{"id":10702,"date":"2018-04-14T10:31:47","date_gmt":"2018-04-14T05:01:47","guid":{"rendered":"http:\/\/ucanalytics.com\/blogs\/?p=10702"},"modified":"2018-08-26T12:12:36","modified_gmt":"2018-08-26T06:42:36","slug":"machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1","status":"publish","type":"post","link":"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/","title":{"rendered":"Machine Learning: Non-linear Regression, Regularization &#038; Cross Validation Simplified  (Part 1)"},"content":{"rendered":"<div id=\"attachment_10711\" style=\"width: 360px\" class=\"wp-caption alignright\"><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg\"><img aria-describedby=\"caption-attachment-10711\" data-attachment-id=\"10711\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/machine-learning-regularization-simplified\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?fit=516%2C523&amp;ssl=1\" data-orig-size=\"516,523\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Machine Learning Regularization Simplified\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?fit=296%2C300&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?fit=516%2C523&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"wp-image-10711 \" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?resize=350%2C354\" alt=\"Machine Learning: Regularization to Prevent Over-fitting - by Roopam\" width=\"350\" height=\"354\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?w=516&amp;ssl=1 516w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?resize=247%2C250&amp;ssl=1 247w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Machine-Learning-Regularization-Simplified.jpg?resize=296%2C300&amp;ssl=1 296w\" sizes=\"(max-width: 350px) 100vw, 350px\" data-recalc-dims=\"1\" \/><\/a><p id=\"caption-attachment-10711\" class=\"wp-caption-text\">Building Blocks of Machine Learning &#8211; by Roopam<\/p><\/div>\n<hr \/>\n<p>In this 3-part series of articles, you will gain an intuitive understanding of some fundamental concepts in machine learning such as:<\/p>\n<ul style=\"list-style-type: square;\">\n<li style=\"list-style-type: none;\">\n<ul style=\"list-style-type: square;\">\n<li>Building blocks of curves<\/li>\n<li>Non-linear regression<\/li>\n<li>Curve fitting and overfitting<\/li>\n<li>Regularization to prevent overfitting<\/li>\n<li>Hyper-parameters in machine learning<\/li>\n<li>Cross-validation to fine-tune models<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>You will also get hands-on practice to understand these concepts better.\u00a0For this post, we will primarily focus on the building blocks of curves, non-linear regression, and curve fitting. This introductory article will serve us well when we explore the fundamental concepts of machine learning, i.e. regularization, hyper-parameter tuning, and cross-validation, in the next parts.<\/p>\n<p>Before we start, let&#8217;s appreciate the beauty of curves through this Batman curve formed with different mathematical equations. <a href=\"https:\/\/guangchuangyu.github.io\/2011\/08\/the-batman-equation\/\" target=\"_blank\" rel=\"noopener\">R code for Batman curve<\/a>.<\/p>\n<p><img data-attachment-id=\"10746\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/batman-curve-1\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?fit=843%2C384&amp;ssl=1\" data-orig-size=\"843,384\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Batman Curve 1\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?fit=300%2C137&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?fit=640%2C292&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-10746 aligncenter\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?resize=640%2C292\" alt=\"\" width=\"640\" height=\"292\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?w=843&amp;ssl=1 843w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?resize=250%2C114&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?resize=300%2C137&amp;ssl=1 300w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2017\/12\/Batman-Curve-1.jpg?resize=768%2C350&amp;ssl=1 768w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/p>\n<p>This Batman Curve is a reminder that mathematical equations are capable of forming every possible shape. This flexibility of equations also makes them susceptible to overfitting as we will learn in the next segment.<\/p>\n<h2><span style=\"color: #3366ff;\">Curve Fitting and Overfitting<\/span><\/h2>\n<h2><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg\"><img data-attachment-id=\"10818\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/overfitting\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?fit=510%2C525&amp;ssl=1\" data-orig-size=\"510,525\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"overfitting\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?fit=291%2C300&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?fit=510%2C525&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\" wp-image-10818 alignright\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?resize=296%2C305\" alt=\"\" width=\"296\" height=\"305\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?w=510&amp;ssl=1 510w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?resize=243%2C250&amp;ssl=1 243w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/overfitting.jpg?resize=291%2C300&amp;ssl=1 291w\" sizes=\"(max-width: 296px) 100vw, 296px\" data-recalc-dims=\"1\" \/><\/a><\/h2>\n<p>Andrew Ng, in his brilliant\u00a0lecture series on machine learning and AI, described supervised machine learning as a multidimensional\u00a0curve fitting.