{"id":1921,"date":"2025-07-20T16:38:44","date_gmt":"2025-07-20T15:38:44","guid":{"rendered":"https:\/\/cyberenlightener.com\/?page_id=1921"},"modified":"2025-09-27T16:32:25","modified_gmt":"2025-09-27T15:32:25","slug":"supervised-learning","status":"publish","type":"page","link":"https:\/\/cyberenlightener.com\/?page_id=1921","title":{"rendered":"Supervised Learning"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Regression<\/strong><\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83d\udcca <strong>Introduction to Regression Analysis<\/strong><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">What is Regression?<\/h3>\n\n\n\n<p>Regression is a <strong>statistical method<\/strong> used to study the <strong>relationship between variables<\/strong>. It helps us understand <strong>how one or more independent variables (predictors)<\/strong> influence a <strong>dependent variable (response)<\/strong>.<\/p>\n\n\n\n<p>At its core, regression answers this question:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><em>&#8220;How does a change in one variable affect another?&#8221;<\/em><\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0d Why Study Regression?<\/h3>\n\n\n\n<p>Regression is a powerful tool used to:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Predict future outcomes<\/strong><\/li>\n\n\n\n<li><strong>Identify trends and relationships<\/strong><\/li>\n\n\n\n<li><strong>Test hypotheses<\/strong><\/li>\n\n\n\n<li><strong>Inform decisions and policies<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83c\udf93 Real-World Questions Answered by Regression<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th><strong>Context<\/strong><\/th><th><strong>Question<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Genetics<\/td><td>Are daughters taller than their mothers?<\/td><\/tr><tr><td>Education<\/td><td>Does reducing class size improve student performance?<\/td><\/tr><tr><td>Geology<\/td><td>Can we predict the time of Old Faithful\u2019s next eruption using the last one?<\/td><\/tr><tr><td>Health &amp; Nutrition<\/td><td>Do dietary changes reduce cholesterol levels?<\/td><\/tr><tr><td>Economics &amp; Demographics<\/td><td>Do wealthier countries have lower birth rates?<\/td><\/tr><tr><td>Transportation<\/td><td>Can better highway designs lower accident rates?<\/td><\/tr><tr><td>Environmental Science<\/td><td>Is water usage increasing over the years?<\/td><\/tr><tr><td>Real Estate &amp; Conservation<\/td><td>Do conservation easements reduce land values?<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83e\uddee Linear Regression: The Foundation<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>Here we will focus on <strong>Linear Regression<\/strong>, the most commonly used regression technique.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\">What is Linear Regression?<\/h3>\n\n\n\n<p>Linear regression models the relationship between variables by fitting a <strong>straight line<\/strong> to the data. It assumes the response is a <strong>linear function<\/strong> of the predictors.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is easy to interpret.<\/li>\n\n\n\n<li>It forms the basis for more advanced regression techniques.<\/li>\n\n\n\n<li>It is widely applicable in fields like economics, biology, engineering, and social sciences.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83c\udfaf Objective of Regression Analysis<\/h2>\n\n\n\n<p>The main goal of regression is to:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Summarize complex data<\/strong> in a <strong>simple and meaningful way<\/strong><\/li>\n\n\n\n<li><strong>Understand relationships<\/strong> between variables<\/li>\n\n\n\n<li><strong>Make predictions<\/strong> and <strong>inform decisions<\/strong><\/li>\n\n\n\n<li>Represent relationships <strong>elegantly and effectively<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Sometimes, a <strong>theory or prior knowledge<\/strong> may guide the form of the relationship (e.g., linear, quadratic).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u270d\ufe0f Key Takeaways<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Regression studies <strong>dependence<\/strong> between variables.<\/li>\n\n\n\n<li>It helps answer a wide range of <strong>real-world questions<\/strong>.<\/li>\n\n\n\n<li><strong>Linear regression<\/strong> is the cornerstone of most regression techniques.<\/li>\n\n\n\n<li>Simplicity, interpretability, and utility are at the heart of regression analysis.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>Thank you! Here&#8217;s the <strong>previously structured version<\/strong> with <strong>all heading numbers removed<\/strong>, making it visually clean and ideal for student-friendly materials like slides, handouts, or websites.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83d\udccc Scatterplots \u2013 A First Look at Regression<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udcd8 Understanding the Basics<\/h3>\n\n\n\n<p>In simple regression, we study <strong>how one variable (X)<\/strong>\u2014called the <strong>predictor<\/strong> influences another variable (Y), the <strong>response<\/strong>.<br>We observe data in pairs:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"390\" height=\"121\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1-4.jpg\" alt=\"\" class=\"wp-image-1922\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1-4.jpg 390w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1-4-300x93.jpg 300w\" sizes=\"auto, (max-width: 390px) 100vw, 390px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>X: Independent variable (e.g., <strong>mother\u2019s height<\/strong>)<\/li>\n\n\n\n<li>Y: Dependent variable (e.g., <strong>daughter\u2019s height<\/strong>)<\/li>\n<\/ul>\n\n\n\n<p>To explore the relationship visually, we use a <strong>scatterplot<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83d\udcca What is a Scatterplot?<\/h2>\n\n\n\n<p>A <strong>scatterplot<\/strong> is a graph that shows each observation as a point:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>X-axis<\/strong>: Predictor variable (e.g., mother\u2019s height)<\/li>\n\n\n\n<li><strong>Y-axis<\/strong>: Response variable (e.g., daughter\u2019s height)<\/li>\n<\/ul>\n\n\n\n<p>It helps in visualizing:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Trends<\/li>\n\n\n\n<li>Correlations<\/li>\n\n\n\n<li>Outliers<\/li>\n\n\n\n<li>Patterns suggesting linearity or non-linearity<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83d\udc68\u200d\ud83d\udc69\u200d\ud83d\udc67 Inheritance of Height: A Historical Example<\/h2>\n\n\n\n<p>Karl Pearson (1893\u20131898) collected data on <strong>1375<\/strong> mother\u2013daughter pairs in the UK.