{"id":1293,"date":"2023-08-25T14:57:59","date_gmt":"2023-08-25T14:57:59","guid":{"rendered":"https:\/\/mlinsightscentral.com\/?page_id=1293"},"modified":"2023-09-18T15:16:53","modified_gmt":"2023-09-18T15:16:53","slug":"feedforward-neural-networks","status":"publish","type":"page","link":"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/","title":{"rendered":"Feedforward Neural Networks"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"1293\" class=\"elementor elementor-1293\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-19761d6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"19761d6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f90614c\" data-id=\"f90614c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6a4f78d elementor-widget elementor-widget-heading\" data-id=\"6a4f78d\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h5 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Artificial_Neural_Networks\"><\/span>Artificial Neural Networks<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d30dda0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d30dda0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cde3bcb\" data-id=\"cde3bcb\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b6c4ee6 elementor-widget elementor-widget-text-editor\" data-id=\"b6c4ee6\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_53 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title \" >Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\" role=\"button\"><label for=\"item-69e304f05aff6\" ><span class=\"\"><span style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input aria-label=\"Toggle\" aria-label=\"item-69e304f05aff6\"  type=\"checkbox\" id=\"item-69e304f05aff6\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Artificial_Neural_Networks\" title=\"Artificial Neural Networks\">Artificial Neural Networks<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Biological_Neuron\" title=\"Biological Neuron\">Biological Neuron<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Artificial_Neuron\" title=\"Artificial Neuron\">Artificial Neuron<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Feed_Forward_Neural_Network_%E2%80%93_Multilayer_Perceptrons\" title=\"Feed Forward Neural Network &#8211; Multilayer Perceptrons\">Feed Forward Neural Network &#8211; Multilayer Perceptrons<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Training_ANNs_%E2%80%93_The_Backpropagation_method\" title=\"Training ANNs &#8211;  The Backpropagation method\">Training ANNs &#8211;  The Backpropagation method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Python_Implementation_%E2%80%93_Nonlinear_regression\" title=\"Python Implementation &#8211; Nonlinear regression\">Python Implementation &#8211; Nonlinear regression<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Goodness_of_fit_%E2%80%93_Coefficient_of_determination_R2_score\" title=\"Goodness of fit &#8211; Coefficient of determination R2 score\">Goodness of fit &#8211; Coefficient of determination R2 score<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#Conclusion\" title=\"Conclusion\">Conclusion<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/#References\" title=\"References\">References<\/a><\/li><\/ul><\/nav><\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-dd5168b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"dd5168b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-05c9592\" data-id=\"05c9592\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-48267bb elementor-widget elementor-widget-text-editor\" data-id=\"48267bb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Artificial Neural Networks (ANNs) are learning architectures that loosely mimic biological neural networks whereby many neurons, the basic unit of information processing are interconnected and acquire knowledge or skill by a process called &#8220;synaptic plasticity&#8221; which is the ability of neuronal junctions to modify their electrochemical compositions to adapt to experiences and consolidate knowledge.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5474233 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5474233\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f59313c\" data-id=\"f59313c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6ee83e2 elementor-widget elementor-widget-heading\" data-id=\"6ee83e2\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Biological_Neuron\"><\/span>Biological Neuron<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-821df4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"821df4a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cc1273b\" data-id=\"cc1273b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fc05fbd elementor-widget elementor-widget-text-editor\" data-id=\"fc05fbd\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>A biological neuron is made of four important parts: the <strong>nucleus<\/strong> which is the central repository of neuronal electrochemistry, the <strong>dendrites<\/strong>, neuronal input membranes, the <strong>axon<\/strong>, the neuron output membrane that ignites a signal when the accumulated nucleus voltage reaches a certain threshold and <strong>synapses<\/strong> which are inhibition and excitation joints in between neuron-to-neuron connections (i.e. dendritic to dendritic, axon to dendritic synapses).