\u00a0The idea is to identify the best curve or function of the x-variables (aka predictors) to estimate the y-variable (aka response) with minimum error. Notably, supervised learning dominates most machine learning\u00a0applications for business.<\/p>\n<p>Curve fitting, essentially, is similar to the game of connecting the dots where you try to complete a picture. In case of data, dots are the data points and idea is to produce the best fit for the observed data which is generalizable for the unobserved data. Non-linear curves are highly flexible and can fit any observed data to perfection. To understand this have a look at this Calvin and Hobbes comic strip.<\/p>\n<h2><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg\"><img data-attachment-id=\"10828\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/non-linear-regression-and-overfitting\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?fit=1489%2C479&amp;ssl=1\" data-orig-size=\"1489,479\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Non Linear Regression and Overfitting\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?fit=300%2C97&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?fit=640%2C206&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"wp-image-10828 aligncenter\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?resize=640%2C206\" alt=\"\" width=\"640\" height=\"206\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?w=1489&amp;ssl=1 1489w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?resize=250%2C80&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?resize=300%2C97&amp;ssl=1 300w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?resize=768%2C247&amp;ssl=1 768w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?resize=1024%2C329&amp;ssl=1 1024w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-and-Overfitting.jpg?w=1280 1280w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><\/h2>\n<p>Essentially, with non-linear curve fitting, you are providing the algorithm the same level of creativity Calvin has opted to exercise at the end of the comic strip. This level of creativity is capable of producing a masterpiece\u00a0but is equally capable of creating a grand mess. Before we learn how to make masterpieces through non-linear equations, let&#8217;s understand the reason why non-linear equations fit and over-fit so well. The root of this is in the Taylor series which you might remember from high-school.<\/p>\n<h2><span style=\"color: #3366ff;\">Building Block of Curves &#8211; Taylor Series<\/span><\/h2>\n<p><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg\"><img data-attachment-id=\"10905\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/elements\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?fit=628%2C810&amp;ssl=1\" data-orig-size=\"628,810\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Elements\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?fit=233%2C300&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?fit=628%2C810&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\" wp-image-10905 alignright\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?resize=317%2C409\" alt=\"\" width=\"317\" height=\"409\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?w=628&amp;ssl=1 628w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?resize=194%2C250&amp;ssl=1 194w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/04\/Elements.jpg?resize=233%2C300&amp;ssl=1 233w\" sizes=\"(max-width: 317px) 100vw, 317px\" data-recalc-dims=\"1\" \/><\/a>One of the primary goals for all fields of science is to identify the building blocks of nature. For instance, physicists identified\u00a0atoms as the building blocks of matter. You, me, and everything we see around, dead or alive, are formed out of atoms. Atoms are further divided into electrons, protons, and neutrons. The combination of these elementary particles is responsible\u00a0for everything in the known universe. Things that are extremely fundamental and simple can produce the complexity of the order of the universe.<\/p>\n<p>Like the elementary particles that formed the entire universe, could there be elementary units of all mathematical functions and curves? You must appreciate that functions could have infinite forms such as the sine function, the exponential function, the Batman function etc. Moreover, a combination of functions is capable of creating complexity of the magnitude of the universe. So, is there a fundamental unit of functions? The answer is yes. The Taylor Series states that any complicated function <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%28x%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f(x)\" class=\"latex\" \/>, including the Batman curve, could be described as a linear combination of x and its infinite powers i.e x<sup>2<\/sup>,\u00a0x<sup>3<\/sup>&#8230;, x<sup>\u221e<\/sup>.\u00a0At this point, let me throw in a daunting looking equation but trust me it will become quite simple when you will continue reading this article. The special form of the Taylor series at x=0, aka\u00a0Maclaurin Series,\u00a0is represented as:<\/p>\n<pre style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%28x%29+%3D+f%280%29%2Bf%27%280%29x%2B%5Cfrac%7Bf%27%27%280%29%7D%7B2%21%7D%2B%5Ccdots%2B%5Cfrac%7Bf%5E%7B%28n%29%7D%280%29%7D%7Bn%21%7Dx%5E%7Bn%7D%2B%5Ccdots&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"f(x) = f(0)+f&#039;(0)x+&#92;frac{f&#039;&#039;(0)}{2!