<br>He wanted to understand:<br>\ud83d\udc49 <em>\u201cDo taller mothers tend to have taller daughters?\u201d<\/em><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Predictor<\/th><th>Response<\/th><\/tr><\/thead><tbody><tr><td>Mother&#8217;s Height (mheight)<\/td><td>Daughter\u2019s Height (dheight)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>We visualize the data using a scatterplot:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"678\" height=\"440\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/2-4.jpg\" alt=\"\" class=\"wp-image-1923\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/2-4.jpg 678w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/2-4-300x195.jpg 300w\" sizes=\"auto, (max-width: 678px) 100vw, 678px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0e Figure: Jittered Scatterplot<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Adds slight randomness to avoid overplotting (many points at same location).<\/li>\n\n\n\n<li>Gives clearer view of point density.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0e Figure: Original Data<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Shows exact height values (rounded to nearest inch).<\/li>\n\n\n\n<li>Suffers from <strong>overplotting<\/strong>\u2014multiple data points overlap.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u2728 Key Insights from the Scatterplot<\/h2>\n\n\n\n<p><strong>Equal Axes for Fair Comparison<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Since mothers and daughters have similar height ranges, both axes should be scaled equally.<\/li>\n\n\n\n<li>A perfect 45\u00b0 line would represent identical heights.<\/li>\n<\/ul>\n\n\n\n<p><strong>Jittering Removes Overlap<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Helps in showing all data points.<\/li>\n\n\n\n<li>A small random value (\u00b10.5) is added to each value to unstack the points.<\/li>\n<\/ul>\n\n\n\n<p><strong>Detecting Dependence<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Scatter of points changes with the predictor.<\/li>\n\n\n\n<li>See visual comparison using mheight = 58, 64, 68:<br>\u27a4 Average daughter\u2019s height increases with mother\u2019s height.<\/li>\n<\/ul>\n\n\n\n<p><strong>Elliptical Shape Suggests Linearity<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Points form an ellipse tilted upward.<\/li>\n\n\n\n<li>Implies a <strong>positive linear trend<\/strong>.<\/li>\n\n\n\n<li>A good candidate for <strong>simple linear regression<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Special Data Points<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Type<\/th><th>Description<\/th><th>Role<\/th><\/tr><\/thead><tbody><tr><td><strong>Leverage Points<\/strong><\/td><td>Unusually high or low X-values<\/td><td>Strong influence on regression line<\/td><\/tr><tr><td><strong>Outliers<\/strong><\/td><td>Unusually high or low Y-values for given X<\/td><td>May indicate anomalies or errors<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>A scatterplot helps check if the response depends on the predictor\u2014here, taller mothers generally have taller daughters, showing an upward trend. Figure 1.2 shows that as mother\u2019s height increases (58, 64, 68 inches), the average daughter\u2019s height also increases. Points far from others horizontally are leverage points; vertically distant ones are potential outliers.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"673\" height=\"473\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/3-4.jpg\" alt=\"\" class=\"wp-image-1924\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/3-4.jpg 673w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/3-4-300x211.jpg 300w\" sizes=\"auto, (max-width: 673px) 100vw, 673px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83e\udde0 Summary: Why Scatterplots Matter<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Offer <strong>visual insight<\/strong> before any formal modeling.<\/li>\n\n\n\n<li>Help assess:\n<ul class=\"wp-block-list\">\n<li>Strength and direction of relationship<\/li>\n\n\n\n<li>Suitability for regression (e.g., linearity)<\/li>\n\n\n\n<li>Presence of outliers or influential points<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u2705 Pro Tips<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Always plot your data first<\/strong>\u2014regression comes later!<\/li>\n\n\n\n<li>Use <strong>jittering<\/strong> when data is rounded.<\/li>\n\n\n\n<li>Maintain equal axis scaling when variables are measured on similar scales.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Forbes\u2019s Data (Figures 1.3 &amp; 1.4):<\/strong><br>Collected in the <strong>19th century (1800s)<\/strong> by James D. Forbes, the data show the relationship between boiling point and atmospheric pressure. Figure 1.3a reveals a curved trend, and residuals in 1.3b show a poor linear fit. After applying a log transformation to pressure, Figure 1.4a becomes linear, and residuals in 1.4b scatter evenly\u2014indicating a good linear model.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"707\" height=\"395\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.3.jpg\" alt=\"\" class=\"wp-image-1925\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.3.jpg 707w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.3-300x168.jpg 300w\" sizes=\"auto, (max-width: 707px) 100vw, 707px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"703\" height=\"400\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.4-2.jpg\" alt=\"\" class=\"wp-image-1932\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.4-2.jpg 703w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.4-2-300x171.jpg 300w\" sizes=\"auto, (max-width: 703px) 100vw, 703px\" \/><\/figure>\n\n\n\n<p><strong>Smallmouth Bass Growth (Figure 1.5):<\/strong><br>Collected from <strong>Lake Ontario<\/strong> in the <strong>1970s<\/strong>, this data shows how length increases with age. However, there&#8217;s large variability among fish of the same age, meaning regression estimates the average size, not individual growth. Predictions carry uncertainty due to biological variation.