with Python: ",[88,18759,18762],{"href":18760,"target":280,"rel":18761},"https:\u002F\u002Fcolab.research.google.com\u002Fdrive\u002F1yi8-fVw2Ak7pqYOzZsiT7NO_zccZrQir?usp=sharing",[282,283],"linear_regression",[18,18764],{},[1442,18766,266],{"id":18767},"classification",[11,18769,18770],{},"A classification problem aims to predict a categorical output variable based on a set of input variables. For example, predicting whether an email is spam or not spam based on its content, or determining whether an image contains a cat or a dog.",[11,18772,18773],{},"There are three main types of classification:",[49,18775,18776,18782,18788],{},[52,18777,18778,18781],{},[29,18779,18780],{},"Binary Classification",": When there are two possible classes. For example, classifying whether a patient has a disease (yes\u002Fno).",[52,18783,18784,18787],{},[29,18785,18786],{},"Multi-class Classification",": When there are more than two possible classes. For example, classifying the type of flower (could be setosa, versicolor or virginica) based on its characteristics.",[52,18789,18790,18793],{},[29,18791,18792],{},"Multi-label Classification",": When each example can belong to multiple classes, such as classifying the labels of a news article (politics, economy, sports) where an article can belong to several categories.",[78,18795,18796],{},[11,18797,18798],{},"The difference between multi-class and multi-label classification is that in the former each example can only belong to one class, while in the latter an example can belong to multiple classes simultaneously.",[11,18800,18801],{},"Let's look a bit more at binary classification. In this case, the objective is to find a function that maps the inputs to one of the two possible classes.",[11,18803,18804,18805,18808,18809,18923],{},"The question that the binary classification model attempts to answer is: ",[29,18806,18807],{},"What is the probability that an example belongs to class 1 given a set of features?"," This can be expressed mathematically as:\n",[805,18810,18812,18852],{"className":18811},[7389],[805,18813,18815],{"className":18814},[7393],[7395,18816,18817],{"xmlns":7397},[7399,18818,18819,18849],{},[7402,18820,18821,18824,18826,18828,18830,18832,18835,18837,18839,18841,18843,18845,18847],{},[7414,18822,18823],{},"P",[7405,18825,2651],{"stretchy":1435},[7414,18827,828],{},[7405,18829,2823],{},[7418,18831,7420],{},[7414,18833,18834],{"mathvariant":7457},"∣",[7414,18836,7416],{},[7405,18838,3993],{"stretchy":1435},[7405,18840,2823],{},[7414,18842,7996],{},[7405,18844,2651],{"stretchy":1435},[7414,18846,7416],{},[7405,18848,3993],{"stretchy":1435},[7487,18850,18851],{"encoding":7489},"P(y=1|x) = f(x)",[805,18853,18855,18880,18905],{"className":18854,"ariaHidden":7423},[7494],[805,18856,18858,18861,18865,18868,18871,18874,18877],{"className":18857},[7498],[805,18859],{"className":18860,"style":7503},[7502],[805,18862,18823],{"className":18863,"style":18864},[7512,7516],"margin-right:0.1389em;",[805,18866,2651],{"className":18867},[7507],[805,18869,828],{"className":18870,"style":7582},[7512,7516],[805,18872],{"className":18873,"style":8028},[7574],[805,18875,2823],{"className":18876},[8032],[805,18878],{"className":18879,"style":8028},[7574],[805,18881,18883,18886,18890,18893,18896,18899,18902],{"className":18882},[7498],[805,18884],{"className":18885,"style":7503},[7502],[805,18887,18889],{"className":18888},[7512],"1∣",[805,18891,7416],{"className":18892},[7512,7516],[805,18894,3993],{"className":18895},[7621],[805,18897],{"className":18898,"style":8028},[7574],[805,18900,2823],{"className":18901},[8032],[805,18903],{"className":18904,"style":8028},[7574],[805,18906,18908,18911,18914,18917,18920],{"className":18907},[7498],[805,18909],{"className":18910,"style":7503},[7502],[805,18912,7996],{"className":18913,"style":8024},[7512,7516],[805,18915,2651],{"className":18916},[7507],[805,18918,7416],{"className":18919},[7512,7516],[805,18921,3993],{"className":18922},[7621],"\nWhere:",[49,18925,18926,19002],{},[52,18927,18928,19001],{},[805,18929,18931,18959],{"className":18930},[7389],[805,18932,18934],{"className":18933},[7393],[7395,18935,18936],{"xmlns":7397},[7399,18937,18938,18956],{},[7402,18939,18940,18942,18944,18946,18948,18950,18952,18954],{},[7414,18941,18823],{},[7405,18943,2651],{"stretchy":1435},[7414,18945,828],{},[7405,18947,2823],{},[7418,18949,7420],{},[7414,18951,18834],{"mathvariant":7457},[7414,18953,7416],{},[7405,18955,3993],{"stretchy":1435},[7487,18957,18958],{"encoding":7489},"P(y=1|x)",[805,18960,18962,18986],{"className":18961,"ariaHidden":7423},[7494],[805,18963,18965,18968,18971,18974,18977,18980,18983],{"className":18964},[7498],[805,18966],{"className":18967,"style":7503},[7502],[805,18969,18823],{"className":18970,"style":18864},[7512,7516],[805,18972,2651],{"className":18973},[7507],[805,18975,828],{"className":18976,"style":7582},[7512,7516],[805,18978],{"className":18979,"style":8028},[7574],[805,18981,2823],{"className":18982},[8032],[805,18984],{"className":18985,"style":8028},[7574],[805,18987,18989,18992,18995,18998],{"className":18988},[7498],[805,18990],{"className":18991,"style":7503},[7502],[805,18993,18889],{"className":18994},[7512],[805,18996,7416],{"className":18997},[7512,7516],[805,18999,3993],{"className":19000},[7621]," is the probability that the class is 1 given the feature vector x.",[52,19003,19004,19048],{},[805,19005,19007,19027],{"className":19006},[7389],[805,19008,19010],{"className":19009},[7393],[7395,19011,19012],{"xmlns":7397},[7399,19013,19014,19024],{},[7402,19015,19016,19018,19020,19022],{},[7414,19017,7996],{},[7405,19019,2651],{"stretchy":1435},[7414,19021,7416],{},[7405,19023,3993],{"stretchy":1435},[7487,19025,19026],{"encoding":7489},"f(x)",[805,19028,19030],{"className":19029,"ariaHidden":7423},[7494],[805,19031,19033,19036,19039,19042,19045],{"className":19032},[7498],[805,19034],{"className":19035,"style":7503},[7502],[805,19037,7996],{"className":19038,"style":8024},[7512,7516],[805,19040,2651],{"className":19041},[7507],[805,19043,7416],{"className":19044},[7512,7516],[805,19046,3993],{"className":19047},[7621]," is the function that maps the features to the probability.",[11,19050,19051,19052,19055],{},"The most common model for binary classification is the ",[29,19053,19054],{},"logistic regression",", which has the following process:",[317,19057,19058],{},[52,19059,19060,19061,19064],{},"We calculate a ",[29,19062,19063],{},"linear combination"," of the features:",[805,19066,19068],{"className":19067},[7980],[805,19069,19071,19144],{"className":19070},[7389],[805,19072,19074],{"className":19073},[7393],[7395,19075,19076],{"xmlns":7397,"display":7989},[7399,19077,19078,19141],{},[7402,19079,19080,19083,19085,19091,19093,19099,19105,19107,19113,19119,19121,19123,19125,19127,19129,19135],{},[7414,19081,19082],{},"z",[7405,19084,2823],{},[7411,19086,19087,19089],{},[7414,19088,8389],{},[7418,19090,8392],{},[7405,19092,8395],{},[7411,19094,19095,19097],{},[7414,19096,8389],{},[7418,19098,7420],{},[7411,19100,19101,19103],{},[7414,19102,7416],{},[7418,19104,7420],{},[7405,19106,8395],{},[7411,19108,19109,19111],{},[7414,19110,8389],{},[7418,19112,7442],{},[7411,19114,19115,19117],{},[7414,19116,7416],{},[7418,19118,7442],{},[7405,19120,8395],{},[7414,19122,44],{"mathvariant":7457},[7414,19124,44],{"mathvariant":7457},[7414,19126,44],{"mathvariant":7457},[7405,19128,8395],{},[7411,19130,19131,19133],{},[7414,19132,8389],{},[7414,19134,11],{},[7411,19136,19137,19139],{},[7414,19138,7416],{},[7414,19140,11],{},[7487,19142,19143],{"encoding":7489},"z = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + ... + \\beta_p x_p",[805,19145,19147,19166,19221,19316,19411,19429],{"className":19146,"ariaHidden":7423},[7494],[805,19148,19150,19153,19157,19160,19163],{"className":19149},[7498],[805,19151],{"className":19152,"style":8843},[7502],[805,19154,19082],{"className":19155,"style":19156},[7512,7516],"margin-right:0.044em;",[805,19158],{"className":19159,"style":8028},[7574],[805,19161,2823],{"className":19162},[8032],[805,19164],{"className":19165,"style":8028},[7574],[805,19167,19169,19172,19212,19215,19218],{"className":19168},[7498],[805,19170],{"className":19171,"style":8020},[7502],[805,19173,19175,19178],{"className":19174},[7512],[805,19176,8389],{"className":19177,"style":8485},[7512,7516],[805,19179,19181],{"className":19180},[7520],[805,19182,19184,19204],{"className":19183},[7524,7525],[805,19185,19187,19201],{"className":19186},[7529],[805,19188,19190],{"className":19189,"style":7534},[7533],[805,19191,19192,19195],{"style":8500},[805,19193],{"className":19194,"style":7542},[7541],[805,19196,19198],{"className":19197},[7546,7547,7548,7549],[805,19199,8392],{"className":19200},[7512,7549],[805,19202,7557],{"className":19203},[7556],[805,19205,19207],{"className":19206},[7529],[805,19208,19210],{"className":19209,"style":7564},[7533],[805,19211],{},[805,19213],{"className":19214,"style":8065},[7574],[805,19216,8395],{"className":19217},[8527],[805,19219],{"className":19220,"style":8065},[7574],[805,19222,19224,19227,19267,19307,19310,19313],{"className":19223},[7498],[805,19225],{"className":19226,"style":8020},[7502],[805,19228,19230,19233],{"className":19229},[7512],[805,19231,8389],{"className":19232,"style":8485},[7512,7516],[805,19234,19236],{"className":19235},[7520],[805,19237,19239,19259],{"className":19238},[7524,7525],[805,19240,19242,19256],{"className":19241},[7529],[805,19243,19245],{"className":19244,"style":7534},[7533],[805,19246,19247,19250],{"style":8500},[805,19248],{"className":19249,"style":7542},[7541],[805,19251,19253],{"className":19252},[7546,7547,7548,7549],[805,19254,7420],{"className":19255},[7512,7549],[805,19257,7557],{"className":19258},[7556],[805,19260,19262],{"className":19261},[7529],[805,19263,19265],{"className":19264,"style":7564},[7533],[805,19266],{},[805,19268,19270,19273],{"className":19269},[7512],[805,19271,7416],{"className":19272},[7512,7516],[805,19274,19276],{"className":19275},[7520],[805,19277,19279,19299],{"className":19278},[7524,7525],[805,19280,19282,19296],{"className":19281},[7529],[805,19283,19285],{"className":19284,"style":7534},[7533],[805,19286,19287,19290],{"style":7537},[805,19288],{"className":19289,"style":7542},[7541],[805,19291,19293],{"className":19292},[7546,7547,7548,7549],[805,19294,7420],{"className":19295},[7512,7549],[805,19297,7557],{"className":19298},[7556],[805,19300,19302],{"className":19301},[7529],[805,19303,19305],{"className":19304,"style":7564},[7533],[805,19306],{},[805,19308],{"className":19309,"style":8065},[7574],[805,19311,8395],{"className":19312},[8527],[805,19314],{"className":19315,"style":8065},[7574],[805,19317,19319,19322,19362,19402,19405,19408],{"className":19318},[7498],[805,19320],{"className":19321,"style":8020},[7502],[805,19323,19325,19328],{