So as before, we have a set of inputs<\/p>\n
x<\/strong> = {x1<\/sub>, x2<\/sub>, . . . , xn<\/sub>}T<\/sup> where N = 6<\/p>\n corresponding to a set of target variables:<\/p>\n t<\/strong> = {t1<\/sub>, t2<\/sub>, . . . , tN<\/sub>}T<\/sup> where N = 6<\/p>\n Our objective is to find a function that relates each of the input variables to each of the target values.<\/p>\n If we assume that the relationship is a linear one, then we can use the equation of a straight line given as:<\/p>\n y =\u00a0\u00df0<\/sub> +\u00a0\u00df1<\/sub>x<\/p>\n Then we simply calculates the coefficients\u00a0 \u00df0<\/sub>\u00a0and\u00a0 \u00df1<\/sub><\/p>\n However since we don’t know the nature of this relationship, we would extend this equation to cover more options.<\/p>\n So we would have it as:<\/p>\n <\/p>\n which is the same as:<\/p>\n This is similar to what we already have.\u00a0 Just like y depends on x<\/em> and \u00df<\/em> in the linear model, also here, y depends on x<\/em> and w<\/strong>.<\/p>\n M is the order of the polynomial. So if M is 1, then we have the linear model. If M is 2, then we have a quadratic function and so on.<\/p>\n <\/p>\n So what is w?<\/strong><\/p>\n w<\/strong> is simply the polynomial coefficients. So w0<\/sub>, w1<\/sub>, . . . , wM<\/sub><\/em> are denoted by the vector w<\/strong>.<\/p>\n So the problem reduces to simply determining the polynomial coefficients. Once we have it, we simple plug it into the polynomial and solve for x.<\/em><\/p>\n <\/p>\n How do we determine w?<\/strong><\/p>\n We determine w by fitting the polynomial to the training data set.\u00a0 This is achieved by minimizing the error function that measures the difference between the function y(x,w), for any given value of w and the corresponding point in the training data set.<\/p>\n To perform minimization, we need the error function. A good choice is to use the sum of squares error between the predicted values y(xn<\/sub>, w<\/strong>)<\/em>\u00a0 for each training data point and the corresponding target values tn<\/sub>.<\/em><\/p>\n This error function is given by:<\/p>\n where 1\/2 is a factor added that would be explained later.<\/p>\n The value of this function would always be non-negative.\u00a0 It can also be zero, but rarely. It is zero if and only if\u00a0 the function produces exactly same output as the training set.<\/p>\n In summary, we need to choose the values of w<\/strong> that would produce the least value of E(w<\/strong>).<\/p>\n But another question is: How to\u00a0 we choose M,<\/strong> that is the order of the polynomial?<\/p>\n I would explain this in the next lecture 6<\/p>\n <\/p>\n Python Code for Polynomial Regression<\/strong><\/p>\n <\/p>\n This is Lecture 6 of Machine Learning 101. We would discuss Polynomial Curve Fitting. Now don’t bother if the name makes it appear tough. This … <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[16],"tags":[],"class_list":["post-1900","post","type-post","status-publish","format-standard","hentry","category-machine-learning"],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1900","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/comments?post=1900"}],"version-history":[{"count":1,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1900\/revisions"}],"predecessor-version":[{"id":2068,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1900\/revisions\/2068"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=1900"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=1900"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=1900"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"Polynomial](\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Polynormial-Curve-Fitting-Equation.jpg\")
<\/p>\n![\"\"](\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Polynormial-Curve-Fitting-Equation2-300x126.jpg\")
<\/p>\n![\"Sum](\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Sum-of-Squares-Error-Function.jpg\")
<\/p>\nimport<\/span> numpy<\/span> as<\/span> np<\/span>\nimport<\/span> pandas<\/span> as<\/span> pd<\/span>\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\nfrom<\/span> sklearn.linear_model<\/span> import<\/span> LinearRegression\nfrom<\/span> sklearn.preprocessing<\/span> import<\/span> PolynomialFeatures\n\n# Read the dataset into a dataframe<\/span>\ndf =<\/span> pd.<\/span>read_excel(r'C:\\ML101\\Curve.xlsx'<\/span>)\n\n# Place the ranges in a variable and preprocess<\/span>\nx =<\/span> df['x'<\/span>].<\/span>values\ny =<\/span> df['y'<\/span>].<\/span>values\nx =<\/span> x.<\/span>reshape(-<\/span>1<\/span>, 1<\/span>)\n\n# Change the order here. degree is same as M<\/span>\npoly =<\/span> PolynomialFeatures(degree=<\/span>11<\/span>)\n\n# Fit a Polynomial Curve<\/span>\nX_poly =<\/span> poly.<\/span>fit_transform(x)\npoly.<\/span>fit(X_poly, y)\nlinreg =<\/span> LinearRegression()\nlinreg.<\/span>fit(X_poly, y)\ny_pred =<\/span> linreg.<\/span>predict(X_poly)\n\n\n# Plot the curves. The regression line is in red<\/span>\nplt.<\/span>scatter(x,y, color=<\/span>'blue'<\/span>)\nplt.<\/span>plot(x, y_pred, color=<\/span>'red'<\/span>)\nplt.<\/span>show()\n<\/pre>\n","protected":false},"excerpt":{"rendered":"