<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ba9c966 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ba9c966\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-334a005\" data-id=\"334a005\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a8f66bb elementor-widget elementor-widget-image\" data-id=\"a8f66bb\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"646\" height=\"287\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/neuron.png\" class=\"attachment-large size-large wp-image-1316\" alt=\"A sketch of a biological neuron\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/neuron.png 646w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/neuron-300x133.png 300w\" sizes=\"auto, (max-width: 646px) 100vw, 646px\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\">A sketch of a biological neuron (Jain et al, 1996)<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-7626c7d\" data-id=\"7626c7d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-67dc39a elementor-widget elementor-widget-image\" data-id=\"67dc39a\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"423\" height=\"286\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/neural_neg.png\" class=\"attachment-large size-large wp-image-1333\" alt=\"\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/neural_neg.png 423w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/neural_neg-300x203.png 300w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\">Simple neural net (Park et Lek, 2016)<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4454b63 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4454b63\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7c51497\" data-id=\"7c51497\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7b6a7d0 elementor-widget elementor-widget-text-editor\" data-id=\"7b6a7d0\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Interconnected neurons will thus form biological neural networks whose learning process is based on the guided adaptability of synaptic junctions which are responsible for skill or knowledge acquisition. Thus, <em>synaptic plasticity<\/em>. Artificial neural networks will thus mimic that process.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0fbc057 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0fbc057\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8952a13\" data-id=\"8952a13\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5595201 elementor-widget elementor-widget-heading\" data-id=\"5595201\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Artificial_Neuron\"><\/span>Artificial Neuron<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-813b948 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"813b948\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-53e2460\" data-id=\"53e2460\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dca287a elementor-widget elementor-widget-text-editor\" data-id=\"dca287a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>An Artificial neuron is a simplified biological neuron that focuses on the learning paradigm. The dendrites are replaced by <strong>input variables<\/strong> (x_i), the synapses are replaced by numerical <strong>weights<\/strong>, that adaptively accumulate input signals as a <strong>weighted sum<\/strong>.\u00a0<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-23beba5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"23beba5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-35c1133\" data-id=\"35c1133\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9e382a7 elementor-widget elementor-widget-image\" data-id=\"9e382a7\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"291\" height=\"184\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/articial_neuron.png\" class=\"attachment-large size-large wp-image-1348\" alt=\"Artificial Neuron\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\">Artificial Neuron (Park et Lek, 2016)<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6b33dd8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6b33dd8\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-58779a3\" data-id=\"58779a3\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-854de90 elementor-widget elementor-widget-heading\" data-id=\"854de90\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\">Feed Forward Neural Network - Multilayer Perceptrons<\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9271f37 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9271f37\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a8d3c50\" data-id=\"a8d3c50\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-eb405c1 elementor-widget elementor-widget-text-editor\" data-id=\"eb405c1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Also called multilayer perceptrons (MLP), feed-forward neural networks are very popular artificial neural networks (ANN) with only forward connections that are used for a wide variety of problems, they are universal approximators that can be used for both regression and classification and learn very complex nonlinear relationships.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f41c701 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f41c701\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0977d46\" data-id=\"0977d46\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-90e2e92 elementor-widget elementor-widget-image\" data-id=\"90e2e92\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"374\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/mlp.png\" class=\"attachment-large size-large wp-image-1378\" alt=\"Multilayer Perceptron\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/mlp.png 602w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/mlp-300x186.png 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\">Multilayer Perceptrons (Park et Lek, 2016)<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-051081f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"051081f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9398d09\" data-id=\"9398d09\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f130fbd elementor-widget elementor-widget-heading\" data-id=\"f130fbd\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\">Training ANNs -  The Backpropagation method<\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0bb9eaf elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0bb9eaf\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-82e73bd\" data-id=\"82e73bd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9bfaa2b elementor-widget elementor-widget-text-editor\" data-id=\"9bfaa2b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Similarly, to common parametric supervised learning models, training MLPs refers to minimising the discrepancy between the prediction and the real data output across all data records.