"className":19324},[7512],[805,19326,8389],{"className":19327,"style":8485},[7512,7516],[805,19329,19331],{"className":19330},[7520],[805,19332,19334,19354],{"className":19333},[7524,7525],[805,19335,19337,19351],{"className":19336},[7529],[805,19338,19340],{"className":19339,"style":7534},[7533],[805,19341,19342,19345],{"style":8500},[805,19343],{"className":19344,"style":7542},[7541],[805,19346,19348],{"className":19347},[7546,7547,7548,7549],[805,19349,7442],{"className":19350},[7512,7549],[805,19352,7557],{"className":19353},[7556],[805,19355,19357],{"className":19356},[7529],[805,19358,19360],{"className":19359,"style":7564},[7533],[805,19361],{},[805,19363,19365,19368],{"className":19364},[7512],[805,19366,7416],{"className":19367},[7512,7516],[805,19369,19371],{"className":19370},[7520],[805,19372,19374,19394],{"className":19373},[7524,7525],[805,19375,19377,19391],{"className":19376},[7529],[805,19378,19380],{"className":19379,"style":7534},[7533],[805,19381,19382,19385],{"style":7537},[805,19383],{"className":19384,"style":7542},[7541],[805,19386,19388],{"className":19387},[7546,7547,7548,7549],[805,19389,7442],{"className":19390},[7512,7549],[805,19392,7557],{"className":19393},[7556],[805,19395,19397],{"className":19396},[7529],[805,19398,19400],{"className":19399,"style":7564},[7533],[805,19401],{},[805,19403],{"className":19404,"style":8065},[7574],[805,19406,8395],{"className":19407},[8527],[805,19409],{"className":19410,"style":8065},[7574],[805,19412,19414,19417,19420,19423,19426],{"className":19413},[7498],[805,19415],{"className":19416,"style":8727},[7502],[805,19418,7729],{"className":19419},[7512],[805,19421],{"className":19422,"style":8065},[7574],[805,19424,8395],{"className":19425},[8527],[805,19427],{"className":19428,"style":8065},[7574],[805,19430,194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apply the ",[29,19521,19522],{},"sigmoid function"," to obtain a probability:",[805,19525,19527],{"className":19526},[7980],[805,19528,19530,19573],{"className":19529},[7389],[805,19531,19533],{"className":19532},[7393],[7395,19534,19535],{"xmlns":7397,"display":7989},[7399,19536,19537,19570],{},[7402,19538,19539,19542,19544,19546,19548,19550],{},[7414,19540,19541],{},"σ",[7405,19543,2651],{"stretchy":1435},[7414,19545,19082],{},[7405,19547,3993],{"stretchy":1435},[7405,19549,2823],{},[10504,19551,19552,19554],{},[7418,19553,7420],{},[7402,19555,19556,19558,19560],{},[7418,19557,7420],{},[7405,19559,8395],{},[10547,19561,19562,19564],{},[7414,19563,17772],{},[7402,19565,19566,19568],{},[7405,19567,9899],{},[7414,19569,19082],{},[7487,19571,19572],{"encoding":7489},"\\sigma(z) = \\frac{1}{1 + 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intercept.",[52,20426,20427,9410],{},[805,20428,20430,20471],{"className":20429},[7389],[805,20431,20433],{"className":20432},[7393],[7395,20434,20435],{"xmlns":7397},[7399,20436,20437,20469],{},[7402,20438,20439,20445,20447,20453,20455,20457,20459,20461,20463],{},[7411,20440,20441,20443],{},[7414,20442,8389],{},[7418,20444,7420],{},[7405,20446,3096],{"separator":7423},[7411,20448,20449,20451],{},[7414,20450,8389],{},[7418,20452,7442],{},[7405,20454,3096],{"separator":7423},[7414,20456,44],{"mathvariant":7457},[7414,20458,44],{"mathvariant":7457},[7414,20460,44],{"mathvariant":7457},[7405,20462,3096],{"separator":7423},[7411,20464,20465,20467],{},[7414,20466,8389],{},[7414,20468,11],{},[7487,20470,9259],{"encoding":7489},[805,20472,20474],{"className":20473,"ariaHidden":7423},[7494],[805,20475,20477,20480,20520,20523,20526,20566,20569,20572,20575,20578,20581],{"className":20476},[7498],[805,20478],{"className":20479,"style":8746},[7502],[805,20481,20483,20486],{"className":20482},[7512],[805,20484,8389],{"className":20485,"style":8485},[7512,7516],[805,20487,20489],{"className":20488},[7520],[805,20490,20492,20512],{"className":20491},[7524,7525],[805,20493,20495,20509],{"className":20494},[7529],[805,20496,20498],{"className":20497,"style":7534},[7533],[805,20499,20500,20503],{"style":8500},[805,20501],{"className":20502,"style":7542},[7541],[805,20504,20506],{"className":20505},[7546,7547,7548,7549],[805,20507,7420],{"className":20508},[7512,7549],[805,20510,7557],{"className":20511},[7556],[805,20513,20515],{"className":20514},[7529],[805,20516,20518],{"className":20517,"style":7564},[7533],[805,20519],{},[805,20521,3096],{"className":20522},[7570],[805,20524],{"className":20525,"style":7575},[7574],[805,20527,20529,20532],{"className":20528},[7512],[805,20530,8389],{"className":20531,"style":8485},[7512,7516],[805,20533,20535],{"className":20534},[7520],[805,20536,20538,20558],{"className":20537},[7524,7525],[805,20539,20541,20555],{"className":20540},[7529],[805,20542,20544],{"className":20543,"style":7534},[7533],[805,20545,20546,20549],{"style":8500},[805,20547],{"className":20548,"style":7542},[7541],[805,20550,20552],{"className":20551},[7546,7547,7548,7549],[805,20553,7442],{"className":20554},[7512,7549],[805,20556,7557],{"className":20557},[7556],[805,20559,20561],{"className":20560},[7529],[805,20562,20564],{"className":20563,"style":7564},[7533],[805,20565],{},[805,20567,3096],{"className":20568},[7570],[805,20570],{"className":20571,"style":7575},[7574],[805,20573,7729],{"className":20574},[7512],[805,20576,3096],{"className":20577},[7570],[805,20579],{"className":20580,"style":7575},[7574],[805,20582,20584,20587],{"className":20583},[7512],[805,20585,8389],{"className":20586,"style":8485},[7512,7516],[805,20588,20590],{"className":20589},[7520],[805,20591,20593,20613],{"className":20592},[7524,7525],[805,20594,20596,20610],{"className":20595},[7529],[805,20597,20599],{"className":20598,"style":7757},[7533],[805,20600,20601,20604],{"style":8500},[805,20602],{"className":20603,"style":7542},[7541],[805,20605,20607],{"className":20606},[7546,7547,7548,7549],[805,20608,11],{"className":20609},[7512,7516,7549],[805,20611,7557],{"className":20612},[7556],[805,20614,20616],{"className":20615},[7529],[805,20617,20619],{"className":20618,"style":8785},[7533],[805,20620],{},[52,20622,20623,20817],{},[805,20624,20626,20667],{"className":20625},[7389],[805,20627,20629],{"className":20628},[7393],[7395,20630,20631],{"xmlns":7397},[7399,20632,20633,20665],{},[7402,20634,20635,20641,20643,20649,20651,20653,20655,20657,20659],{},[7411,20636,20637,20639],{},[7414,20638,7416],{},[7418,20640,7420],{},[7405,20642,3096],{"separator":7423},[7411,20644,20645,20647],{},[7414,20646,7416],{},[7418,20648,7442],{},[7405,20650,3096],{"separator":7423},[7414,20652,44],{"mathvariant":7457},[7414,20654,44],{"mathvariant":7457},[7414,20656,44],{"mathvariant":7457},[7405,20658,3096],{"separator":7423},[7411,20660,20661,20663],{},[7414,20662,7416],{},[7414,20664,11],{},[7487,20666,8929],{"encoding":7489},[805,20668,20670],{"className":20669,"ariaHidden":7423},[7494],[805,20671,20673,20676,20716,20719,20722,20762,20765,20768,20771,20774,20777],{"className":20672},[7498],[805,20674],{"className":20675,"style":8939},[7502],[805,20677,20679,20682],{"className":20678},[7512],[805,20680,7416],{"className":20681},[7512,7516],[805,20683,20685],{"className":20684},[7520],[805,20686,20688,20708],{"className":20687},[7524,7525],[805,20689,20691,20705],{"className":20690},[7529],[805,20692,20694],{"className":20693,"style":7534},[7533],[805,20695,20696,20699],{"style":7537},[805,20697],{"className":20698,"style":7542},[7541],[805,20700,20702],{"className":20701},[7546,7547,7548,7549],[805,20703,7420],{"className":20704},[7512,7549],[805,20706,7557],{"className":20707},[7556],[805,20709,20711],{"className":20710},[7529],[805,20712,20714],{"className":20713,"style":7564},[7533],[805,20715],{},[805,20717,3096],{"className":20718},[7570],[80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are the features or independent variables.",[52,20819,20820,19001],{},[805,20821,20823,20850],{"className":20822},[7389],[805,20824,20826],{"className":20825},[7393],[7395,20827,20828],{"xmlns":7397},[7399,20829,20830,20848],{},[7402,20831,20832,20834,20836,20838,20840,20842,20844,20846],{},[7414,20833,18823],{},[7405,20835,2651],{"stretchy":1435},[7414,20837,828],{},[7405,20839,2823],{},[7418,20841,7420],{},[7414,20843,18834],{"mathvariant":7457},[7414,20845,7416],{},[7405,20847,3993],{"stretchy":1435},[7487,20849,18958],{"encoding":7489},[805,20851,20853,20877],{"className":20852,"ariaHidden":7423},[7494],[805,20854,20856,20859,20862,20865,20868,20871,20874],{"className":20855},[7498],[805,20857],{"className":20858,"style":7503},[7502],[805,20860,18823],{"className":20861,"style":18864},[7512,7516],[805,20863,2651],{"className":20864},[7507],[805,20866,828],{"className":20867,"style":7582},[7512,7516],[805,20869],{"className":20870,"style":8028},[7574],[805,20872,2823],{"className":20873},[8032],[805,20875],{"className":20876,"style":8028},[7574],[805,20878,20880,20883,20886,20889],{"className":20879},[7498],[805,20881],{"className":20882,"style":7503},[7502],[805,20884,18889],{"className":20885},[7512],[805,20887,7416],{"className":20888},[7512,7516],[805,20890,3993],{"className":20891},[7621],[11,20893,20894],{},"Each feature provides evidence for or against a particular class. For example, if the word \"free\" appears in an email, it might increase the probability that it is spam. On the other hand, if the word \"meeting\" appears, it might decrease the probability of it being spam.",[78,20896,20897],{},[11,20898,20899],{},"This gives us not only a classification but also a measure of confidence in that classification through the probability calculated by the sigmoid function.",[11,20901,20902],{},"The sigmoid function transforms any real value into a value between 0 and 1. The formula as shown above 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is the linear combination of the features, that