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"712\" height=\"492\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.5-1.jpg\" alt=\"\" class=\"wp-image-1931\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.5-1.jpg 712w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.5-1-300x207.jpg 300w\" sizes=\"auto, (max-width: 712px) 100vw, 712px\" \/><\/figure>\n\n\n\n<p><strong>Snowfall Prediction (Figure 1.6):<\/strong><br>From <strong>Flagstaff, Arizona<\/strong>, data from <strong>1910 to 1960<\/strong> compares early and late seasonal snowfall. The scatterplot shows no meaningful trend\u2014early snowfall (Sept\u2013Dec) does not predict later snowfall (Jan\u2013May). Correlation is weak or absent.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"746\" height=\"492\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.6-1.jpg\" alt=\"\" class=\"wp-image-1929\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.6-1.jpg 746w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.6-1-300x198.jpg 300w\" sizes=\"auto, (max-width: 746px) 100vw, 746px\" \/><\/figure>\n\n\n\n<p><strong>Turkey Weight Gain (Figure 1.7):<\/strong><br>From an <strong>animal nutrition study in the 1960s<\/strong>, the data explore how turkey weight gain varies with methionine dose and its source. Gain increases with dose, but overlapping variability and different slopes among sources suggest potential interactions and biological complexity.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"809\" height=\"535\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/TURKEY.jpg\" alt=\"\" class=\"wp-image-1969\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/TURKEY.jpg 809w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/TURKEY-300x198.jpg 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/TURKEY-768x508.jpg 768w\" sizes=\"auto, (max-width: 809px) 100vw, 809px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>MEAN FUNCTION<\/strong><\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>The <strong>mean function<\/strong> describes how the <strong>average value of a response variable (Y)<\/strong> changes with a <strong>predictor variable (X)<\/strong>. It is written as <strong>E(Y | X = x)<\/strong>, meaning the <strong>expected value of Y<\/strong> given that X takes a specific value x. In many cases, we model this relationship using a <strong>straight line<\/strong>, such as <strong>E(Y | X = x) = \u03b2\u2080 + \u03b2\u2081x<\/strong>, where \u03b2\u2080 is the intercept and \u03b2\u2081 is the slope. This linear form represents a simple mean function and helps us understand the trend between variables.<\/p>\n\n\n\n<p>In the <strong>Galton height dataset<\/strong> (collected in <strong>1885\u20131886<\/strong>), we examine how a daughter\u2019s height (dheight) depends on her mother\u2019s height (mheight). If mothers and daughters were exactly the same height, we\u2019d expect a <strong>mean function with slope 1<\/strong>. This is shown as a <strong>dashed line<\/strong> in <strong>Figure 1.8<\/strong>. However, the <strong>observed data<\/strong> shows a <strong>solid line<\/strong> with <strong>slope less than 1<\/strong>, indicating that tall mothers tend to have slightly shorter daughters, and short mothers slightly taller daughters. This effect is known as <strong>regression to the mean<\/strong>, a phenomenon first noted by <strong>Francis Galton<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"685\" height=\"497\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.8.jpg\" alt=\"\" class=\"wp-image-1933\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.8.jpg 685w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/1.8-300x218.jpg 300w\" sizes=\"auto, (max-width: 685px) 100vw, 685px\" \/><\/figure>\n\n\n\n<p>In another example, <strong>Figure 1.5<\/strong> uses smallmouth bass data to illustrate the concept. The <strong>dashed curve<\/strong> connects the <strong>average length of fish<\/strong> at each age, forming an empirical estimate of <strong>E(length | age)<\/strong>. This curve acts as the mean function, summarizing how expected fish length increases with age. The individual data points still show variation around the curve, reminding us that not all fish grow at the same rate. <\/p>\n\n\n\n<h3 class=\"wp-block-heading\">What does &#8220;mean&#8221; have to do with it?<\/h3>\n\n\n\n<p>The <strong>mean<\/strong> (average) serves as a <strong>reference point<\/strong> for understanding the relationship between mothers&#8217; and daughters&#8217; heights. In the context of Galton&#8217;s data:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If height were perfectly inherited, we&#8217;d expect that a mother <strong>one inch taller than average<\/strong> would have a daughter also <strong>one inch taller than average<\/strong>.<\/li>\n\n\n\n<li>This would result in a <strong>mean function<\/strong> (i.e., the expected daughter height given mother\u2019s height) with a <strong>slope of 1<\/strong> \u2014 meaning daughters&#8217; heights track perfectly with mothers&#8217;.<\/li>\n<\/ul>\n\n\n\n<p>But what was observed?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>regression line<\/strong> (the solid line) had a <strong>slope less than 1<\/strong> \u2014 meaning:\n<ul class=\"wp-block-list\">\n<li><strong>Tall mothers<\/strong> tend to have daughters who are tall <strong>but not quite as tall<\/strong>.<\/li>\n\n\n\n<li><strong>Short mothers<\/strong> tend to have daughters who are short <strong>but not quite as short<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Over many cases, <strong>extreme values (very tall or very short)<\/strong> regress <strong>toward the mean<\/strong> height.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udcc9 Why is the mean function important here?<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>It defines expectations<\/strong>:\n<ul class=\"wp-block-list\">\n<li>The mean function tells us the <strong>expected daughter\u2019s height<\/strong> for a given mother\u2019s height.<\/li>\n\n\n\n<li>Without the concept of the <strong>mean<\/strong>, we can&#8217;t define or detect <strong>regression to the mean<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>It reveals the pattern<\/strong>:\n<ul class=\"wp-block-list\">\n<li>The deviation from the dashed line (slope = 1) shows that heredity is <strong>not perfect<\/strong>.<\/li>\n\n\n\n<li>This pattern <strong>emerges statistically<\/strong> when analyzing many mother\u2013daughter pairs and comparing them to the <strong>mean<\/strong> of the population.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83e\udde0 What does &#8220;regression to the mean&#8221; actually mean?