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bbed44f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bbed44f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-af5a2b7\" data-id=\"af5a2b7\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-289a55d elementor-widget elementor-widget-image\" data-id=\"289a55d\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"601\" height=\"110\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/sse_aan_eq.png\" class=\"attachment-large size-large wp-image-1391\" alt=\"Objective function, artificial neural networks\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/sse_aan_eq.png 601w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/sse_aan_eq-300x55.png 300w\" sizes=\"auto, (max-width: 601px) 100vw, 601px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-66cb8f2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"66cb8f2\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c80cf27\" data-id=\"c80cf27\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4f8fce1 elementor-widget elementor-widget-text-editor\" data-id=\"4f8fce1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>The backpropagation method, an adapted gradient descent algorithm that uses the chain rule is used to estimate the huge number of parameters int the feedforward neural network.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bf90004 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bf90004\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d78e83a\" data-id=\"d78e83a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9a213f7 elementor-widget elementor-widget-image\" data-id=\"9a213f7\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"167\" height=\"55\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/gradient_desc_alg.png\" class=\"attachment-large size-large wp-image-1393\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\">Gradient descent algorithm - weight update equation<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3d88a43 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3d88a43\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-36f9b09\" data-id=\"36f9b09\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bcd72f9 elementor-widget elementor-widget-heading\" data-id=\"bcd72f9\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\">Python Implementation - Nonlinear regression<\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0e1d8d5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0e1d8d5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3ba6277\" data-id=\"3ba6277\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0011588 elementor-widget elementor-widget-text-editor\" data-id=\"0011588\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>To illustrate the pattern discovery prowess of neural networks, an MLP will be trained to estimate the nonlinear relationship in regression data. A typical linear regression will not accurately estimate the existing relationship in the current dataset. However, using an MLP as a universal regression approximator, the inherent pattern can be deciphered.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1dcb28c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1dcb28c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d3ecc62\" data-id=\"d3ecc62\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8fa3630 elementor-widget elementor-widget-image\" data-id=\"8fa3630\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"565\" height=\"453\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/nonlinear_regression_data.png\" class=\"attachment-large size-large wp-image-1404\" alt=\"\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/nonlinear_regression_data.png 565w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/nonlinear_regression_data-300x241.png 300w\" sizes=\"auto, (max-width: 565px) 100vw, 565px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-36c2b78 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"36c2b78\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a801258\" data-id=\"a801258\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3395541 elementor-widget elementor-widget-elementor-syntax-highlighter\" data-id=\"3395541\" data-element_type=\"widget\" data-widget_type=\"elementor-syntax-highlighter.