is, ",[805,21128,21130,21201],{"className":21129},[7389],[805,21131,21133],{"className":21132},[7393],[7395,21134,21135],{"xmlns":7397},[7399,21136,21137,21199],{},[7402,21138,21139,21141,21143,21149,21151,21157,21163,21165,21171,21177,21179,21181,21183,21185,21187,21193],{},[7414,21140,19082],{},[7405,21142,2823],{},[7411,21144,21145,21147],{},[7414,21146,8389],{},[7418,21148,8392],{},[7405,21150,8395],{},[7411,21152,21153,21155],{},[7414,21154,8389],{},[7418,21156,7420],{},[7411,21158,21159,21161],{},[7414,21160,7416],{},[7418,21162,7420],{},[7405,21164,8395],{},[7411,21166,21167,21169],{},[7414,21168,8389],{},[7418,21170,7442],{},[7411,21172,21173,21175],{},[7414,21174,7416],{},[7418,21176,7442],{},[7405,21178,8395],{},[7414,21180,44],{"mathvariant":7457},[7414,21182,44],{"mathvariant":7457},[7414,21184,44],{"mathvariant":7457},[7405,21186,8395],{},[7411,21188,21189,21191],{},[7414,21190,8389],{},[7414,21192,11],{},[7411,21194,21195,21197],{},[7414,21196,7416],{},[7414,21198,11],{},[7487,21200,19143],{"encoding":7489},[805,21202,21204,21222,21277,21372,21467,21485],{"className":21203,"ariaHidden":7423},[7494],[805,21205,21207,21210,21213,21216,21219],{"className":21206},[7498],[805,21208],{"className":21209,"style":8843},[7502],[805,21211,19082],{"className":21212,"style":19156},[7512,7516],[805,21214],{"className":21215,"style":8028},[7574],[805,21217,2823],{"className":21218},[8032],[805,21220],{"className":21221,"style":8028},[7574],[805,21223,21225,21228,21268,21271,21274],{"className":21224},[7498],[805,21226],{"className":21227,"style":8020},[7502],[805,21229,21231,21234],{"className":21230},[7512],[805,21232,8389],{"className":21233,"style":8485},[7512,7516],[805,21235,21237],{"className":21236},[7520],[805,21238,21240,21260],{"className":21239},[7524,7525],[805,21241,21243,21257],{"className":21242},[7529],[805,21244,21246],{"className":21245,"style":7534},[7533],[805,21247,21248,21251],{"style":8500},[805,21249],{"className":21250,"style":7542},[7541],[805,21252,21254],{"className":21253},[7546,7547,7548,7549],[805,21255,8392],{"className":21256},[7512,7549],[805,21258,7557],{"className":21259},[7556],[805,21261,21263],{"className":21262},[7529],[805,21264,21266],{"className":21265,"style":7564},[7533],[805,21267],{},[805,21269],{"className":21270,"style":8065},[7574],[805,21272,8395],{"className":21273},[8527],[805,21275],{"className":21276,"style":8065},[7574],[805,21278,21280,21283,21323,21363,21366,21369],{"className":21279},[7498],[805,21281],{"className":21282,"style":8020},[7502],[805,21284,21286,21289],{"className":21285},[7512],[805,21287,8389],{"className":21288,"style":8485},[7512,7516],[805,21290,21292],{"className":21291},[7520],[805,21293,21295,21315],{"className":21294},[7524,7525],[805,21296,21298,21312],{"className":21297},[7529],[805,21299,21301],{"className":21300,"style":7534},[7533],[805,21302,21303,21306],{"style":8500},[805,21304],{"className":21305,"style":7542},[7541],[805,21307,21309],{"className":21308},[7546,7547,7548,7549],[805,21310,7420],{"className":21311},[7512,7549],[805,21313,7557],{"className":21314},[7556],[805,21316,21318],{"className":21317},[7529],[805,21319,21321],{"className":21320,"style":7564},[7533],[805,21322],{},[805,21324,21326,21329],{"className":21325},[7512],[805,21327,7416],{"className":21328},[7512,7516],[805,21330,21332],{"className":21331},[7520],[805,21333,21335,21355],{"className":21334},[7524,7525],[805,21336,21338,21352],{"className":21337},[7529],[805,21339,21341],{"className":21340,"style":7534},[7533],[805,21342,21343,21346],{"style":7537},[805,21344],{"className":21345,"style":7542},[7541],[805,21347,21349],{"className":21348},[7546,7547,7548,7549],[805,21350,7420],{"className":21351},[7512,7549],[805,21353,7557],{"className":21354},[7556],[805,21356,21358],{"className":21357},[7529],[805,21359,21361],{"className":21360,"style":7564},[7533],[805,21362],{},[805,21364],{"className":21365,"style":8065},[7574],[805,21367,8395],{"className":21368},[8527],[805,21370],{"className":21371,"style":8065},[7574],[805,21373,21375,21378,21418,21458,21461,21464],{"className":21374},[7498],[805,21376],{"className":21377,"style":8020},[7502],[805,21379,21381,21384],{"className":21380},[7512],[805,21382,8389],{"className":21383,"style":8485},[7512,7516],[805,21385,21387],{"className":21386},[7520],[805,21388,21390,21410],{"className":21389},[7524,7525],[805,21391,21393,21407],{"className":21392},[7529],[805,21394,21396],{"className":21395,"style":7534},[7533],[805,21397,21398,21401],{"style":8500},[805,21399],{"className":21400,"style":7542},[7541],[805,21402,21404],{"className":21403},[7546,7547,7548,7549],[805,21405,7442],{"className":21406},[7512,7549],[805,21408,7557],{"className":21409},[7556],[805,21411,21413],{"className":21412},[7529],[805,21414,21416],{"className":21415,"style":7564},[7533],[805,21417],{},[805,21419,21421,21424],{"className":21420},[7512],[805,21422,7416],{"className":21423},[7512,7516],[805,21425,21427],{"className":21426},[7520],[805,21428,21430,21450],{"className":21429},[7524,7525],[805,21431,21433,21447],{"className":21432},[7529],[805,21434,21436],{"className":21435,"style":7534},[7533],[805,21437,21438,21441],{"style":7537},[805,21439],{"className":21440,"style":7542},[7541],[805,21442,21444],{"className":21443},[7546,7547,7548,7549],[805,21445,7442],{"className":21446},[7512,7549],[805,21448,7557],{"className":21449},[7556],[805,21451,21453],{"className":21452},[7529],[805,21454,21456],{"className":21455,"style":7564},[7533],[805,21457],{},[805,21459],{"className":21460,"style":8065},[7574],[805,21462,8395],{"className":21463},[8527],[805,21465],{"className":21466,"style":8065},[7574],[805,21468,21470,21473,21476,21479,21482],{"className":21469},[7498],[805,21471],{"className":21472,"style":8727},[7502],[805,21474,7729],{"className":21475},[7512],[805,21477],{"className":21478,"style":8065},[7574],[805,21480,8395],{"className":21481},[8527],[805,21483],{"className":21484,"style":8065},[7574],[805,21486,21488,21491,21531],{"className":21487},[7498],[805,21489],{"className":21490,"style":8746},[7502],[805,21492,21494,21497],{"className":21493},[7512],[805,21495,8389],{"className":21496,"style":8485},[7512,7516],[805,21498,21500],{"className":21499},[7520],[805,21501,21503,21523],{"className":21502},[7524,7525],[805,21504,21506,21520],{"className":21505},[7529],[805,21507,21509],{"className":21508,"style":7757},[7533],[805,21510,21511,21514],{"style":8500},[805,21512],{"className":21513,"style":7542},[7541],[805,21515,21517],{"className":21516},[7546,7547,7548,7549],[805,21518,11],{"className":21519},[7512,7516,7549],[805,21521,7557],{"className":21522},[7556],[805,21524,21526],{"className":21525},[7529],[805,21527,21529],{"className":21528,"style":8785},[7533],[805,21530],{},[805,21532,21534,21537],{"className":21533},[7512],[805,21535,7416],{"className":21536},[7512,7516],[805,21538,21540],{"className":21539},[7520],[805,21541,21543,21563],{"className":21542},[7524,7525],[805,21544,21546,21560],{"className":21545},[7529],[805,21547,21549],{"className":21548,"style":7757},[7533],[805,21550,21551,21554],{"style":7537},[805,21552],{"className":21553,"style":7542},[7541],[805,21555,21557],{"className":21556},[7546,7547,7548,7549],[805,21558,11],{"className":21559},[7512,7516,7549],[805,21561,7557],{"className":21562},[7556],[805,21564,21566],{"className":21565},[7529],[805,21567,21569],{"className":21568,"style":8785},[7533],[805,21570],{},[52,21572,21573,21601],{},[805,21574,21576,21589],{"className":21575},[7389],[805,21577,21579],{"className":21578},[7393],[7395,21580,21581],{"xmlns":7397},[7399,21582,21583,21587],{},[7402,21584,21585],{},[7414,21586,17772],{},[7487,21588,17772],{"encoding":7489},[805,21590,21592],{"className":21591,"ariaHidden":7423},[7494],[805,21593,21595,21598],{"className":21594},[7498],[805,21596],{"className":21597,"style":8843},[7502],[805,21599,17772],{"className":21600},[7512,7516]," is Euler's number, approximately equal to 2.71828.",[52,21603,21604,21648],{},[805,21605,21607,21627],{"className":21606},[7389],[805,21608,21610],{"className":21609},[7393],[7395,21611,21612],{"xmlns":7397},[7399,21613,21614,21624],{},[7402,21615,21616,21618,21620,21622],{},[7414,21617,19541],{},[7405,21619,2651],{"stretchy":1435},[7414,21621,19082],{},[7405,21623,3993],{"stretchy":1435},[7487,21625,21626],{"encoding":7489},"\\sigma(z)",[805,21628,21630],{"className":21629,"ariaHidden":7423},[7494],[805,21631,21633,21636,21639,21642,21645],{"className":21632},[7498],[805,21634],{"className":21635,"style":7503},[7502],[805,21637,19541],{"className":21638,"style":7582},[7512,7516],[805,21640,2651],{"className":21641},[7507],[805,21643,19082],{"className":21644,"style":19156},[7512,7516],[805,21646,3993],{"className":21647},[7621]," is the output of the sigmoid function, which represents the probability of the class being 1 given the value of z (range between 0 and 1).",[11,21650,21651,21652,21695,21696,21700],{},"Visually it looks something like this (with z on the X-axis and ",[805,21653,21655,21674],{"className":21654},[7389],[805,21656,21658],{"className":21657},[7393],[7395,21659,21660],{"xmlns":7397},[7399,21661,21662,21672],{},[7402,21663,21664,21666,21668,21670],{},[7414,21665,19541],{},[7405,21667,2651],{"stretchy":1435},[7414,21669,19082],{},[7405,21671,3993],{"stretchy":1435},[7487,21673,21626],{"encoding":7489},[805,21675,21677],{"className":21676,"ariaHidden":7423},[7494],[805,21678,21680,21683,21686,21689,21692],{"className":21679},[7498],[805,21681],{"className":21682,"style":7503},[7502],[805,21684,19541],{"className":21685,"style":7582},[7512,7516],[805,21687,2651],{"className":21688},[7507],[805,21690,19082],{"className":21691,"style":19156},[7512,7516],[805,21693,3993],{"className":21694},[7621]," on the Y-axis):\n",[833,21697],{"alt":21698,"src":21699},"Sigmoid Function Graph","\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations\u002Fshared\u002Fsigmoid-function.webp",[838,21701,21698],{},[11,21703,21704],{},"From this we can draw some important conclusions:",[49,21706,21707,21754,21801],{},[52,21708,21709,21710,21753],{},"When z is very negative, ",[805,21711,21713,21732],{"className":21712},[7389],[805,21714,21716],{"className":21715},[7393],[7395,21717,21718],{"xmlns":7397},[7399,21719,21720,21730],{},[7402,21721,21722,21724,21726,21728],{},[7414,21723,19541],{},[7405,21725,2651],{"stretchy":1435},[7414,21727,19082],{},[7405,21729,3993],{"stretchy":1435},[7487,21731,21626],{"encoding":7489},[805,21733,21735],{"className":21734,"ariaHidden":7423},[7494],[805,21736,21738,21741,21744,21747,21750],{"className":21737},[7498],[805,21739],{"className":21740,"style":7503},[7502],[805,21742,19541],{"className":21743,"style":7582},[7512,7516],[805,21745,2651],{"className":21746},[7507],[805,21748,19082],{"className":21749,"style":19156},[7512,7516],[805,21751,3993],{"className":21752},[7621]," approaches 0, indicating a low probability that the class is 1.",[52,21755,21756,21757,21800],{},"When z is very positive, ",[805,21758,21760,21779],{"className":21759},[7389],[805,21761,21763],{"className":21762},[7393],[7395,21764,21765],{"xmlns":7397},[7399,21766,21767,21777],{},[7402,21768,21769,21771,21773,21775],{},[7414,21770,19541],{},[7405,21772,2651],{"stretchy":1435},[7414,21774,19082],{},[7405,21776,3993],{"stretchy":1435},[7487,21778,21626],{"encoding":7489},[805,21780,21782],{"className":21781,"ariaHidden":7423},[7494],[805,21783,21785,21788,21791,21794,21797],{"className":21784},[7498],[805,21786],{"className":21787,"style":7503},[7502],[805,21789,19541],{"className":21790,"style":7582},[7512,7516],[805,21792,2651],{"className":21793},[7507],[805,21795,19082],{"className":21796,"style":19156},[7512,7516],[805,21798,3993],{"className":21799},[7621]," approaches 1, indicating a high probability that the class is 1.",[52,21802,21803,21804,21847],{},"When z is 