}+&#92;cdots+&#92;frac{f^{(n)}(0)}{n!}x^{n}+&#92;cdots\" class=\"latex\" \/><\/pre>\n<p>Here,\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%27%28x%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f&#039;(x)\" class=\"latex\" \/> is the first differential of the function and\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%5E%7B%28n%29%7D%28x%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f^{(n)}(x)\" class=\"latex\" \/> is the nth differential. Essentially, we can mimic a complicated function <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%28x%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f(x)\" class=\"latex\" \/>, like the Batman curve, with just x and its polynomials. If we do the Taylor Series calculation long enough i.e. higher powers of x (yeah, like we have nothing better to do) then we can find the exact value of the function at any point. To make things easy to understand let&#8217;s have a competition between&#8230;<\/p>\n<h2><span style=\"color: #3366ff;\">Non-Linear Regression Vs. Taylor Series<\/span><\/h2>\n<p>Everyone loves a good competition and today in the ring are non-linear regression and Taylor series. They will compete against each other to approximate the sine function between -3\u03c0 to 3\u03c0 which looks like. You could use this R-code to reproduce the results you will see in the subsequent sections:<strong> <a href=\"http:\/\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Non-Linear-Regression-Vs.-Taylor-Series.txt\" target=\"_blank\" rel=\"noopener\">Non-linear regression Vs. Taylor series<\/a>.<\/strong><\/p>\n<p><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg\"><img data-attachment-id=\"10842\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/plot-1-sine-curve-jpg\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?fit=633%2C266&amp;ssl=1\" data-orig-size=\"633,266\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1516204448&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Plot 1 &#8211; sine curve.jpg\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?fit=300%2C126&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?fit=633%2C266&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"wp-image-10842 aligncenter\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?resize=640%2C269\" alt=\"\" width=\"640\" height=\"269\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?w=633&amp;ssl=1 633w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?resize=250%2C105&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-1-sine-curve.jpg.jpeg?resize=300%2C126&amp;ssl=1 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><br \/>\nThis competition will help us to understand the intuition behind non-linear regression which often has polynomial forms of the predictor variables (X). It will help us understand how these two different methods of approximation have different objectives and how these objectives rule the way these methods approximate response variable (Y). The objectives for 2 methods can be described as:<\/p>\n<p><strong>Objective non-linear regression: <\/strong>to minimize the overall error for the entire sample. This equation describes the generalized polynomial (non-linear) form of the regression model.<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%28x%29+%3D+%5Cbeta+_%7B0%7D%2B%5Cbeta+_%7B1%7Dx%2B+%5Cbeta+_%7B2%7Dx%5E%7B2%7D%2B%5Cbeta+_%7B3%7Dx%5E%7B3%7D%5Ccdots+&#038;bg=ffffff&#038;fg=000&#038;s=3&#038;c=20201002\" alt=\"f(x) = &#92;beta _{0}+&#92;beta _{1}x+ &#92;beta _{2}x^{2}+&#92;beta _{3}x^{3}&#92;cdots \" class=\"latex\" \/><\/pre>\n<p>Here, the idea is to identify beta (\u03b2) parameters with the help of observed data (x and its polynomial forms).<\/p>\n<p><strong>Objective Taylor Series (at x=0): <\/strong>to find the best approximation for y using x around x=0<\/p>\n<p>Taylor Series for sine function at x=0 is described as:<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=Sin%28x%29+%3D+x-+%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%21%7D%2B%5Cfrac%7Bx%5E%7B5%7D%7D%7B5%21%7D-+%5Cfrac%7Bx%5E%7B7%7D%7D%7B7%21%7D%5Ccdots+&#038;bg=ffffff&#038;fg=000&#038;s=3&#038;c=20201002\" alt=\"Sin(x) = x- &#92;frac{x^{3}}{3!}+&#92;frac{x^{5}}{5!}- &#92;frac{x^{7}}{7!}&#92;cdots \" class=\"latex\" \/><\/pre>\n<p>This series expands to infinite powers of x and it can be written in the general form as:<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=Sin%28x%29+%3D%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D%5Cfrac%7B-1%5E%7Bk%7D%7D%7B%282k%2B1%29%21%7Dx%5E%7B2k%2B1%7D&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"Sin(x) =&#92;sum_{k=0}^{&#92;infty }&#92;frac{-1^{k}}{(2k+1)!}x^{2k+1}\" class=\"latex\" \/><\/pre>\n<p>When we use a high degree of polynomials of x, the beta coefficients of regression and Taylor series coefficients will look identical. Keep this in mind when you see curves for non-linear regression and Taylor series converge. So, let the competition begin&#8230;<\/p>\n<h2><span style=\"color: #3366ff;\">Non-Linear Regression and Taylor Series &#8211; Race to the Finish<\/span><\/h2>\n<p>Remember, we are trying to fit both Taylor series and non-linear regression on data generated by the sine function. To begin with, let&#8217;s just use the first power of x for both the methods. With just x as the parameter, this is a linear regression. Remember, Taylor series is approximating the curve near x=0 and regression is using the entire dataset between\u00a0 -3\u03c0 to 3\u03c0.