<\/h3>\n\n\n\n<p>It means that:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Children of extreme parents (very tall or very short) tend to be closer to the average height<\/strong> than their parents. <\/p>\n\n\n\n<p>It\u2019s not because of any active &#8220;pull&#8221; toward the mean \u2014 it\u2019s a <strong>statistical effect<\/strong> that arises because traits like height are influenced by both <strong>genetics<\/strong> and <strong>environment<\/strong>, and some of the extremes are due to <strong>random variation<\/strong>. <\/p>\n\n\n\n<p>Thus, the <strong>mean function<\/strong> is central to regression\u2014it captures how <strong>the average response<\/strong> changes with the predictor and provides the basis for building statistical models.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"629\" height=\"162\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/MEAN2.jpg\" alt=\"\" class=\"wp-image-1973\" style=\"width:742px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/MEAN2.jpg 629w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/MEAN2-300x77.jpg 300w\" sizes=\"auto, (max-width: 629px) 100vw, 629px\" \/><\/figure>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83d\udcca Understanding Variance Functions in Regression<\/h2>\n\n\n\n<p>When analyzing how a response variable (like a daughter&#8217;s height) depends on a predictor (like a mother\u2019s height), we don\u2019t just look at the <strong>average trend<\/strong>\u2014we also care about how <strong>spread out<\/strong> the data is. This is where the <strong>variance function<\/strong> comes in.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0d What Is a Variance Function?<\/h3>\n\n\n\n<p>The <strong>variance function<\/strong> tells us how <strong>variable<\/strong> the response is when we fix the predictor at a certain value. Mathematically, it is written as:<\/p>\n\n\n\n<p><strong>Var(Y | X = x)<\/strong><br>(read as: <em>the variance of Y given X equals x<\/em>)<\/p>\n\n\n\n<p>This function gives insight into how consistent or spread out the response values are around their mean, for each fixed value of the predictor.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"603\" height=\"331\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR1.jpg\" alt=\"\" class=\"wp-image-1964\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR1.jpg 603w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR1-300x165.jpg 300w\" sizes=\"auto, (max-width: 603px) 100vw, 603px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"647\" height=\"312\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR2.jpg\" alt=\"\" class=\"wp-image-1965\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR2.jpg 647w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR2-300x145.jpg 300w\" sizes=\"auto, (max-width: 647px) 100vw, 647px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"630\" height=\"411\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR3.jpg\" alt=\"\" class=\"wp-image-1966\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR3.jpg 630w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/VAR3-300x196.jpg 300w\" sizes=\"auto, (max-width: 630px) 100vw, 630px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udcc8 Visual Examples<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Figure 1.2 \u2013 Daughters\u2019 Height vs. Mothers\u2019 Height<\/strong><\/h4>\n\n\n\n<p>For the Galton height data:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The variability in <strong>daughter&#8217;s height<\/strong> for given <strong>mother\u2019s heights<\/strong> (58, 64, 68 inches) appears roughly <strong>constant<\/strong>.<\/li>\n\n\n\n<li>This means the spread of daughter heights doesn\u2019t change much across different mother heights.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Figure 1.5 \u2013 Smallmouth Bass Length by Age<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Here too, the <strong>variance in fish lengths<\/strong> across ages seems fairly consistent.<\/li>\n\n\n\n<li>Although it&#8217;s not guaranteed, assuming <strong>constant variance<\/strong> is <strong>reasonable<\/strong> in this case.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Turkey Data Example<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In the turkey weight dataset, only <strong>treatment means<\/strong> are plotted\u2014not individual pen-level values.<\/li>\n\n\n\n<li>So, we <strong>can\u2019t evaluate<\/strong> the variance function directly, since the graph doesn\u2019t show within-treatment variability.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udccf Common Assumption in Linear Models<\/h3>\n\n\n\n<p>In many simple linear regression models, we <strong>assume<\/strong> the variance stays the same for all values of x:<\/p>\n\n\n\n<p><strong>Var(Y | X = x) = \u03c3\u00b2<\/strong><br>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u03c3\u00b2<\/strong> (<em>sigma squared<\/em>) is a <strong>positive constant<\/strong>,<\/li>\n\n\n\n<li>It reflects a uniform level of variability across all predictor values.<\/li>\n<\/ul>\n\n\n\n<p>This assumption helps simplify model fitting, although <strong>more advanced models<\/strong> (explored in <strong>Chapter 7<\/strong>) can allow variance to change with x.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83e\udde0 Key Takeaway<\/h3>\n\n\n\n<p>While the <strong>mean function<\/strong> tells us the average trend, the <strong>variance function<\/strong> reveals how <strong>spread out<\/strong> the data is at each level of the predictor. Both are essential for building and interpreting effective regression models.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\ud83d\udcda <strong>Disclaimer<\/strong>:<br>The concepts, explanations, and figures presented on this webpage are <strong>adapted from<\/strong> the textbook <em>Applied Linear Regression<\/em> by <strong>Sanford Weisberg<\/strong>, 4th Edition, Wiley-Interscience (2013). All rights and credits belong to the original author and publisher<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Simple Linear Regression<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Linear-Regression_Estimations.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Linear-Regression_Estimations.\"><\/object><a id=\"wp-block-file--media-01de4aff-ef54-45b0-8ca1-05fbb7aa7c88\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Linear-Regression_Estimations.pdf\">Linear-Regression_Estimations<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Linear-Regression_Estimations.