0, ",[805,21805,21807,21826],{"className":21806},[7389],[805,21808,21810],{"className":21809},[7393],[7395,21811,21812],{"xmlns":7397},[7399,21813,21814,21824],{},[7402,21815,21816,21818,21820,21822],{},[7414,21817,19541],{},[7405,21819,2651],{"stretchy":1435},[7414,21821,19082],{},[7405,21823,3993],{"stretchy":1435},[7487,21825,21626],{"encoding":7489},[805,21827,21829],{"className":21828,"ariaHidden":7423},[7494],[805,21830,21832,21835,21838,21841,21844],{"className":21831},[7498],[805,21833],{"className":21834,"style":7503},[7502],[805,21836,19541],{"className":21837,"style":7582},[7512,7516],[805,21839,2651],{"className":21840},[7507],[805,21842,19082],{"className":21843,"style":19156},[7512,7516],[805,21845,3993],{"className":21846},[7621]," is 0.5, indicating an equal probability that the class is 0 or 1.",[11,21849,21850,21851,21854],{},"The value of 0.5 ",[29,21852,21853],{},"is commonly used as a threshold",", so that:",[49,21856,21857,21928],{},[52,21858,21859,21860,21927],{},"If ",[805,21861,21863,21888],{"className":21862},[7389],[805,21864,21866],{"className":21865},[7393],[7395,21867,21868],{"xmlns":7397},[7399,21869,21870,21885],{},[7402,21871,21872,21874,21876,21878,21880,21883],{},[7414,21873,19541],{},[7405,21875,2651],{"stretchy":1435},[7414,21877,19082],{},[7405,21879,3993],{"stretchy":1435},[7405,21881,21882],{},"≥",[7418,21884,6210],{},[7487,21886,21887],{"encoding":7489},"\\sigma(z) \\geq 0.5",[805,21889,21891,21918],{"className":21890,"ariaHidden":7423},[7494],[805,21892,21894,21897,21900,21903,21906,21909,21912,21915],{"className":21893},[7498],[805,21895],{"className":21896,"style":7503},[7502],[805,21898,19541],{"className":21899,"style":7582},[7512,7516],[805,21901,2651],{"className":21902},[7507],[805,21904,19082],{"className":21905,"style":19156},[7512,7516],[805,21907,3993],{"className":21908},[7621],[805,21910],{"className":21911,"style":8028},[7574],[805,21913,21882],{"className":21914},[8032],[805,21916],{"className":21917,"style":8028},[7574],[805,21919,21921,21924],{"className":21920},[7498],[805,21922],{"className":21923,"style":16038},[7502],[805,21925,6210],{"className":21926},[7512],", it is classified as class 1.",[52,21929,21859,21930,21996],{},[805,21931,21933,21957],{"className":21932},[7389],[805,21934,21936],{"className":21935},[7393],[7395,21937,21938],{"xmlns":7397},[7399,21939,21940,21954],{},[7402,21941,21942,21944,21946,21948,21950,21952],{},[7414,21943,19541],{},[7405,21945,2651],{"stretchy":1435},[7414,21947,19082],{},[7405,21949,3993],{"stretchy":1435},[7405,21951,2815],{},[7418,21953,6210],{},[7487,21955,21956],{"encoding":7489},"\\sigma(z) \u003C 0.5",[805,21958,21960,21987],{"className":21959,"ariaHidden":7423},[7494],[805,21961,21963,21966,21969,21972,21975,21978,21981,21984],{"className":21962},[7498],[805,21964],{"className":21965,"style":7503},[7502],[805,21967,19541],{"className":21968,"style":7582},[7512,7516],[805,21970,2651],{"className":21971},[7507],[805,21973,19082],{"className":21974,"style":19156},[7512,7516],[805,21976,3993],{"className":21977},[7621],[805,21979],{"className":21980,"style":8028},[7574],[805,21982,2815],{"className":21983},[8032],[805,21985],{"className":21986,"style":8028},[7574],[805,21988,21990,21993],{"className":21989},[7498],[805,21991],{"className":21992,"style":16038},[7502],[805,21994,6210],{"className":21995},[7512],", it is classified as class 0.",[78,21998,21999,22002],{},[11,22000,22001],{},"Key properties:",[49,22003,22004,22007,22054],{},[52,22005,22006],{},"Bounded range between 0 and 1",[52,22008,22009,22010,22053],{},"Monotonicity: if z increases, ",[805,22011,22013,22032],{"className":22012},[7389],[805,22014,22016],{"className":22015},[7393],[7395,22017,22018],{"xmlns":7397},[7399,22019,22020,22030],{},[7402,22021,22022,22024,22026,22028],{},[7414,22023,19541],{},[7405,22025,2651],{"stretchy":1435},[7414,22027,19082],{},[7405,22029,3993],{"stretchy":1435},[7487,22031,21626],{"encoding":7489},[805,22033,22035],{"className":22034,"ariaHidden":7423},[7494],[805,22036,22038,22041,22044,22047,22050],{"className":22037},[7498],[805,22039],{"className":22040,"style":7503},[7502],[805,22042,19541],{"className":22043,"style":7582},[7512,7516],[805,22045,2651],{"className":22046},[7507],[805,22048,19082],{"className":22049,"style":19156},[7512,7516],[805,22051,3993],{"className":22052},[7621]," also increases",[52,22055,22056,22057,22123,22124,36,22176,22123,22242],{},"Asymptotic: ",[805,22058,22060,22084],{"className":22059},[7389],[805,22061,22063],{"className":22062},[7393],[7395,22064,22065],{"xmlns":7397},[7399,22066,22067,22081],{},[7402,22068,22069,22071,22073,22075,22077,22079],{},[7414,22070,19541],{},[7405,22072,2651],{"stretchy":1435},[7414,22074,19082],{},[7405,22076,3993],{"stretchy":1435},[7405,22078,8004],{},[7418,22080,7420],{},[7487,22082,22083],{"encoding":7489},"\\sigma(z) \\to 1",[805,22085,22087,22114],{"className":22086,"ariaHidden":7423},[7494],[805,22088,22090,22093,22096,22099,22102,22105,22108,22111],{"className":22089},[7498],[805,22091],{"className":22092,"style":7503},[7502],[805,22094,19541],{"className":22095,"style":7582},[7512,7516],[805,22097,2651],{"className":22098},[7507],[805,22100,19082],{"className":22101,"style":19156},[7512,7516],[805,22103,3993],{"className":22104},[7621],[805,22106],{"className":22107,"style":8028},[7574],[805,22109,8004],{"className":22110},[8032],[805,22112],{"className":22113,"style":8028},[7574],[805,22115,22117,22120],{"className":22116},[7498],[805,22118],{"className":22119,"style":16038},[7502],[805,22121,7420],{"className":22122},[7512]," when ",[805,22125,22127,22146],{"className":22126},[7389],[805,22128,22130],{"className":22129},[7393],[7395,22131,22132],{"xmlns":7397},[7399,22133,22134,22143],{},[7402,22135,22136,22138,22140],{},[7414,22137,19082],{},[7405,22139,8004],{},[7414,22141,22142],{"mathvariant":7457},"∞",[7487,22144,22145],{"encoding":7489},"z \\to \\infty",[805,22147,22149,22167],{"className":22148,"ariaHidden":7423},[7494],[805,22150,22152,22155,22158,22161,22164],{"className":22151},[7498],[805,22153],{"className":22154,"style":8843},[7502],[805,22156,19082],{"className":22157,"style":19156},[7512,7516],[805,22159],{"className":22160,"style":8028},[7574],[805,22162,8004],{"className":22163},[8032],[805,22165],{"className":22166,"style":8028},[7574],[805,22168,22170,22173],{"className":22169},[7498],[805,22171],{"className":22172,"style":8843},[7502],[805,22174,22142],{"className":22175},[7512],[805,22177,22179,22203],{"className":22178},[7389],[805,22180,22182],{"className":22181},[7393],[7395,22183,22184],{"xmlns":7397},[7399,22185,22186,22200],{},[7402,22187,22188,22190,22192,22194,22196,22198],{},[7414,22189,19541],{},[7405,22191,2651],{"stretchy":1435},[7414,22193,19082],{},[7405,22195,3993],{"stretchy":1435},[7405,22197,8004],{},[7418,22199,8392],{},[7487,22201,22202],{"encoding":7489},"\\sigma(z) \\to 0",[805,22204,22206,22233],{"className":22205,"ariaHidden":7423},[7494],[805,22207,22209,22212,22215,22218,22221,22224,22227,22230],{"className":22208},[7498],[805,22210],{"className":22211,"style":7503},[7502],[805,22213,19541],{"className":22214,"style":7582},[7512,7516],[805,22216,2651],{"className":22217},[7507],[805,22219,19082],{"className":22220,"style":19156},[7512,7516],[805,22222,3993],{"className":22223},[7621],[805,22225],{"className":22226,"style":8028},[7574],[805,22228,8004],{"className":22229},[8032],[805,22231],{"className":22232,"style":8028},[7574],[805,22234,22236,22239],{"className":22235},[7498],[805,22237],{"className":22238,"style":16038},[7502],[805,22240,8392],{"className":22241},[7512],[805,22243,22245,22265],{"className":22244},[7389],[805,22246,22248],{"className":22247},[7393],[7395,22249,22250],{"xmlns":7397},[7399,22251,22252,22262],{},[7402,22253,22254,22256,22258,22260],{},[7414,22255,19082],{},[7405,22257,8004],{},[7405,22259,9899],{},[7414,22261,22142],{"mathvariant":7457},[7487,22263,22264],{"encoding":7489},"z \\to -\\infty",[805,22266,22268,22286],{"className":22267,"ariaHidden":7423},[7494],[805,22269,22271,22274,22277,22280,22283],{"className":22270},[7498],[805,22272],{"className":22273,"style":8843},[7502],[805,22275,19082],{"className":22276,"style":19156},[7512,7516],[805,22278],{"className":22279,"style":8028},[7574],[805,22281,8004],{"className":22282},[8032],[805,22284],{"className":22285,"style":8028},[7574],[805,22287,22289,22292,22295],{"className":22288},[7498],[805,22290],{"className":22291,"style":8727},[7502],[805,22293,9899],{"className":22294},[7512],[805,22296,22142],{"className":22297},[7512],[11,22299,22300,22303,22304,927,22307,3573],{},[29,22301,22302],{},"How do we measure error in classification?"," Here we use metrics like ",[29,22305,22306],{},"cross-entropy",[29,22308,22309],{},"log loss",[11,22311,22312],{},"Loss for a single sample:",[805,22314,22316],{"className":22315},[7980],[805,22317,22319,22397],{"className":22318},[7389],[805,22320,22322],{"className":22321},[7393],[7395,22323,22324],{"xmlns":7397,"display":7989},[7399,22325,22326,22394],{},[7402,22327,22328,22331,22333,22335,22338,22340,22343,22346,22349,22351,22357,22359,22361,22363,22365,22367,22369,22371,22373,22375,22377,22379,22381,22383,22389,22391],{},[7414,22329,22330],{},"L",[7405,22332,2823],{},[7405,22334,9899],{},[7405,22336,22337],{"stretchy":1435},"[",[7414,22339,828],{},[7405,22341,22342],{},"⋅",[7414,22344,22345],{},"log",[7405,22347,22348],{},"⁡",[7405,22350,2651],{"stretchy":1435},[9901,22352,22353,22355],{"accent":7423},[7414,22354,828],{},[7405,22356,9911],{},[7405,22358,3993],{"stretchy":1435},[7405,22360,8395],{},[7405,22362,2651],{"stretchy":1435},[7418,22364,7420],{},[7405,22366,9899],{},[7414,22368,828],{},[7405,22370,3993],{"stretchy":1435},[7405,22372,22342],{},[7414,22374,22345],{},[7405,22376,22348],{},[7405,22378,2651],{"stretchy":1435},[7418,22380,7420],{},[7405,22382,9899],{},[9901,22384,22385,22387],{"accent":7423},[7414,22386,828],{},[7405,22388,9911],{},[7405,22390,3993],{"stretchy":1435},[7405,22392,22393],{"stretchy":1435},"]",[7487,22395,22396],{"encoding":7489},"L = -[y \\cdot \\log(\\hat{y}) + (1 - y) \\cdot \\log(1 - \\hat{y})]",[805,22398,22400,22418,22442,22513,22534,22555,22581],{"className":22399,"ariaHidden":7423},[7494],[805,22401,22403,22406,22409,22412,22415],{"className":22402},[7498],[805,22404],{"className":22405,"style":8042},[7502],[805,22407,22330],{"className":22408},[7512,7516],[805,22410],{"className":22411,"style":8028},[7574],[805,22413,2823],{"className":22414},[8032],[805,22416],{"className":22417,"style":8028},[7574],[805,22419,22421,22424,22427,22430,22433,22436,22439],{"className":22420},[7498],[805,22422],{"className":22423,"style":7503},[7502],[805,22425,9899],{"className":22426},[7512],[805,22428,22337],{"className":22429},[7507],[805,22431,828],{"className":22432,"style":7582},[7512,7516],[805,22434],{"className":22435,"style":8065},[7574],[805,22437,22342],{"className":22438},[8527],[805,22440],{"className":22441,"style":8065},[7574],[805,22443,22445,22448,22456,22459,22501,22504,22507,22510],{"className":22444},[7498],[805,22446],{"className":22447,"style":7503},[7502],[805,22449,22451,22452],{"className":22450},[10668],"lo",[805,22453,22455],{"style":22454},"margin-right:0.0139em;","g",[805,22457,2651],{"className":22458},[7507],[805,22460,22462],{"className":22461},[7512,10049],[805,22463,22465,22493],{"className":22464},[7524,7525],[805,22466,22468,22490],{"className":22467},[7529],[805,22469,22471,22479],{"className":22470,"style":10813},[7533],[805,22472,22473,22476],{"style":10062},[805,22474],{"className":22475,"style":10066},[7541],[805,22477,828],{"className":22478,"style":7582},[7512,7516],[805,22480,22481,22484],{"style":10062},[805,22482],{"className":22483,"style":10066},[7541],[805,22485,22487],{"className":22486,"style":11115},[10116],[805,22488,9911],{"className":22489},[7512],[805,22491,7557],{"className":22492},[7556],[805,22494,22496],{"className":22495},[7529],[805,22497,22499],{"className":22498,"style":10130},[7533],[805,22500],{},[805,22502,3993],{"className":22503},[7621],[805,22505],{"className":22506,"style":8065},[7574],[805,22508,8395],{"className":22509},[8527],[805,22511],{"className":22512,"style":8065},[7574],[805,22514,22516,22519,22522,22525,22528,22531],{"className":22515},[7498],[805,22517],{"className":22518,"style":7503},[7502],[805,22520,2651],{"className":22521},[7507],[805,22523,7420],{"className":22524},[7512],[805,22526],{"className":22527,"style":8065},[7574],[805,22529,9899],{"className":22530},[8527],[805,22532],{"className":22533,"style":8065},[7574],[805,22535,22537,22540,22543,22546,22549,22552],{"className":22536},[7498],[805,22538],{"className":22539,"style":7503},[7502],[805,22541,828],{"className":22542,"style":7582},[7512,7516],[805,22544,3993],{"className":22545},[7621],[805,22547],{"className":22548,"style":8065},[7574],[805,22550,22342],{"className":22551},[8527],[805,22553],{"className":22554,"style":8065},[7574],[805,22556,22558,22561,22566,22569,22572,22575,22578],{"className":22557},[7498],[805,22559],{"className":22560,"style":7503},[7502],[805,22562,22451,22564],{"className":22563},[10668],[805,22565,22455],{"style":22454},[805,22567,2651],{"className":22568},[7507],[805,22570,7420],{"className":22571},[7512],[805,22573],{"className":22574,"style":8065},[7574],[805,22576,9899],{"className":22577},[8527],[805,22579],{"className":22580,"style":8065},[7574],[805,22582,22584,22587,22629],{"className":22583},[7498],[805,22585],{"className":22586,"style":7503},[7502],[805,22588,22590],{"className":22589},[7512,10049],[805,22591,22593,22621],{"className":22592},[7524,7525],[805,22594,22596,22618],{"className":22595},[7529],[805,22597,22599,22607],{"className":22598,"style":10813},[7533],[805,22600,22601,22604],{"style":10062},[805,22602],{"className":22603,"style":10066},[7541],[805,22605,828],{"className":22606,"style":7582},[7512,7516],[805,22608,22609,22612],{"style":10062},[805,22610],{"className":22611,"style":10066},[7541],[805,22613,22615],{"className":22614,"style":11115},[10116],[805,22616,9911],{"className":22617},[7512],[805,22619,7557],{"className":22620},[7556],[805,22622,22624],{"className":22623},[7529],[805,22625,22627],{"className":22626,"style":10130},[7533],[805,22628],{},[805,22630,22632],{"className":22631},[7621],")]",[11,22634,22635],{},"Average loss (or cost function) for the entire dataset:",[805,22637,22639],{"className":22638},[7980],[805,22640,22642,22751],{"className":22641},[7389],[805,22643,22645],{"className":22644},[7393],[7395,22646,22647],{"xmlns":7397,"display":7989},[7399,22648,22649,22748],{},[7402,22650,22651,22654,22656,22658,22664,22678,22680,22686,22688,22690,22692,22694,22704,22706,22708,22710,22712,22714,22720,22722,22724,22726,22728,22730,22732,22734,22744,22746],{},[7414,22652,22653],{},"J",[7405,22655,2823],{},[7405,22657,9899],{},[10504,22659,22660,22662],{},[7418,22661,7420],{},[7414,22663,7472],{},[10511,22665,22666,22668,22676],{},[7405,22667,10515],{},[7402,22669,22670,22672,22674],{},[7414,22671,7849],{},[7405,22673,2823],{},[7418,22675,7420],{},[7414,22677,7472],{},[7405,22679,22337],{"stretchy":1435},[7411,22681,22682,22684],{},[7414,22683,828],{},[7414,22685,7849],{},[7405,22687,22342],{},[7414,22689,22345],{},[7405,22691,22348],{},[7405,22693,2651],{"stretchy":1435},[9901,22695,22696,22702],{"accent":7423},[7411,22697,22698,22700],{},[7414,22699,828],{},[7414,22701,7849],{},[7405,22703,9911],{},[7405,22705,3993],{"stretchy":1435},[7405,22707,8395],{},[7405,22709,2651],{"stretchy":1435},[7418,22711,7420],{},[7405,22713,9899],{},[7411,22715,22716,22718],{},[7414,22717,828],{},[7414,22719,7849],{},[7405,22721,3993],{"stretchy":1435},[7405,22723,22342],{},[7414,22725,22345],{},[7405,22727,22348],{},[7405,22729,2651],{"stretchy":1435},[7418,22731,7420],{},[7405,22733,9899],{},[9901,22735,22736,22742],{"accent":7423},[7411,22737,22738,22740],{},[7414,22739,828],{},[7414,22741,7849],{},[7405,22743,9911],{},[7405,22745,3993],{"stretchy":1435},[7405,22747,22393],{"stretchy":1435},[7487,22749,22750],{"encoding":7489},"J = -\\frac{1}{n} \\sum_{i=1}^{n} [y_i \\cdot \\log(\\hat{y_i}) + (1 - y_i) \\cdot \\log(1 - \\hat{y_i})]",[805,22752,22754,22773,22966,23071,23092,23150,23176],{"className":22753,"ariaHidden":7423},[7494],[805,22755,22757,22760,22764,22767,22770],{"className":22756},[7498],[805,22758],{"className":22759,"style":8042},[7502],[805,22761,22653],{"className":22762,"style":22763},[7512,7516],"margin-right:0.0962em;",[805,22765],{"className":22766,"style":8028},[7574],[805,22768,2823],{"className":22769},[8032],[805,22771],{"className":22772,"style":8028},[7574],[805,22774,22776,22779,22782,22844,22847,22914,22917,22957,22960,22963],{"className":22775},[7498],[805,22777],{"className":22778,"style":10591},[7502],[805,22780,9899],{"className":22781},[7512],[805,22783,22785,22788,22841],{"className":22784},[7512],[805,22786],{"className":22787},[7507,10598],[805,22789,22791],{"className":22790},[10504],[805,22792,22794,22833],{"className":22793},[7524,7525],[805,22795,22797,22830],{"className":22796},[7529],[805,22798,22800,22811,22819],{"className":22799,"style":10611},[7533],[805,22801,22802,22805],{"style":10614},[805,22803],{"className":22804,"style":10066},[7541],[805,22806,22808],{"className":22807},[7512],[805,22809,7472],{"className":22810},[7512,7516],[805,22812,22813,22816],{"style":10626},[805,22814],{"className":22815,"style":10066},[7541],[805,22817],{"className":22818,"style":10634},[10633],[805,22820,22821,22824],{"style":10637},[805,22822],{"className":22823,"style":10066},[7541],[805,22825,22827],{"className":22826},[7512],[805,22828,7420],{"className":22829},[7512],[805,22831,7557],{"className":22832},[7556],[805,22834,22836],{"className":22835},[7529],[805,22837,22839],{"className":22838,"style":10656},[7533],[805,22840],{},[805,22842],{"className":22843},[7621,10598],[805,22845],{"className":22846,"style":7575},[7574],[805,22848,22850],{"className":22849},[10668,10669],[805,22851,22853,22906],{"className":22852},[7524,7525],[805,22854,22856,22903],{"className":22855},[7529],[805,22857,22859,22879,22889],{"className":22858,"style":10679},[7533],[805,22860,22861,22864],{"style":10682},[805,22862],{"className":22863,"style":10686},[7541],[805,22865,22867],{"className":22866},[7546,7547,7548,7549],[805,22868,22870,22873,22876],{"className":22869},[7512,7549],[805,22871,7849],{"className":22872},[7512,7516,7549],[805,22874,2823],{"className":22875},[8032,7549],[805,22877,7420],{"className":22878},[7512,7549],[805,22880,22881,22884],{"style":10704},[805,22882],{"className":22883,"style":10686},[7541],[805,22885,22886],{},[805,22887,10515],{"className":22888},[10668,10713,10714],[805,22890,22891,22894],{"style":10717},[805,22892],{"className":22893,"style":10686},[7541],[805,22895,22897],{"className":22896},[7546,7547,7548,7549],[805,22898,22900],{"className":22899},[7512,7549],[805,22901,7472],{"className":22902},[7512,7516,7549],[805,22904,7557],{"className":22905},[7556],[805,22907,22909],{"className":22908},[7529],[805,22910,22912],{"className":22911,"style":10739},[7533],[805,22913],{},[805,22915,22337],{"className":22916},[7507],[805,22918,22920,22923],{"className":22919},[7512],[805,22921,828],{"className":22922,"style":7582},[7512,7516],[805,22924,22926],{"className":22925},[7520],[805,22927,22929,22949],{"className":22928},[7524,7525],[805,22930,22932,22946],{"className":22931},[7529],[805,22933,22935],{"className":22934,"style":7881},[7533],[805,22936,22937,22940],{"style":7597},[805,22938],{"className":22939,"style":7542},[7541],[805,22941,22943],{"className":22942},[7546,7547,7548,7549],[805,22944,7849],{"className":22945},[7512,7516,7549],[805,22947,7557],{"className":22948},[7556],[805,22950,22952],{"className":22951},[7529],[805,22953,22955],{"className":22954,"style":7564},[7533],[805,22956],{},[805,22958],{"className":22959,"style":8065},[7574],[805,22961,22342],{"className":22962},[8527],[805,22964],{"className":22965,"style":8065},[7574],[805,22967,22969,22972,22977,22980,23059,23062,23065,23068],{"className":22968},[7498],[805,22970],{"className":22971,"style":7503},[7502],[805,22973,22451,22975],{"className":22974},[10668],[805,22976,22455],{"style":22454},[805,22978,2651],{"className":22979},[7507],[805,22981,22983],{"className":22982},[7512,10049],[805,22984,22986,23051],{"className":22985},[7524,7525],[805,22987,22989,23048],{"className":22988},[7529],[805,22990,22992,23037],{"className":22991,"style":10813},[7533],[805,22993,22994,22997],{"style":10062},[805,22995],{"className":22996,"style":10066},[7541],[805,22998,23000,23003],{"className":22999},[7512],[805,23001,828],{"className":23002,"style":7582},[7512,7516],[805,23004,23006],{"className":23005},[7520],[805,23007,23009,23029],{"className":23008},[7524,7525],[805,23010,23012,23026],{"className":23011},[7529],[805,23013,23015],{"className":23014,"style":7881},[7533],[805,23016,23017,23020],{"style":7597},[805,23018],{"className":23019,"style":7542},[7541],[805,23021,23023],{"className":23022},[7546,7547,7548,7549],[805,23024,7849],{"className":23025},[7512,7516,7549],[805,23027,7557],{"className":23028},[7556],[805,23030,23032],{"className":23031},[7529],[805,23033,23035],{"className":23034,"style":7564},[7533],[805,23036],{},[805,23038,23039,23042],{"style":10062},[805,23040],{"className":23041,"style":10066},[7541],[805,23043,23045],{"className":23044,"style":10117},[10116],[805,23046,9911],{"className":23047},[7512],[805,23049,7557],{"className":23050},[7556],[805,23052,23054],{"className":23053},[7529],[805,23055,23057],{"className":23056,"style":10130},[7533],[805,23058],{},[805,23060,3993],{"className":23061},[7621],[805,23063],{"className":23