<\/p>\n<p><a class=\"\" href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg\"><img data-attachment-id=\"10843\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/plot-2-sine-curve-jpg\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?fit=629%2C262&amp;ssl=1\" data-orig-size=\"629,262\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1516205350&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Plot 2 &#8211; sine curve.jpg\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?fit=300%2C125&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?fit=629%2C262&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"aligncenter size-full wp-image-10843 gr-progress\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?resize=629%2C262\" alt=\"\" width=\"629\" height=\"262\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?w=629&amp;ssl=1 629w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?resize=250%2C104&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-2-sine-curve.jpg.jpeg?resize=300%2C125&amp;ssl=1 300w\" sizes=\"(max-width: 629px) 100vw, 629px\" data-recalc-dims=\"1\" \/><\/a>With just x,\u00a0these are not very good approximations. The regression line in blue has the following equation (you can get this easily with the R code provided at the top)<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y%3D0.0000094%2B0.037x&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"y=0.0000094+0.037x\" class=\"latex\" \/><\/pre>\n<p>Taylor series, on the other hand, has this form with just x.<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y%3Dx&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"y=x\" class=\"latex\" \/><\/pre>\n<p>These two equations are nothing like each other. But you will notice that they will become identical when we will increase the polynomial powers of x in the calculation.<\/p>\n<h4><span style=\"color: #3366ff;\">5th-degree Polynomials of x<\/span><\/h4>\n<p>Let&#8217;s turn up the heat a bit in this competition. We will give both Taylor series and regression all the polynomials of x till the power of 5. This is the equation we are fitting with both the methods.<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y+%3D+%5Cbeta+_%7B0%7D%2B%5Cbeta+_%7B1%7Dx%2B+%5Cbeta+_%7B2%7Dx%5E%7B2%7D%2B%5Cbeta+_%7B3%7Dx%5E%7B3%7D%2B%5Cbeta+_%7B4%7Dx%5E%7B4%7D%2B%5Cbeta+_%7B5%7Dx%5E%7B5%7D+&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"y = &#92;beta _{0}+&#92;beta _{1}x+ &#92;beta _{2}x^{2}+&#92;beta _{3}x^{3}+&#92;beta _{4}x^{4}+&#92;beta _{5}x^{5} \" class=\"latex\" \/><\/pre>\n<p>The Taylor series, in this case, will be:<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y%3Dx-0.167x%5E3%2B0.0084x%5E5&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"y=x-0.167x^3+0.0084x^5\" class=\"latex\" \/><\/pre>\n<p>The regression equation is:<\/p>\n<pre><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y%3D0.000025-0.079x%2B0.0026x%5E3-0.0000077x%5E5&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"y=0.000025-0.079x+0.0026x^3-0.0000077x^5\" class=\"latex\" \/><\/pre>\n<p>See how regression has started to agree with Taylor series here. It has set the\u00a0\u03b2 values for even powers of x, i.e x<sup>2<\/sup> and x<sup>4<\/sup>, as zero. The magnitude for coefficients are still different but that will also converge when we will add higher polynomials of x to the equations. This is how the fit looks like for 5th-degree polynomials.<\/p>\n<p><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg\"><img data-attachment-id=\"10844\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/plot-3-sine-curve-jpg\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?fit=631%2C260&amp;ssl=1\" data-orig-size=\"631,260\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1516205364&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Plot 3 &#8211; sine curve.jpg\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?fit=300%2C124&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?fit=631%2C260&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"aligncenter size-full wp-image-10844\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?resize=631%2C260\" alt=\"\" width=\"631\" height=\"260\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?w=631&amp;ssl=1 631w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?resize=250%2C103&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-3-sine-curve.jpg.jpeg?resize=300%2C124&amp;ssl=1 300w\" sizes=\"(max-width: 631px) 100vw, 631px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<h4><span style=\"color: #3366ff;\">15th-degree Polynomials of x<\/span><\/h4>\n<p>Ok enough with this club level competition, let&#8217;s take both these methods to the Olympics. We will provide them with 15th-degree polynomials of x.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y+%3D+%5Cbeta+_%7B0%7D%2B%5Cbeta+_%7B1%7Dx%2B+%5Cbeta+_%7B2%7Dx%5E%7B2%7D%2B%5Cbeta+_%7B3%7Dx%5E%7B3%7D%2B%5Ccdots%5Cbeta+_%7B14%7Dx%5E%7B14%7D%2B%5Cbeta+_%7B15%7Dx%5E%7B15%7D+&#038;bg=ffffff&#038;fg=000&#038;s=2&#038;c=20201002\" alt=\"y = &#92;beta _{0}+&#92;beta _{1}x+ &#92;beta _{2}x^{2}+&#92;beta _{3}x^{3}+&#92;cdots&#92;beta _{14}x^{14}+&#92;beta _{15}x^{15} \" class=\"latex\" \/><\/p>\n<p>The regression equation has fit the data almost perfectly. Taylor series is still trying to catch up. Remember, they had different objectives.