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-01de4aff-ef54-45b0-8ca1-05fbb7aa7c88\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Simple Linear Regression<\/strong> : <strong>ERROR CALCULATION<\/strong>S<\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Final_Linear_Regression_AllErrorsSSROY.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Final_Linear_Regression_AllErrorsSSROY.\"><\/object><a id=\"wp-block-file--media-4b341f49-9d2f-46cb-b6d9-fae39ec6bb76\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Final_Linear_Regression_AllErrorsSSROY.pdf\">Final_Linear_Regression_AllErrorsSSROY<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Final_Linear_Regression_AllErrorsSSROY.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-4b341f49-9d2f-46cb-b6d9-fae39ec6bb76\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Multiple linear regression<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Multiple_Linear_RegressionSSROY.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Multiple_Linear_RegressionSSROY.\"><\/object><a id=\"wp-block-file--media-69c6eee3-61b9-41a5-9294-519ea2c6cdf5\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Multiple_Linear_RegressionSSROY.pdf\">Multiple_Linear_RegressionSSROY<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Multiple_Linear_RegressionSSROY.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-69c6eee3-61b9-41a5-9294-519ea2c6cdf5\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Na\u00efve Bayes Classification<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/Roy_Naive-Bayes-tutorial_ssroy2025NAIVE-BAYSE.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Roy_Na\u00efve Bayes tutorial_ssroy2025NAIVE-BAYSE.\"><\/object><a id=\"wp-block-file--media-4f34c8bc-ad9a-413f-b738-c86f9ccec121\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/Roy_Naive-Bayes-tutorial_ssroy2025NAIVE-BAYSE.pdf\">Roy_Na\u00efve Bayes tutorial_ssroy2025NAIVE-BAYSE<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/Roy_Naive-Bayes-tutorial_ssroy2025NAIVE-BAYSE.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-4f34c8bc-ad9a-413f-b738-c86f9ccec121\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Na\u00efve Bayes with Continuous Attributes<\/strong><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Why Gaussian Distribution?<\/h3>\n\n\n\n<p>When attributes are <strong>continuous<\/strong>, we assume they follow a <strong>normal (Gaussian) distribution<\/strong>.<br>This allows us to use the <strong>Gaussian Probability Density Function (PDF)<\/strong> to calculate probabilities.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/SSROY_Naive-Bayes-tutorial_REALVALUES1-1-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of SSROY_Na\u00efve Bayes tutorial_REALVALUES1 (1) (1).\"><\/object><a id=\"wp-block-file--media-1f5647c7-e516-4d2f-ba32-d63bc9c5ce83\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/SSROY_Naive-Bayes-tutorial_REALVALUES1-1-1.pdf\">SSROY_Na\u00efve Bayes tutorial_REALVALUES1 (1) (1)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/SSROY_Naive-Bayes-tutorial_REALVALUES1-1-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-1f5647c7-e516-4d2f-ba32-d63bc9c5ce83\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>DECISION TREE : ID3<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/Decision-Tree-ID3-Example-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Decision Tree ID3 Example.\"><\/object><a id=\"wp-block-file--media-3cce21ac-77d1-41ca-8ae4-24de92666b70\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/Decision-Tree-ID3-Example-1.pdf\">Decision Tree ID3 Example<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/07\/Decision-Tree-ID3-Example-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-3cce21ac-77d1-41ca-8ae4-24de92666b70\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>DIFFERENCE BETWEEN ID3 and CART<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Feature \/ Criterion<\/th><th><strong>ID3<\/strong><\/th><th><strong>CART<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Developed by<\/td><td>Quinlan (1986)<\/td><td>Breiman et al. (1986)<\/td><\/tr><tr><td>Classification Support<\/td><td>\u2705<\/td><td>\u2705<\/td><\/tr><tr><td>Regression Support<\/td><td>\u274c<\/td><td>\u2705<\/td><\/tr><tr><td>Splitting Criterion<\/td><td>Entropy \/ Info Gain \u2705<\/td><td>Gini Index \/ MSE \u2705<\/td><\/tr><tr><td>Binary Splits Only<\/td><td>\u274c (Multi-way)<\/td><td>\u2705<\/td><\/tr><tr><td>Handles Continuous Attributes<\/td><td>\u274c (Needs preprocessing)<\/td><td>\u2705<\/td><\/tr><tr><td>Handles Categorical Attributes<\/td><td>\u2705<\/td><td>\u2705<\/td><\/tr><tr><td>Pruning Supported<\/td><td>\u274c<\/td><td>\u2705 (Cost-Complexity)<\/td><\/tr><tr><td>Missing Value Handling<\/td><td>\u274c<\/td><td>\u2705<\/td><\/tr><tr><td>Tree Interpretability<\/td><td>\u2705<\/td><td>\u2705<\/td><\/tr><tr><td>Output Labels<\/td><td>Discrete only \u2705<\/td><td>Discrete \/ Numeric \u2705<\/td><\/tr><tr><td>Computational Efficiency<\/td><td>Moderate \u26a0\ufe0f<\/td><td>Efficient \u2705<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>DECISION TREE : CART and C4.5-&gt;When to use GINI INDEX and when to use GINI RATIO?<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Decision_Tree_CART_C4.5_SSROY_2025-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Decision_Tree_CART_C4.5_SSROY_2025.\"><\/object><a id=\"wp-block-file--media-6fc19869-2ae8-4deb-91a2-bf170904e06b\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Decision_Tree_CART_C4.5_SSROY_2025-1.pdf\">Decision_Tree_CART_C4.5_SSROY_2025<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Decision_Tree_CART_C4.5_SSROY_2025-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-6fc19869-2ae8-4deb-91a2-bf170904e06b\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>K-Nearest Neighbor Algorithm<\/strong><\/h2>\n\n\n\n<p>The <strong>k-nearest neighbors (k-NN)<\/strong> algorithm is a simple yet powerful supervised learning technique that does not make assumptions about the underlying data distribution, making it a non-parametric method. Introduced by Evelyn Fix and Joseph Hodges in 1951 and further developed by Thomas Cover, k-NN is widely applied in classification tasks, where an input is assigned the class most common among its k closest points in the training data. In the case of k=1, the label is directly taken from the nearest neighbor. Beyond classification, k-NN also supports regression tasks by averaging the target values of nearby data points, a method often referred to as nearest neighbor interpolation. To enhance prediction accuracy, particularly in scenarios with varying neighbor distances, weights can be applied \u2014 frequently in the form of an inverse relationship with distance (e.g., 1\/d). Importantly, since the algorithm depends heavily on distance metrics, proper scaling or normalization of features is crucial, especially when the data includes heterogeneous units or differing value ranges.