064,"style":8065},[7574],[805,23066,8395],{"className":23067},[8527],[805,23069],{"className":23070,"style":8065},[7574],[805,23072,23074,23077,23080,23083,23086,23089],{"className":23073},[7498],[805,23075],{"className":23076,"style":7503},[7502],[805,23078,2651],{"className":23079},[7507],[805,23081,7420],{"className":23082},[7512],[805,23084],{"className":23085,"style":8065},[7574],[805,23087,9899],{"className":23088},[8527],[805,23090],{"className":23091,"style":8065},[7574],[805,23093,23095,23098,23138,23141,23144,23147],{"className":23094},[7498],[805,23096],{"className":23097,"style":7503},[7502],[805,23099,23101,23104],{"className":23100},[7512],[805,23102,828],{"className":23103,"style":7582},[7512,7516],[805,23105,23107],{"className":23106},[7520],[805,23108,23110,23130],{"className":23109},[7524,7525],[805,23111,23113,23127],{"className":23112},[7529],[805,23114,23116],{"className":23115,"style":7881},[7533],[805,23117,23118,23121],{"style":7597},[805,23119],{"className":23120,"style":7542},[7541],[805,23122,23124],{"className":23123},[7546,7547,7548,7549],[805,23125,7849],{"className":23126},[7512,7516,7549],[805,23128,7557],{"className":23129},[7556],[805,23131,23133],{"className":23132},[7529],[805,23134,23136],{"className":23135,"style":7564},[7533],[805,23137],{},[805,23139,3993],{"className":23140},[7621],[805,23142],{"className":23143,"style":8065},[7574],[805,23145,22342],{"className":23146},[8527],[805,23148],{"className":23149,"style":8065},[7574],[805,23151,23153,23156,23161,23164,23167,23170,23173],{"className":23152},[7498],[805,23154],{"className":23155,"style":7503},[7502],[805,23157,22451,23159],{"className":23158},[10668],[805,23160,22455],{"style":22454},[805,23162,2651],{"className":23163},[7507],[805,23165,7420],{"className":23166},[7512],[805,23168],{"className":23169,"style":8065},[7574],[805,23171,9899],{"className":23172},[8527],[805,23174],{"className":23175,"style":8065},[7574],[805,23177,23179,23182,23261],{"className":23178},[7498],[805,23180],{"className":23181,"style":7503},[7502],[805,23183,23185],{"className":23184},[7512,10049],[805,23186,23188,23253],{"className":23187},[7524,7525],[805,23189,23191,23250],{"className":23190},[7529],[805,23192,23194,23239],{"className":23193,"style":10813},[7533],[805,23195,23196,23199],{"style":10062},[805,23197],{"className":23198,"style":10066},[7541],[805,23200,23202,23205],{"className":23201},[7512],[805,23203,828],{"className":23204,"style":7582},[7512,7516],[805,23206,23208],{"className":23207},[7520],[805,23209,23211,23231],{"className":23210},[7524,7525],[805,23212,23214,23228],{"className":23213},[7529],[805,23215,23217],{"className":23216,"style":7881},[7533],[805,23218,23219,23222],{"style":7597},[805,23220],{"className":23221,"style":7542},[7541],[805,23223,23225],{"className":23224},[7546,7547,7548,7549],[805,23226,7849],{"className":23227},[7512,7516,7549],[805,23229,7557],{"className":23230},[7556],[805,23232,23234],{"className":23233},[7529],[805,23235,23237],{"className":23236,"style":7564},[7533],[805,23238],{},[805,23240,23241,23244],{"style":10062},[805,23242],{"className":23243,"style":10066},[7541],[805,23245,23247],{"className":23246,"style":10117},[10116],[805,23248,9911],{"className":23249},[7512],[805,23251,7557],{"className":23252},[7556],[805,23254,23256],{"className":23255},[7529],[805,23257,23259],{"className":23258,"style":10130},[7533],[805,23260],{},[805,23262,22632],{"className":23263},[7621],[11,23265,8849],{},[49,23267,23268,23298,23398],{},[52,23269,23270,10946],{},[805,23271,23273,23286],{"className":23272},[7389],[805,23274,23276],{"className":23275},[7393],[7395,23277,23278],{"xmlns":7397},[7399,23279,23280,23284],{},[7402,23281,23282],{},[7414,23283,7472],{},[7487,23285,7472],{"encoding":7489},[805,23287,23289],{"className":23288,"ariaHidden":7423},[7494],[805,23290,23292,23295],{"className":23291},[7498],[805,23293],{"className":23294,"style":8843},[7502],[805,23296,7472],{"className":23297},[7512,7516],[52,23299,23300,23369,23370,44],{},[805,23301,23303,23320],{"className":23302},[7389],[805,23304,23306],{"className":23305},[7393],[7395,23307,23308],{"xmlns":7397},[7399,23309,23310,23318],{},[7402,23311,23312],{},[7411,23313,23314,23316],{},[7414,23315,828],{},[7414,23317,7849],{},[7487,23319,7925],{"encoding":7489},[805,23321,23323],{"className":23322,"ariaHidden":7423},[7494],[805,23324,23326,23329],{"className":23325},[7498],[805,23327],{"className":23328,"style":7935},[7502],[805,23330,23332,23335],{"className":23331},[7512],[805,23333,828],{"className":23334,"style":7582},[7512,7516],[805,23336,23338],{"className":23337},[7520],[805,23339,23341,23361],{"className":23340},[7524,7525],[805,23342,23344,23358],{"className":23343},[7529],[805,23345,23347],{"className":23346,"style":7881},[7533],[805,23348,23349,23352],{"style":7597},[805,23350],{"className":23351,"style":7542},[7541],[805,23353,23355],{"className":23354},[7546,7547,7548,7549],[805,23356,7849],{"className":23357},[7512,7516,7549],[805,23359,7557],{"className":23360},[7556],[805,23362,23364],{"className":23363},[7529],[805,23365,23367],{"className":23366,"style":7564},[7533],[805,23368],{}," is the true label (0 or 1) for example ",[805,23371,23373,23386],{"className":23372},[7389],[805,23374,23376],{"className":23375},[7393],[7395,23377,23378],{"xmlns":7397},[7399,23379,23380,23384],{},[7402,23381,23382],{},[7414,23383,7849],{},[7487,23385,7849],{"encoding":7489},[805,23387,23389],{"className":23388,"ariaHidden":7423},[7494],[805,23390,23392,23395],{"className":23391},[7498],[805,23393],{"className":23394,"style":11044},[7502],[805,23396,7849],{"className":23397},[7512,7516],[52,23399,23400,23513,23514,23542],{},[805,23401,23403,23425],{"className":23402},[7389],[805,23404,23406],{"className":23405},[7393],[7395,23407,23408],{"xmlns":7397},[7399,23409,23410,23422],{},[7402,23411,23412],{},[9901,23413,23414,23420],{"accent":7423},[7411,23415,23416,23418],{},[7414,23417,828],{},[7414,23419,7849],{},[7405,23421,9911],{},[7487,23423,23424],{"encoding":7489},"\\hat{y_i}",[805,23426,23428],{"className":23427,"ariaHidden":7423},[7494],[805,23429,23431,23434],{"className":23430},[7498],[805,23432],{"className":23433,"style":8020},[7502],[805,23435,23437],{"className":23436},[7512,10049],[805,23438,23440,23505],{"className":23439},[7524,7525],[805,23441,23443,23502],{"className":23442},[7529],[805,23444,23446,23491],{"className":23445,"style":10813},[7533],[805,23447,23448,23451],{"style":10062},[805,23449],{"className":23450,"style":10066},[7541],[805,23452,23454,23457],{"className":23453},[7512],[805,23455,828],{"className":23456,"style":7582},[7512,7516],[805,23458,23460],{"className":23459},[7520],[805,23461,23463,23483],{"className":23462},[7524,7525],[805,23464,23466,23480],{"className":23465},[7529],[805,23467,23469],{"className":23468,"style":7881},[7533],[805,23470,23471,23474],{"style":7597},[805,23472],{"className":23473,"style":7542},[7541],[805,23475,23477],{"className":23476},[7546,7547,7548,7549],[805,23478,7849],{"className":23479},[7512,7516,7549],[805,23481,7557],{"className":23482},[7556],[805,23484,23486],{"className":23485},[7529],[805,23487,23489],{"className":23488,"style":7564},[7533],[805,23490],{},[805,23492,23493,23496],{"style":10062},[805,23494],{"className":23495,"style":10066},[7541],[805,23497,23499],{"className":23498,"style":10117},[10116],[805,23500,9911],{"className":23501},[7512],[805,23503,7557],{"className":23504},[7556],[805,23506,23508],{"className":23507},[7529],[805,23509,23511],{"className":23510,"style":10130},[7533],[805,23512],{}," is the predicted probability by the model for example ",[805,23515,23517,23530],{"className":23516},[7389],[805,23518,23520],{"className":23519},[7393],[7395,23521,23522],{"xmlns":7397},[7399,23523,23524,23528],{},[7402,23525,23526],{},[7414,23527,7849],{},[7487,23529,7849],{"encoding":7489},[805,23531,23533],{"className":23532,"ariaHidden":7423},[7494],[805,23534,23536,23539],{"className":23535},[7498],[805,23537],{"className":23538,"style":11044},[7502],[805,23540,7849],{"className":23541},[7512,7516]," (value between 0 and 1).",[11,23544,23545],{},"Why is cross-entropy used?",[317,23547,23548,23551,23554],{},[52,23549,23550],{},"It penalizes incorrect predictions with high confidence more heavily.",[52,23552,23553],{},"It is a convex loss function, which makes optimization easier using methods like gradient descent.",[52,23555,23556],{},"It has a probabilistic interpretation, as it is based on the predicted probability by the model.",[11,23558,23559],{},"The behavior of the loss function is shown in the following graph:",[11,23561,23562,23566],{},[833,23563],{"alt":23564,"src":23565},"Cross-Entropy Loss Function Graph","\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations\u002Fshared\u002Fcross-entropy.webp",[838,23567,23564],{},[11,23569,23570],{},"The function penalizes more heavily the incorrect predictions with high confidence, which is reflected in the shape of the curve.",[317,23572,23573],{},[52,23574,23575],{},"When the true label is 1 (y=1) (blue line):",[49,23577,23578,23702,23777],{},[52,23579,23580,23581],{},"The formula simplifies to ",[805,23582,23584,23616],{"className":23583},[7389],[805,23585,23587],{"className":23586},[7393],[7395,23588,23589],{"xmlns":7397},[7399,23590,23591,23613],{},[7402,23592,23593,23595,23597,23599,23601,23603,23605,23611],{},[7414,23594,22330],{},[7405,23596,2823],{},[7405,23598,9899],{},[7414,23600,22345],{},[7405,23602,22348],{},[7405,23604,2651],{"stretchy":1435},[9901,23606,23607,23609],{"accent":7423},[7414,23608,828],{},[7405,23610,9911],{},[7405,23612,3993],{"stretchy":1435},[7487,23614,23615],{"encoding":7489},"L = -\\log(\\hat{y})",[805,23617,23619,23637],{"className":23618,"ariaHidden":7423},[7494],[805,23620,23622,23625,23628,23631,23634],{"className":23621},[7498],[805,23623],{"className":23624,"style":8042},[7502],[805,23626,22330],{"className":23627},[7512,7516],[805,23629],{"className":23630,"style":8028},[7574],[805,23632,2823],{"className":23633},[8032],[805,23635],{"className":23636,"style":8028},[7574],[805,23638,23640,23643,23646,23649,23654,23657,23699],{"className":23639},[7498],[805,23641],{"className":23642,"style":7503},[7502],[805,23644,9899],{"className":23645},[7512],[805,23647],{"className":23648,"style":7575},[7574],[805,23650,22451,23652],{"className":23651},[10668],[805,23653,22455],{"style":22454},[805,23655,2651],{"className":23656},[7507],[805,23658,23660],{"className":23659},[7512,10049],[805,23661,23663,23691],{"className":23662},[7524,7525],[805,23664,23666,23688],{"className":23665},[7529],[805,23667,23669,23677],{"className":23668,"style":10813},[7533],[805,23670,23671,23674],{"style":10062},[805,23672],{"className":23673,"style":10066},[7541],[805,23675,828],{"className":23676,"style":7582},[7512,7516],[805,23678,23679,23682],{"style":10062},[805,23680],{"className":23681,"style":10066},[7541],[805,23683,23685],{"className":23684,"style":11115},[10116],[805,23686,9911],{"className":23687},[7512],[805,23689,7557],{"className":23690},[7556],[805,23692,23694],{"className":23693},[7529],[805,23695,23697],{"className":23696,"style":10130},[7533],[805,23698],{},[805,23700,3993],{"className":23701},[7621],[52,23703,21859,23704,23776],{},[805,23705,23707,23725],{"className":23706},[7389],[805,23708,23710],{"className":23709},[7393],[7395,23711,23712],{"xmlns":7397},[7399,23713,23714,23722],{},[7402,23715,23716],{},[9901,23717,23718,23720],{"accent":7423},[7414,23719,828],{},[7405,23721,9911],{},[7487,23723,23724],{"encoding":7489},"\\hat{y}",[805,23726,23728],{"className":23727,"ariaHidden":7423},[7494],[805,23729,23731,23734],{"className":23730},[7498],[805,23732],{"className":23733,"style":8020},[7502],[805,23735,23737],{"className":23736},[7512,10049],[805,23738,23740,23768],{"className":23739},[7524,7525],[805,23741,23743,23765],{"className":23742},[7529],[805,23744,23746,23754],{"className":23745,"style":10813},[7533],[805,23747,23748,23751],{"style":10062},[805,23749],{"className":23750,"style":10066},[7541],[805,23752,828],{"className":23753,"style":7582},[7512,7516],[805,23755,23756,23759],{"style":10062},[805,23757],{"className":23758,"style":10066},[7541],[805,23760,23762],{"className":23761,"style":11115},[10116],[805,23763,9911],{"className":23764},[7512],[805,23766,7557],{"className":23767},[7556],[805,23769,23771],{"className":23770},[7529],[805,23772,23774],{"className":23773,"style":10130},[7533],[805,23775],{}," approaches 1, the loss approaches 0 (good prediction).",[52,23778,21859,23779,23850],{},[805,23780,23782,23799],{"className":23781},[7389],[805,23783,23785],{"className":23784},[7393],[7395,23786,23787],{"xmlns":7397},[7399,23788,23789,23797],{},[7402,23790,23791],{},[9901,23792,23793,23795],{"accent":7423},[7414,23794,828],{},[7405,23796,9911],{},[7487,23798,23724],{"encoding":7489},[805,23800,23802],{"className":23801,"ariaHidden":7423},[7494],[805,23803,23805,23808],{"className":23804},[7498],[805,23806],{"className":23807,"style":8020},[7502],[805,23809,23811],{"className":23810},[7512,10049],[805,23812,23814,23842],{"className":23813},[7524,7525],[805,23815,23817,23839],{"className":23816},[7529],[805,23818,23820,23828],{"className":23819,"style":10813},[7533],[805,23821,23822,23825],{"style":10062},[805,23823],{"className":23824,"style":10066},[7541],[805,23826,828],{"className":23827,"style":7582},[7512,7516],[805,23829,23830,23833],{"style":10062},[805,23831],{"className":23832,"style":10066},[7541],[805,23834,23836],{"className":23835,"style":11115},[10116],[805,23837,9911],{"className":23838},[7512],[805,23840,7557],{"className":23841},[7556],[805,23843,23845],{"className":23844},[7529],[805,23846,23848],{"className":23847,"style":10130},[7533],[805,23849],{}," approaches 0, the loss shoots to infinity (bad prediction).",[317,23852,23853],{"start":670},[52,23854,23855],{},"When the true label is 0 (y=0) (red line):",[49,23857,23858,24003,24077],{},[52,23859,23580,23860],{},[805,23861,23863,23899],{"className":23862},[7389],[805,23864,23866],{"className":23865},[7393],[7395,23867,23868],{"xmlns":7397},[7399,23869,23870,23896],{},[7402,23871,23872,23874,23876,23878,23880,23882,23884,23886,23888,23894],{},[7414,23873,22330],{},[7405,23875,2823],{},[7405,23877,9899],{},[7414,23879,22345],{},[7405,23881,22348],{},[7405,23883,2651],{"stretchy":1435},[7418,23885,7420],{},[7405,23887,9899],{},[9901,23889,23890,23892],{"accent":7423},[7414,23891,828],{},[7405,23893,9911],{},[7405,23895,3993],{"stretchy":1435},[7487,23897,23898],{"encoding":7489},"L = -\\log(1 - \\hat{y})",[805,23900,23902,23920,23952],{"className":23901,"ariaHidden":7423},[7494],[805,23903,23905,23908,23911,23914,23917],{"className":23904},[7498],[805,23906],{"className":23907,"style":8042},[7502],[805,23909,22330],{"className":23910},[7512,7516],[805,23912],{"className":23913,"style":8028},[7574],[805,23915,2823],{"className":23916},[8032],[805,23918],{"className":23919,"style":8028},[7574],[805,23921,23923,23926,23929,23932,23937,23940,23943,23946,23949],{"className":23922},[7498],[805,23924],{"className":23925,"style":7503},[7502],[805,23927,9899],{"className":23928},[7512],[805,23930],{"className":23931,"style":7575},[7574],[805,23933,22451,23935],{"className":23934},[10668],[805,23936,22455],{"style":22454},[805,23938,2651],{"className":23939},[7507],[805,23941,7420],{"className":23942},[7512],[805,23944],{"className":23945,"style":8065},[7574],[805,23947,9899],{"className":23948},[8527],[805,23950],{"className":23951,"style":8065},[7574],[805,23953,23955,23958,24000],{"className":23954},[7498],[805,23956],{"className":23957,"style":7503},[7502],[805,23959,23961],{"className":23960},[7512,10049],[805,23962,23964,23992],{"className":23963},[7524,7525],[805,23965,23967,23989],{"className":23966},[7529],[805,23968,23970,23978],{"className":23969,"style":10813},[7533],[805,23971,23972,23975],{"style":10062},[805,23973],{"className":23974,"style":10066},[7541],[805,23976,828],{"className":23977,"style":7582},[7512,7516],[805,23979,23980,23983],{"style":10062},[805,23981],{"className":23982,"style":10066},[7541],[805,23984,23986],{"className":23985,"style":11115},[10116],[805,23987,9911],{"className":23988},[7512],[805,23990,7557],{"className":23991},[7556],[805,23993,23995],{"className":23994},[7529],[805,23996,23998],{"className":23997,"style":10130},[7533],[805,23999],{},[805,24001,3993],{"className":24002},[7621],[52,24004,21859,24005,24076],{},[805,24006,24008,24025],{"className":24007},[7389],[805,24009,24011],{"className":24010},[7393],[7395,24012,24013],{"xmlns":7397},[7399,24014,24015,24023],{},[7402,24016,24017],{},[9901,24018,24019,24021],{"accent":7423},[7414,24020,828],{},[7405,24022,9911],{},[7487,24024,23724],{"encoding":7489},[805,24026,24028],{"className":24027,"ariaHidden":7423},[7494],[805,24029,24031,24034],{"className":24030},[7498],[805,24032],{"className":24033,"style":8020},[7502],[805,24035,24037],{"className":24036},[7512,10049],[805,24038,24040,24068],{"className":24039},[7524,7525],[805,24041,24043,24065],{"className":24042},[7529],[805,24044,24046,24054],{"className":24045,"style":10813},[7533],[805,24047,24048,24051],{"style":10062},[805,24049],{"className":24050,"style":10066},[7541],[805,24052,828],{"className":24053,"style":7582},[7512,7516],[805,24055,24056,24059],{"style":10062},[805,24057],{"className":24058,"style":10066},[7541],[805,24060,24062],{"className":24061,"style":11115},[10116],[805,24063,9911],{"className":24064},[7512],[805,24066,7557],{"className":24067},[7556],[805,24069,24071],{"className":24070},[7529],[805,24072,24074],{"className":24073,"style":10130},[7533],[805,24075],{}," approaches 0, the loss approaches 0 (good prediction).",[52,24078,21859,24079,24150],{},[805,24080,24082,24099],{"className":24081},[7389],[805,24083,24085],{"className":24084},[7393],[7395,24086,24087],{"xmlns":7397},[7399,24088,24089,24097],{},[7402,24090,24091],{},[9901,24092,24093,24095],{"accent":7423},[7414,24094,828],{},[7405,24096,9911],{},[7487,24098,23724],{"encoding":7489},[805,24100,24102],{"className":24101,"ariaHidden":7423},[7494],[805,24103,24105,24108],{"className":24104},[7498],[805,24106],{"className":24107,"style":8020},[7502],[805,24109,24111],{"className":24110},[7512,10049],[805,24112,24114,24142],{"className":24113},[7524,7525],[805,24115,24117,24139],{"className":24116},[7529],[805,24118,24120,24128],{"className":24119,"style":10813},[7533],[805,24121,24122,24125],{"style":10062},[805,24123],{"className":24124,"style":10066},[7541],[805,24126,828],{"className":24127,"style":7582},[7512,7516],[805,24129,24130,24133],{"style":10062},[805,24131],{"className":24132,"style":10066},[7541],[805,24134,24136],{"className":24135,"style":11115},[10116],[805,24137,9911],{"className":24138},[7512],[805,24140,7557],{"className":24141},[7556],[805,24143,24145],{"className":24144},[7529],[805,24146,24148],{"className":24147,"style":10130},[7533],[805,24149],{}," approaches 1, the loss shoots to infinity (bad prediction).",[11,24152,24153],{},"The logarithmic nature of the function ensures that the model is heavily penalized when it is \"confident but wrong\", forcing it to adjust its weights more aggressively to improve predictions.",[21,24155,24157],{"id":24156},"validation-and-optimization","Validation and Optimization",[1442,24159,24161],{"id":24160},"evaluation-metrics","Evaluation Metrics",[11,24163,24164,24165,24167],{},"For ",[29,24166,8359],{},", the common metrics include:",[49,24169,24170,24176,24182],{},[52,24171,24172,24175],{},[29,24173,24174],{},"Mean Squared Error (MSE)",": Average of the squares of the differences between the actual values and the predictions.",[52,24177,24178,24181],{},[29,24179,24180],{},"Root Mean Squared Error (RMSE)",": Square root of the MSE, which has the same unit as the dependent variable.",[52,24183,24184,24187],{},[29,24185,24186],{},"Coefficient of Determination (R²)",": Proportion of the variance in the dependent variable that is explained by the model.",[11,24189,24164,24190,24167],{},[29,24191,18767],{},[49,24193,24194,24200,24206,24212],{},[52,24195,24196,24199],{},[29,24197,24198],{},"Accuracy",": Proportion of correct predictions over the total number of examples.",[52,24201,24202,24205],{},[29,24203,24204],{},"Precision",": Proportion of true positives over the total number of positive predictions.",[52,24207,24208,24211],{},[29,24209,24210],{},"Recall (Sensitivity)",": Proportion of true positives over the total number of actual positive examples.",[52,24213,24214,24217],{},[29,24215,24216],{},"F1 Score",": Harmonic mean of precision and recall, providing a balanced measure between both.",[78,24219,24220],{},[11,24221,24222],{},"The choice of the appropriate metric depends on the context of the problem and the consequences of classification errors. For example, in a fraud detection problem, it is more important to minimize false negatives (failing to detect fraud) than false positives (marking a legitimate transaction as fraud), so recall could be a more relevant metric than precision.",[1442,24224,24226],{"id":24225},"optimization","Optimization",[11,24228,24229,24230,24233],{},"Optimization of machine learning models refers to the process of adjusting the model parameters to ",[29,24231,24232],{},"minimize the loss function",". This can be achieved through techniques like gradient descent, which iteratively adjusts the model weights in the direction that reduces the loss.",[11,24235,24236],{},"Gradient descent can be mathematically expressed