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<pre><code class='language-python'>import pandas as pd\r\nimport matplotlib.pyplot as plt\r\nimport warnings\r\nwarnings.filterwarnings('ignore') #ignore warnings\r\nfrom sklearn.neural_network import MLPRegressor\r\nfrom sklearn.preprocessing import StandardScaler \r\n\r\n\r\ndf = pd.read_csv('https:\/\/raw.githubusercontent.com\/mlinsights\/freemium\/main\/datasets\/regression-analysis\/1d_nonlinear_regression_data.csv')\r\n\r\nx = df[['x']]\r\ny = df[['y']]\r\n\r\n#standardise the input feature for accurate modelling\r\nscaler = StandardScaler()\r\nscaler.fit(x)\r\nxs = scaler.transform(x) #generate standardised input feature\r\n\r\n#MLP with 1 hidden layer of 20 neurons\r\nclf = MLPRegressor(hidden_layer_sizes=(20),max_iter=1000,\r\n                  early_stopping=False,activation='relu',solver='lbfgs')\r\n\r\n#traing MLP model\r\nclf.fit(xs,y)\r\n\r\n#predict the observation data\r\ny_pred = clf.predict(xs)\r\n\r\nplt.figure()\r\nplt.scatter(x,y,color=&quot;blue&quot;)\r\nplt.plot(x,y_pred,color=&quot;red&quot;)\r\nplt.xlabel('x')\r\nplt.ylabel('y')\r\nplt.title('MLP model estimation')\r\nplt.show() <\/code><\/pre><script>\nif (!document.getElementById('syntaxed-prism')) {\n\tvar my_awesome_script = document.createElement('script');\n\tmy_awesome_script.setAttribute('src','https:\/\/mlinsightscentral.com\/wp-content\/plugins\/syntax-highlighter-for-elementor\/assets\/prism2.js');\n\tmy_awesome_script.setAttribute('id','syntaxed-prism');\n\tdocument.body.appendChild(my_awesome_script);\n} else {\n\twindow.Prism && Prism.highlightAll();\n}\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5db7291 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5db7291\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2843668\" data-id=\"2843668\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-80fe1c2 elementor-widget elementor-widget-image\" data-id=\"80fe1c2\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"565\" height=\"453\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/ann_model_estimation.png\" class=\"attachment-large size-large wp-image-1412\" alt=\"\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/ann_model_estimation.png 565w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/ann_model_estimation-300x241.png 300w\" sizes=\"auto, (max-width: 565px) 100vw, 565px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-08997da elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"08997da\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d037fa7\" data-id=\"d037fa7\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9ba6cf0 elementor-widget elementor-widget-heading\" data-id=\"9ba6cf0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\">Goodness of fit - Coefficient of determination R2 score<\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ca5de04 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ca5de04\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8619e53\" data-id=\"8619e53\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b2ff4f1 elementor-widget elementor-widget-text-editor\" data-id=\"b2ff4f1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Similarly to any regression algorithm, the goodness of fit between the observation data and model prediction including the coefficient of determination (R2 score) will evaluate how well the data fitting process has been. More information of the estimation of this coefficient can be found under <a href=\"https:\/\/mlinsightscentral.com\/index.php\/linear-regression\">linear regression<\/a>.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2880eae elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2880eae\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4d5de52\" data-id=\"4d5de52\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-43419fa elementor-widget elementor-widget-elementor-syntax-highlighter\" data-id=\"43419fa\" data-element_type=\"widget\" data-widget_type=\"elementor-syntax-highlighter.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<pre><code class='language-python'>from sklearn.metrics import r2_score\r\n\r\nr2_score_m = r2_score(y_pred,y)\r\n\r\nplt.figure()\r\nplt.scatter(y, y_pred, color=&quot;b&quot;)\r\nplt.xlabel('observed data: y_d')\r\nplt.ylabel('model prediction: y_p')\r\nplt.title('Goodness of fit -  R2: %.2f'%r2_score_m)\r\nplt.show()  <\/code><\/pre><script>\nif (!document.getElementById('syntaxed-prism')) {\n\tvar my_awesome_script = document.createElement('script');\n\tmy_awesome_script.setAttribute('src','https:\/\/mlinsightscentral.com\/wp-content\/plugins\/syntax-highlighter-for-elementor\/assets\/prism2.js');\n\tmy_awesome_script.setAttribute('id','syntaxed-prism');\n\tdocument.body.appendChild(my_awesome_script);\n} else {\n\twindow.Prism && Prism.highlightAll();\n}\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-070e70e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"070e70e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cd75428\" data-id=\"cd75428\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e245aae elementor-widget elementor-widget-image\" data-id=\"e245aae\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"565\" height=\"453\" src=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/ann_goodness_fit.png\" class=\"attachment-large size-large wp-image-1417\" alt=\"ANN goodness of fit\" srcset=\"https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/ann_goodness_fit.png 565w, https:\/\/mlinsightscentral.com\/wp-content\/uploads\/2023\/08\/ann_goodness_fit-300x241.png 300w\" sizes=\"auto, (max-width: 565px) 100vw, 565px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c1b4ff1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c1b4ff1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-92af3d1\" data-id=\"92af3d1\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c0129d8 elementor-widget elementor-widget-heading\" data-id=\"c0129d8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1fdd5e7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1fdd5e7\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3ab3ca2\" data-id=\"3ab3ca2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d47cd15 elementor-widget elementor-widget-text-editor\" data-id=\"d47cd15\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>This tutorial has introduced the topic of artificial neural networks with a particular emphasis on the popular architecture of feedforward neural networks. Neural networks are both able to detect very complex relationships and are at the same time prompt to overfitting especially for problems that require generalisation on test datasets. Proper training, regularisation and validation approaches are needed to prevent overfitting. Other neural network architectures exist such as convolutional neural networks (CNN), and recurrent neural networks (RNN) are more suited for specific types of problems such as image recognition time series analysis, or audio patterns and more.