<\/p>\n\n\n\n<div class=\"wp-block-cover alignleft\" style=\"min-height:357px;aspect-ratio:unset;\"><span aria-hidden=\"true\" class=\"wp-block-cover__background has-background-dim\" style=\"background-color:#73318c\"><\/span><img loading=\"lazy\" decoding=\"async\" width=\"800\" height=\"723\" class=\"wp-block-cover__image-background wp-image-2003\" alt=\"\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KnnClassification.svg_-3.png\" data-object-fit=\"cover\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KnnClassification.svg_-3.png 800w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KnnClassification.svg_-3-300x271.png 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KnnClassification.svg_-3-768x694.png 768w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-align-center has-large-font-size\"><\/p>\n<\/div><\/div>\n\n\n\n<p> Above figure shows, In k-NN, <strong>if k = 3 (2 triangles, 1 square),<\/strong> the test sample is classified as a triangle; <strong>if k = 5 (3 squares, 2 triangles),<\/strong> it&#8217;s classified as a square.<\/p>\n\n\n\n<p>Fig-KNN , Ref : <a href=\"https:\/\/en.wikipedia.org\/wiki\/K-nearest_neighbors_algorithm#\/media\/File:KnnClassification.svg\">https:\/\/en.wikipedia.org\/wiki\/K-nearest_neighbors_algorithm#\/media\/File:KnnClassification.svg<\/a><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Key Points:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>k-NN is a non-parametric supervised learning algorithm<\/strong> that operates without assuming a specific distribution for the data.<\/li>\n\n\n\n<li><strong>Originally developed in 1951<\/strong>, it assigns class labels based on the majority vote among the k nearest data points in classification tasks.<\/li>\n\n\n\n<li><strong>In regression settings<\/strong>, k-NN predicts a value by averaging the outputs of the k closest neighbors, with k=1 yielding a simple interpolation.<\/li>\n\n\n\n<li><strong>Weighted k-NN can improve performance<\/strong> by giving more influence to closer neighbors, often using weights inversely proportional to distance (like 1\/d).<\/li>\n\n\n\n<li><strong>Normalization of features is essential<\/strong>, as the algorithm\u2019s distance-based nature makes it sensitive to differences in scale or units across attributes.<\/li>\n<\/ul>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KNN_Classifier_SSRoy2025_ML_PA.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of KNN_Classifier_SSRoy2025_ML_PA.\"><\/object><a id=\"wp-block-file--media-cee1b771-3c71-487d-ba4f-6800b3e669c5\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KNN_Classifier_SSRoy2025_ML_PA.pdf\">KNN_Classifier_SSRoy2025_ML_PA<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/KNN_Classifier_SSRoy2025_ML_PA.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-cee1b771-3c71-487d-ba4f-6800b3e669c5\">Download<\/a><\/div>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Mathematics_behind_Knn-1-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Mathematics_behind_Knn (1).\"><\/object><a id=\"wp-block-file--media-bc4dd638-f75c-446d-83d1-e2c147bbbb2f\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Mathematics_behind_Knn-1-1.pdf\">Mathematics_behind_Knn (1)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Mathematics_behind_Knn-1-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-bc4dd638-f75c-446d-83d1-e2c147bbbb2f\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Logistic Regression<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Logistic-regression-tutorial-1_ssroy-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Logistic regression tutorial-1_ssroy (1).\"><\/object><a id=\"wp-block-file--media-f508e752-006f-4f20-895f-7a0c524b3ef8\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Logistic-regression-tutorial-1_ssroy-1.pdf\">Logistic regression tutorial-1_ssroy (1)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Logistic-regression-tutorial-1_ssroy-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-f508e752-006f-4f20-895f-7a0c524b3ef8\">Download<\/a><\/div>\n\n\n\n<p><a href=\"https:\/\/academic.oup.com\/jrsssb\/article\/20\/2\/215\/7027376?login=false\" data-type=\"link\" data-id=\"https:\/\/academic.oup.com\/jrsssb\/article\/20\/2\/215\/7027376?login=false\">Reference : (Above paper)Cox, D. R. (1958). The regression analysis of binary sequences.&nbsp;<em>Journal of the Royal Statistical Society Series B: Statistical Methodology<\/em>,&nbsp;<em>20<\/em>(2), 215-232.<\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Logistic Regression : Further simplified explanation<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/LOGISTIC_REGRESSION-SSROY_EXPLAINED.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of LOGISTIC_REGRESSION-SSROY_EXPLAINED.\"><\/object><a id=\"wp-block-file--media-60a84577-4a48-4101-a826-dd4d3ceab2b7\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/LOGISTIC_REGRESSION-SSROY_EXPLAINED.pdf\">LOGISTIC_REGRESSION-SSROY_EXPLAINED<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/LOGISTIC_REGRESSION-SSROY_EXPLAINED.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-60a84577-4a48-4101-a826-dd4d3ceab2b7\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Exercises on Logistic Regression<\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Logistic-regression-exercise-class-3.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Logistic regression exercise-class (3).\"><\/object><a id=\"wp-block-file--media-3c4cf007-a093-4453-aed3-8e2fad5750c0\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Logistic-regression-exercise-class-3.pdf\">Logistic regression exercise-class (3)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Logistic-regression-exercise-class-3.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-3c4cf007-a093-4453-aed3-8e2fad5750c0\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><em><strong>Simple McCulloch-Pitts neurons can be used to design logical operations<\/strong><\/em><\/h2>\n\n\n\n<p>A <strong>McCulloch\u2013Pitts neural network<\/strong> is a simple mathematical model of a neuron that takes binary inputs, applies weights, sums them, and produces a binary output based on a fixed threshold, used to represent basic logic functions.