<\/p>\n<p><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg\"><img data-attachment-id=\"10845\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/plot-4-sine-curve-jpg\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?fit=629%2C263&amp;ssl=1\" data-orig-size=\"629,263\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1516205380&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Plot 4 &#8211; sine curve.jpg\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?fit=300%2C125&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?fit=629%2C263&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"aligncenter size-full wp-image-10845\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?resize=629%2C263\" alt=\"\" width=\"629\" height=\"263\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?w=629&amp;ssl=1 629w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?resize=250%2C105&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-4-sine-curve.jpg.jpeg?resize=300%2C125&amp;ssl=1 300w\" sizes=\"(max-width: 629px) 100vw, 629px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>The winner, in this competition, is the non-linear regression.<\/p>\n<h4><span style=\"color: #3366ff;\">25th-degree Polynomials of x<\/span><\/h4>\n<p>Remember, the Olympics are about participating not winning. So, let&#8217;s give\u00a0 Taylor series enough time to finish the race. With the 25th-degree polynomial term, the curves look identical.<\/p>\n<p><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg\"><img data-attachment-id=\"10846\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/plot-5-sine-curve-jpg\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?fit=628%2C266&amp;ssl=1\" data-orig-size=\"628,266\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1516205397&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Plot 5 &#8211; sine curve.jpg\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?fit=300%2C127&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?fit=628%2C266&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"aligncenter size-full wp-image-10846\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?resize=628%2C266\" alt=\"\" width=\"628\" height=\"266\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?w=628&amp;ssl=1 628w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?resize=250%2C106&amp;ssl=1 250w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2018\/01\/Plot-5-sine-curve.jpg.jpeg?resize=300%2C127&amp;ssl=1 300w\" sizes=\"(max-width: 628px) 100vw, 628px\" data-recalc-dims=\"1\" \/><\/a>The moral of the story is if you give enough polynomial terms of x to the equation it will be able to fit any curve perfectly. But is perfection good? The answer is almost always no when it comes to data analysis. Perfection reflects overfitting for most models especially if they are not tested against unobserved data.<\/p>\n<p>For instance, if you built the sine curve model with even higher degree polynomials, say 35th-degree, you will see that the R code will return this interesting warning statement:<\/p>\n<blockquote>\n<div><span style=\"color: #ff0000;\"><span style=\"color: #0000ff;\">&gt;<\/span>Warning Messages:<\/span><\/div>\n<div><span style=\"color: #ff0000;\"><span style=\"color: #0000ff;\">&gt;<\/span>1: attempting model selection on an essentially perfect fit is nonsense<\/span><\/div>\n<\/blockquote>\n<p>This means the regression fit is perfect i.e. with zero error. R is highlighting the essential fact in data analysis i.e. perfect fit is nonsense.<\/p>\n<h4><span style=\"color: #3366ff;\">Sign-off Note<\/span><\/h4>\n<p>In data analysis, like every other endeavor in life, perfection is a mirage. Perfection signifies overfitting and in the next articles, we will learn how to avoid the trap of overfitting and sketch masterpieces using non-linear regression. Gear up to learn some essential concepts in machine learning i.e. regularization, hyperparameter tuning, and cross-validation. Hope to see you soon!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this 3-part series of articles, you will gain an intuitive understanding of some fundamental concepts in machine learning such as: Building blocks of curves Non-linear regression Curve fitting and overfitting Regularization to prevent overfitting Hyper-parameters in machine learning Cross-validation to fine-tune models You will also get hands-on practice to understand these concepts better.\u00a0For this<\/p>\n<p><a class=\"excerpt-more blog-excerpt\" href=\"https:\/\/ucanalytics.com\/blogs\/machine-learning-non-linear-regression-regularization-cross-validation-simplified-part-1\/\">Read More&#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":10716,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_newsletter_tier_id":0,"jetpack_publicize_message":"","jetpack_is_tweetstorm":false,"jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[84,85],"tags":[],"jetpack_publicize_connections":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Machine Learning: Non-linear Regression, Regularization &amp; 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