as:",[805,24238,24240],{"className":24239},[7980],[805,24241,24243,24278],{"className":24242},[7389],[805,24244,24246],{"className":24245},[7393],[7395,24247,24248],{"xmlns":7397,"display":7989},[7399,24249,24250,24275],{},[7402,24251,24252,24255,24257,24259,24261,24264,24267,24269,24271,24273],{},[7414,24253,24254],{},"θ",[7405,24256,2823],{},[7414,24258,24254],{},[7405,24260,9899],{},[7414,24262,24263],{},"α",[7414,24265,24266],{"mathvariant":7457},"∇",[7414,24268,22653],{},[7405,24270,2651],{"stretchy":1435},[7414,24272,24254],{},[7405,24274,3993],{"stretchy":1435},[7487,24276,24277],{"encoding":7489},"\\theta = \\theta - \\alpha \\nabla J(\\theta)",[805,24279,24281,24300,24319],{"className":24280,"ariaHidden":7423},[7494],[805,24282,24284,24287,24291,24294,24297],{"className":24283},[7498],[805,24285],{"className":24286,"style":10813},[7502],[805,24288,24254],{"className":24289,"style":24290},[7512,7516],"margin-right:0.0278em;",[805,24292],{"className":24293,"style":8028},[7574],[805,24295,2823],{"className":24296},[8032],[805,24298],{"className":24299,"style":8028},[7574],[805,24301,24303,24307,24310,24313,24316],{"className":24302},[7498],[805,24304],{"className":24305,"style":24306},[7502],"height:0.7778em;vertical-align:-0.0833em;",[805,24308,24254],{"className":24309,"style":24290},[7512,7516],[805,24311],{"className":24312,"style":8065},[7574],[805,24314,9899],{"className":24315},[8527],[805,24317],{"className":24318,"style":8065},[7574],[805,24320,24322,24325,24329,24332,24335,24338,24341],{"className":24321},[7498],[805,24323],{"className":24324,"style":7503},[7502],[805,24326,24263],{"className":24327,"style":24328},[7512,7516],"margin-right:0.0037em;",[805,24330,24266],{"className":24331},[7512],[805,24333,22653],{"className":24334,"style":22763},[7512,7516],[805,24336,2651],{"className":24337},[7507],[805,24339,24254],{"className":24340,"style":24290},[7512,7516],[805,24342,3993],{"className":24343},[7621],[11,24345,8849],{},[49,24347,24348,24380,24412],{},[52,24349,24350,24379],{},[805,24351,24353,24367],{"className":24352},[7389],[805,24354,24356],{"className":24355},[7393],[7395,24357,24358],{"xmlns":7397},[7399,24359,24360,24364],{},[7402,24361,24362],{},[7414,24363,24254],{},[7487,24365,24366],{"encoding":7489},"\\theta",[805,24368,24370],{"className":24369,"ariaHidden":7423},[7494],[805,24371,24373,24376],{"className":24372},[7498],[805,24374],{"className":24375,"style":10813},[7502],[805,24377,24254],{"className":24378,"style":24290},[7512,7516]," represents the model parameters (for example, the coefficients in regression).",[52,24381,24382,24411],{},[805,24383,24385,24399],{"className":24384},[7389],[805,24386,24388],{"className":24387},[7393],[7395,24389,24390],{"xmlns":7397},[7399,24391,24392,24396],{},[7402,24393,24394],{},[7414,24395,24263],{},[7487,24397,24398],{"encoding":7489},"\\alpha",[805,24400,24402],{"className":24401,"ariaHidden":7423},[7494],[805,24403,24405,24408],{"className":24404},[7498],[805,24406],{"className":24407,"style":8843},[7502],[805,24409,24263],{"className":24410,"style":24328},[7512,7516]," is the learning rate, which controls the size of the steps taken in each iteration.",[52,24413,24414,24463],{},[805,24415,24417,24439],{"className":24416},[7389],[805,24418,24420],{"className":24419},[7393],[7395,24421,24422],{"xmlns":7397},[7399,24423,24424,24436],{},[7402,24425,24426,24428,24430,24432,24434],{},[7414,24427,24266],{"mathvariant":7457},[7414,24429,22653],{},[7405,24431,2651],{"stretchy":1435},[7414,24433,24254],{},[7405,24435,3993],{"stretchy":1435},[7487,24437,24438],{"encoding":7489},"\\nabla J(\\theta)",[805,24440,24442],{"className":24441,"ariaHidden":7423},[7494],[805,24443,24445,24448,24451,24454,24457,24460],{"className":24444},[7498],[805,24446],{"className":24447,"style":7503},[7502],[805,24449,24266],{"className":24450},[7512],[805,24452,22653],{"className":24453,"style":22763},[7512,7516],[805,24455,2651],{"className":24456},[7507],[805,24458,24254],{"className":24459,"style":24290},[7512,7516],[805,24461,3993],{"className":24462},[7621]," is the gradient of the loss function with respect to the parameters, indicating the direction of greatest increase in the loss.",[11,24465,24466],{},"The optimization process continues until a convergence criterion is met, such as a maximum number of iterations or a minimum improvement in the loss function.",[49,24468,24469,24504,24558],{},[52,24470,24471,24472,4595,24475,24503],{},"It has a ",[29,24473,24474],{},"learning rate",[805,24476,24478,24491],{"className":24477},[7389],[805,24479,24481],{"className":24480},[7393],[7395,24482,24483],{"xmlns":7397},[7399,24484,24485,24489],{},[7402,24486,24487],{},[7414,24488,24263],{},[7487,24490,24398],{"encoding":7489},[805,24492,24494],{"className":24493,"ariaHidden":7423},[7494],[805,24495,24497,24500],{"className":24496},[7498],[805,24498],{"className":24499,"style":8843},[7502],[805,24501,24263],{"className":24502,"style":24328},[7512,7516],") that controls the size of the steps taken in each iteration (typically a small value like 0.01 or 0.001).",[52,24505,2472,24506,4595,24509,24557],{},[29,24507,24508],{},"gradient",[805,24510,24512,24533],{"className":24511},[7389],[805,24513,24515],{"className":24514},[7393],[7395,24516,24517],{"xmlns":7397},[7399,24518,24519,24531],{},[7402,24520,24521,24523,24525,24527,24529],{},[7414,24522,24266],{"mathvariant":7457},[7414,24524,22653],{},[7405,24526,2651],{"stretchy":1435},[7414,24528,24254],{},[7405,24530,3993],{"stretchy":1435},[7487,24532,24438],{"encoding":7489},[805,24534,24536],{"className":24535,"ariaHidden":7423},[7494],[805,24537,24539,24542,24545,24548,24551,24554],{"className":24538},[7498],[805,24540],{"className":24541,"style":7503},[7502],[805,24543,24266],{"className":24544},[7512],[805,24546,22653],{"className":24547,"style":22763},[7512,7516],[805,24549,2651],{"className":24550},[7507],[805,24552,24254],{"className":24553,"style":24290},[7512,7516],[805,24555,3993],{"className":24556},[7621],") is a vector containing the partial derivatives of the loss function with respect to each parameter, indicating the direction of greatest increase in the loss.",[52,24559,24560,24561,24564],{},"The optimization process continues until a ",[29,24562,24563],{},"convergence"," criterion is met, such as a maximum number of iterations or a minimum improvement in the loss function.",[11,24566,24567],{},"Graphically:",[11,24569,24570,24574],{},[833,24571],{"alt":24572,"src":24573},"Graphical representation of the gradient descent optimization process","\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations\u002Fshared\u002Fgradient-descent.webp",[838,24575,24576],{},"Optimization with Gradient Descent",[317,24578,24579,24582,24585,24588],{},[52,24580,24581],{},"The process starts with a random point on the loss function (INITIAL POINT).",[52,24583,24584],{},"The gradient is calculated at that point, indicating the direction of greatest increase in the loss.",[52,24586,24587],{},"The model parameters are updated in the opposite direction of the gradient, with a step size controlled by the learning rate (LEARNING RATE \u002F STEP SIZE).",[52,24589,24590],{},"This process is repeated iteratively until a local or global minimum of the loss function is reached, indicating that the model has been optimized.",[11,24592,24593],{},"The learning rate is crucial for the success of the optimization process:",[49,24595,24596,24606,24615],{},[52,24597,24598,24601,24602,24605],{},[29,24599,24600],{},"DIVERGENCE",": If the learning rate is too ",[29,24603,24604],{},"high",", the model may diverge, jumping over the minimum and increasing the loss.",[52,24607,24608,24601,24611,24614],{},[29,24609,24610],{},"SLOW CONVERGENCE",[29,24612,24613],{},"low",", the optimization process may be very slow, taking a long time to converge or getting stuck in a local minimum.",[52,24616,24617,24620],{},[29,24618,24619],{},"OPTIMAL CONVERGENCE",": An appropriate learning rate allows the model to converge efficiently towards a global or local minimum, effectively optimizing the loss function.",[21,24622,24624],{"id":24623},"the-machine-learning-process","The Machine Learning Process",[11,24626,24627],{},"We can summarize the ML process in:",[317,24629,24630,24641,24652,24663,24673],{},[52,24631,24632,24635,24636],{},[29,24633,24634],{},"Learning Paradigm",": Choose the type of learning (supervised, unsupervised, reinforcement) based on the problem to solve.\n",[49,24637,24638],{},[52,24639,24640],{},"Define the type of problem and available data.",[52,24642,24643,24646,24647],{},[29,24644,24645],{},"Mathematical Model",": Select an appropriate model (regression, classification, clustering) and understand its mathematical formulation.\n",[49,24648,24649],{},[52,24650,24651],{},"Establish the mathematical relationship between input and output.",[52,24653,24654,24657,24658],{},[29,24655,24656],{},"Loss\u002FCost Function",": Define a loss function that measures the model's error and can be optimized.\n",[49,24659,24660],{},[52,24661,24662],{},"Quantifies how bad the model is in its predictions.",[52,24664,24665,24667,24668],{},[29,24666,24226],{},": Use techniques like gradient descent to adjust the model parameters and minimize the loss function.\n",[49,24669,24670],{},[52,24671,24672],{},"Finds the best parameters for the model to make good predictions.",[52,24674,24675,24678,24679],{},[29,24676,24677],{},"Evaluation",": Measure the model's performance using appropriate metrics for the problem type (MSE for regression, accuracy\u002Frecall for classification, etc.).\n",[49,24680,24681],{},[52,24682,24683],{},"Validates the performance of the model and its ability to generalize to unseen data.",{"title":666,"searchDepth":667,"depth":667,"links":24685},[24686,24691,24695,24699],{"id":7372,"depth":670,"text":7373,"children":24687},[24688,24689,24690],{"id":7379,"depth":667,"text":186},{"id":8085,"depth":667,"text":311},{"id":8312,"depth":667,"text":419},{"id":8345,"depth":670,"text":8346,"children":24692},[24693,24694],{"id":8359,"depth":667,"text":272},{"id":18767,"depth":667,"text":266},{"id":24156,"depth":670,"text":24157,"children":24696},[24697,24698],{"id":24160,"depth":667,"text":24161},{"id":24225,"depth":667,"text":24226},{"id":24623,"depth":670,"text":24624},"2026-04-06","\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations\u002Fshared\u002Fml-paradigms.webp",{},"\u002Fblog\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations",{"title":7353,"description":7358},{"loc":24706,"priority":686,"lastmod":24700},"\u002Fes\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations","machine-learning-paradigms-and-mathematical-foundations","blog\u002Fblog\u002Fmachine-learning-paradigms-and-mathematical-foundations","Types of machine learning, common algorithms and essential mathematical foundations for understanding how machine learning models work.",[109,691,692,693,109],"UebnJOgy7GPnhDI7uhV4sfkeLEvIHGR5vGNjmPhJxHE",1776805794318]