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"225\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/image-9-1024x225.png\" alt=\"\" class=\"wp-image-2120\" style=\"width:471px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/image-9-1024x225.png 1024w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/image-9-300x66.png 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/image-9-768x169.png 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/image-9.png 1301w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Perceptron-tutorial-with-calculationsSSROY1-2.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Perceptron tutorial with calculationsSSROY1 (2).\"><\/object><a id=\"wp-block-file--media-0dcd007c-c965-44b1-a350-a4e3c60f164c\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Perceptron-tutorial-with-calculationsSSROY1-2.pdf\">Perceptron tutorial with calculationsSSROY1 (2)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Perceptron-tutorial-with-calculationsSSROY1-2.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-0dcd007c-c965-44b1-a350-a4e3c60f164c\">Download<\/a><\/div>\n\n\n\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"452\" height=\"538\" data-id=\"2122\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/nn1.jpg\" alt=\"\" class=\"wp-image-2122\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/nn1.jpg 452w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/nn1-252x300.jpg 252w\" sizes=\"auto, (max-width: 452px) 100vw, 452px\" \/><\/figure>\n<\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"399\" height=\"270\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/nn3-1.jpg\" alt=\"\" class=\"wp-image-2125\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/nn3-1.jpg 399w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/nn3-1-300x203.jpg 300w\" sizes=\"auto, (max-width: 399px) 100vw, 399px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>BACK PROPAGATION<\/strong><\/h2>\n\n\n\n<p><strong>Backpropagation<\/strong> is a fundamental algorithm used to train <strong>artificial neural networks by minimizing the error between the network\u2019s predicted output and the actual target output.<\/strong> It works by performing a forward pass to compute the outputs based on current weights and biases, then calculating the error using a loss function such as mean squared error. <strong>The algorithm then propagates this error backward through the network, layer by layer, using the chain rule of calculus to compute gradients of the loss with respect to each weight and bias<\/strong>. These gradients indicate how to adjust the parameters to reduce the error. By iteratively updating the weights and biases in the opposite direction of the gradients\u2014scaled by a learning rate\u2014backpropagation enables the network to learn patterns in the training data and improve its predictions over time.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"449\" height=\"380\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-7.png\" alt=\"\" class=\"wp-image-2255\" style=\"width:1148px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-7.png 449w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-7-300x254.png 300w\" sizes=\"auto, (max-width: 449px) 100vw, 449px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"446\" height=\"157\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-6.png\" alt=\"\" class=\"wp-image-2254\" style=\"width:1148px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-6.png 446w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-6-300x106.png 300w\" sizes=\"auto, (max-width: 446px) 100vw, 446px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Neural Network Backpropagation<\/strong><\/h2>\n\n\n\n<p><strong>Backpropagation <\/strong>is a learning algorithm for neural networks that works in two steps: first, the error between the predicted output and the target is calculated; then, using the <strong>chain rule of calculus<\/strong>, this error is propagated backward through the network to compute gradients for each weight. These gradients are then used to update the weights and improve the model\u2019s accuracy.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"770\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/ROY-3-770x1024.jpeg\" alt=\"\" class=\"wp-image-2298\" style=\"width:1170px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/ROY-3-770x1024.jpeg 770w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/ROY-3-226x300.jpeg 226w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/ROY-3-768x1021.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/ROY-3.jpeg 963w\" sizes=\"auto, (max-width: 770px) 100vw, 770px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Key variables :<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"407\" height=\"267\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-15.png\" alt=\"\" class=\"wp-image-2276\" style=\"width:1044px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-15.png 407w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-15-300x197.png 300w\" sizes=\"auto, (max-width: 407px) 100vw, 407px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"716\" height=\"467\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-8.png\" alt=\"\" class=\"wp-image-2266\" style=\"width:1151px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-8.png 716w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-8-300x196.png 300w\" sizes=\"auto, (max-width: 716px) 100vw, 716px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"642\" height=\"403\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-9.png\" alt=\"\" class=\"wp-image-2267\" style=\"width:1172px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-9.png 642w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-9-300x188.png 300w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"555\" height=\"126\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-10.png\" alt=\"\" class=\"wp-image-2268\" style=\"width:1199px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-10.png 555w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/image-10-300x68.png 300w\" sizes=\"auto, (max-width: 555px) 100vw, 555px\" \/><\/figure>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/NN-BACKPROPAGATION.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of NN-BACKPROPAGATION.