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-52a835d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"52a835d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-599f45b\" data-id=\"599f45b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3cc26b4 elementor-widget elementor-widget-heading\" data-id=\"3cc26b4\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<p class=\"elementor-heading-title elementor-size-default\">Author: Yves Matanga, PhD<\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9c37bfe elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9c37bfe\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-184eb29\" data-id=\"184eb29\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a33175b elementor-widget elementor-widget-heading\" data-id=\"a33175b\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"References\"><\/span>References<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1f3ae45 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1f3ae45\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ad7d17e\" data-id=\"ad7d17e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2b39d3f elementor-widget elementor-widget-text-editor\" data-id=\"2b39d3f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>[1] Jain, A.K., Mao, J. and Mohiuddin, K.M., 1996. Artificial neural networks: A tutorial. <i>Computer<\/i>,\u00a0<i>29<\/i>(3), pp.31-44.<br \/>[2]\u00a0 Park, Y.S. and Lek, S., 2016. Artificial neural networks: Multilayer perceptron for ecological modelling. In <i>Developments in environmental modelling<\/i>\u00a0(Vol. 28, pp. 123-140). Elsevier.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-30487c98 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"30487c98\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1524aee\" data-id=\"1524aee\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b6cb09e elementor-widget elementor-widget-heading\" data-id=\"b6cb09e\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<span class=\"elementor-heading-title elementor-size-default\">Be the first to receive notification, when new content is available!<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-768efab6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"768efab6\" data-element_type=\"section\">\n\t\t\t\t\t\t\t<div class=\"elementor-background-overlay\"><\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-36213a81\" data-id=\"36213a81\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-59bd7ff5 elementor-widget elementor-widget-wpforms\" data-id=\"59bd7ff5\" data-element_type=\"widget\" data-widget_type=\"wpforms.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"wpforms-container inline-fields\" id=\"wpforms-10\"><form id=\"wpforms-form-10\" class=\"wpforms-validate wpforms-form\" data-formid=\"10\" method=\"post\" enctype=\"multipart\/form-data\" action=\"\/index.php\/wp-json\/wp\/v2\/pages\/1293\" data-token=\"bbd3c4710160bb8dec80be36f260a4f7\"><noscript class=\"wpforms-error-noscript\">Please enable JavaScript in your browser to complete this form.<\/noscript><div class=\"wpforms-field-container\"><div id=\"wpforms-10-field_1-container\" class=\"wpforms-field wpforms-field-email wpforms-mobile-full\" data-field-id=\"1\"><label class=\"wpforms-field-label wpforms-label-hide\" for=\"wpforms-10-field_1\">Email <span class=\"wpforms-required-label\">*<\/span><\/label><input type=\"email\" id=\"wpforms-10-field_1\" class=\"wpforms-field-large wpforms-field-required\" name=\"wpforms[fields][1]\" placeholder=\"Email address\" spellcheck=\"false\" required><\/div><\/div><!-- .wpforms-field-container --><div class=\"wpforms-submit-container\" ><input type=\"hidden\" name=\"wpforms[id]\" value=\"10\"><input type=\"hidden\" name=\"wpforms[author]\" value=\"1\"><button type=\"submit\" name=\"wpforms[submit]\" id=\"wpforms-submit-10\" class=\"wpforms-submit\" data-alt-text=\"Sending...\" data-submit-text=\"Subscribe\" aria-live=\"assertive\" value=\"wpforms-submit\">Subscribe<\/button><\/div><\/form><\/div>  <!-- .wpforms-container -->\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Artificial Neural Networks Artificial Neural Networks (ANNs) are learning architectures that loosely mimic biological neural networks whereby many neurons, the basic unit of information processing are interconnected and acquire knowledge or skill by a process called &#8220;synaptic plasticity&#8221; which is the ability of neuronal junctions to modify their electrochemical compositions to adapt to experiences and &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mlinsightscentral.com\/index.php\/feedforward-neural-networks\/\"> <span class=\"screen-reader-text\">Feedforward Neural Networks<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"no-sidebar","site-content-layout":"page-builder","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":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","footnotes":""},"wf_page_folders":[18],"class_list":["post-1293","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.11 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Feedforward Neural Networks - MLInsightsCentral<\/title>\n<meta name=\"description\" content=\"Artificial Neural Networks (ANNs) are biological inspired learning systems that mimics the human neural networks in a bid to discover complex data patterns. 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