\"><\/object><a id=\"wp-block-file--media-2c4cad7b-5aea-410b-870e-dc491c0c70ea\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/NN-BACKPROPAGATION.pdf\">NN-BACKPROPAGATION<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/NN-BACKPROPAGATION.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-2c4cad7b-5aea-410b-870e-dc491c0c70ea\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Exercise solved<\/strong><\/h2>\n\n\n<a class=\"wp-block-read-more\" href=\"https:\/\/cyberenlightener.com\/?page_id=1921\" target=\"_self\">https:\/\/mattmazur.com\/2015\/03\/17\/a-step-by-step-backpropagation-example\/<span class=\"screen-reader-text\">: Supervised Learning<\/span><\/a>\n\n\n<h2 class=\"wp-block-heading\"><strong>SVM(Support Vector Machine)<\/strong><\/h2>\n\n\n\n<p><\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/SVMSSROY.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of SVMSSROY.\"><\/object><a id=\"wp-block-file--media-85320c21-de46-4d91-a34e-5ec918c81fe0\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/SVMSSROY.pdf\">SVMSSROY<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/SVMSSROY.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-85320c21-de46-4d91-a34e-5ec918c81fe0\">Download<\/a><\/div>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Explanation_of_the_above_questions1ssroy.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Explanation_of_the_above_questions1ssroy.\"><\/object><a id=\"wp-block-file--media-dab6989e-d0b6-44ae-9ad8-86391bf3e134\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Explanation_of_the_above_questions1ssroy.pdf\">Explanation_of_the_above_questions1ssroy<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/Explanation_of_the_above_questions1ssroy.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-dab6989e-d0b6-44ae-9ad8-86391bf3e134\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Support Vector Machine<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"707\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/1_roy-707x1024.jpeg\" alt=\"\" class=\"wp-image-2071\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/1_roy-707x1024.jpeg 707w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/1_roy-207x300.jpeg 207w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/1_roy-768x1113.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/1_roy-1060x1536.jpeg 1060w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/1_roy.jpeg 1104w\" sizes=\"auto, (max-width: 707px) 100vw, 707px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"721\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/2_roy-721x1024.jpeg\" alt=\"\" class=\"wp-image-2073\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/2_roy-721x1024.jpeg 721w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/2_roy-211x300.jpeg 211w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/2_roy-768x1091.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/2_roy-1081x1536.jpeg 1081w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/2_roy.jpeg 1126w\" sizes=\"auto, (max-width: 721px) 100vw, 721px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"727\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/3_roy-727x1024.jpeg\" alt=\"\" class=\"wp-image-2074\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/3_roy-727x1024.jpeg 727w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/3_roy-213x300.jpeg 213w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/3_roy-768x1082.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/3_roy-1091x1536.jpeg 1091w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/3_roy.jpeg 1136w\" sizes=\"auto, (max-width: 727px) 100vw, 727px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"718\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/4_roy-718x1024.jpeg\" alt=\"\" class=\"wp-image-2075\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/4_roy-718x1024.jpeg 718w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/4_roy-210x300.jpeg 210w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/4_roy-768x1095.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/4_roy-1077x1536.jpeg 1077w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/4_roy.jpeg 1122w\" sizes=\"auto, (max-width: 718px) 100vw, 718px\" \/><\/figure>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/SVM_SSROY2.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of SVM_SSROY2.\"><\/object><a id=\"wp-block-file--media-6dafd1fd-6f7e-4d82-a4f3-abaf7decd61a\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/SVM_SSROY2.pdf\">SVM_SSROY2<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/08\/SVM_SSROY2.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-6dafd1fd-6f7e-4d82-a4f3-abaf7decd61a\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>K-MEANS CLUSTERING <\/strong><\/h2>\n\n\n<a class=\"wp-block-read-more\" href=\"https:\/\/cyberenlightener.com\/?page_id=1921\" target=\"_self\">https:\/\/cyberenlightener.com\/?page_id=1324<span class=\"screen-reader-text\">: Supervised Learning<\/span><\/a>\n\n\n\n\n<h2 class=\"wp-block-heading\"><strong>K-MODES CLUSTERING<\/strong><\/h2>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/K-Modes-clustering-tutorial-2.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of K-Modes clustering tutorial (2).\"><\/object><a id=\"wp-block-file--media-535f5bc9-e31b-4d29-8c10-26b3123cfd8c\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/K-Modes-clustering-tutorial-2.pdf\">K-Modes clustering tutorial (2)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/09\/K-Modes-clustering-tutorial-2.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-535f5bc9-e31b-4d29-8c10-26b3123cfd8c\">Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Regression \ud83d\udcca Introduction to Regression Analysis What is Regression? Regression is a statistical method used to study the relationship between variables. It helps us understand how one or more independent variables (predictors) influence a dependent variable (response). At its core, regression answers this question: &#8220;How does a change in one variable affect another?&#8221; \ud83d\udd0d Why [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-1921","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages\/1921","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1921"}],"version-history":[{"count":39,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages\/1921\/revisions"}],"predecessor-version":[{"id":2299,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages\/1921\/revisions\/2299"}],"wp:attachment":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}