{"id":3367,"date":"2025-10-25T10:59:22","date_gmt":"2025-10-25T08:59:22","guid":{"rendered":"https:\/\/neuraldesigner.com\/blog\/3_methods_to_deal_with_outliers\/"},"modified":"2025-11-27T15:17:36","modified_gmt":"2025-11-27T14:17:36","slug":"effective-outlier-treatment-methods-machine-learning","status":"publish","type":"blog","link":"https:\/\/www.neuraldesigner.com\/blog\/effective-outlier-treatment-methods-machine-learning\/","title":{"rendered":"3 methods to treat outliers in machine learning"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"3367\" class=\"elementor elementor-3367\" data-elementor-post-type=\"blog\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6d71d4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6d71d4\" 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-2f4f081\" data-id=\"2f4f081\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5e0afe55 elementor-widget elementor-widget-text-editor\" data-id=\"5e0afe55\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>This post examines three key methods for handling outliers in machine learning.&nbsp;<\/p>\n<p><a style=\"background-color: #ffffff;\" href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Outliers\">Outliers<\/a> are data points that are distant from the rest.&nbsp;<\/p>\n<p>They may be due to variability in the measurement or may indicate experimental errors.&nbsp;<\/p>\n<p><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\">If possible, outliers should be excluded from the <\/span><a style=\"font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight ); background-color: #ffffff;\" href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\">dataset<\/a><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\">.&nbsp;<\/span><\/p>\n<p><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\">However, detecting anomalous <\/span><a style=\"font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight ); background-color: #ffffff;\" href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Instances\">instances<\/a><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-size: var( --e-global-typography-text-font-size ); font-weight: var( --e-global-typography-text-font-weight );\"> might be difficult and is not always possible.<\/span><\/p>\n<p>The data science and machine learning platform&nbsp;<a href=\"https:\/\/www.neuraldesigner.com\/\" style=\"background-color: rgb(255, 255, 255);\">Neural Designer<\/a>&nbsp;encompasses all these methods, enabling their practical application. You can download a free trial&nbsp;<a href=\"https:\/\/www.neuraldesigner.com\/free-trial\" style=\"background-color: rgb(255, 255, 255);\">here<\/a>.<\/p>\n<section><\/section>\n<section>\n<ul>\n<li><a href=\"#Introduction\">Introduction<\/a>.<\/li>\n<li><a href=\"#UnivariateMethod\">1. Univariate method<\/a>.<\/li>\n<li><a href=\"#MultivariateMethod\">2. Multivariate method<\/a>.<\/li>\n<li><a href=\"#MinkowskiError\">3. Minkowski error<\/a>.<\/li>\n<li><a href=\"#Conclusions\">Conclusions<\/a>.<\/li>\n<\/ul>\n<div>&nbsp;<\/div>\n<div>\n<p><span style=\"color: inherit; font-family: inherit; font-size: 2rem;\">Introduction<\/span><\/p>\n<\/div>\n<\/section>\n<section>\n<p>Machine learning algorithms are susceptible to the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Statistics\">statistics<\/a> and <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Distributions\">distribution<\/a> of the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#InputVariables\">input variables<\/a>.&nbsp;<\/p>\n<p>Data outliers can spoil and mislead the training process.&nbsp;<\/p>\n<p>That results in longer training times, less accurate models, and poor results.<\/p>\n<p>In this post, we introduce three different methods of dealing with outliers:<\/p>\n<ol>\n<li><a href=\"#UnivariateMethod\">Univariate method<\/a>: This method identifies data points with extreme values on a single variable.<\/li>\n<li><a href=\"#MultivariateMethod\">Multivariate method<\/a>: Here, we look for unusual combinations of all the variables.<\/li>\n<li><a href=\"#MinkowskiError\">Minkowski error<\/a>: This method reduces the contribution of potential outliers in the training process.<\/li>\n<\/ol>\n<p><br><\/p>\n<p>To illustrate those methods, we generate a <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\">data set<\/a> from the following function<\/p>\n<p>$$y = sin{(pi x)}$$<\/p>\n<p>Then, we replace two (y) values for those far from our function. The following chart depicts this data set.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/outliers-data.webp\" alt=\"Outliers data\" width=\"600\" height=\"400\"><\/p>\n<p>The points A=(-0.5,-1.5) and B=(0.5,0.5) are outliers.&nbsp;<\/p>\n<p>Point (A) is outside the range defined by the (y) data, while Point (B) is inside that range. As we will see, that makes them different, and we will need different methods to detect and treat them.<\/p>\n<\/section>\n<section>\n<h2>1. Univariate method<\/h2>\n<p>One of the simplest methods for detecting outliers is using <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#BoxPlots\">box plots<\/a>.&nbsp;<\/p>\n<p>A box plot is a graphical display that describes the data distribution. Box plots use the median and the lower and upper quartiles.<\/p>\n<p><span style=\"margin: 0px; padding: 0px;\">Tukey&#8217;s method defines an outlier as a value of a&nbsp;<a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Variables\" target=\"_blank\" rel=\"noopener\">variable<\/a>&nbsp;that falls far from the central point, the median.<\/span><\/p>\n<p>The <em>cleaning parameter<\/em> is the maximum distance to the median that will be allowed. The test becomes less sensitive to outliers if the cleaning parameter is large. On the contrary, many values are detected as outliers if they are too small.<\/p>\n<p>The following chart shows the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#BoxPlots\">box plot<\/a> for the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Variables\">variable<\/a> (y).<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/outliers-box-plot-1.webp\" alt=\"treat outliers \" width=\"600\" height=\"400\"><\/p>\n<p>As we can see, the minimum is far away from the first quartile and the median. If we set the cleaning parameter to 0.6, Tukey&#8217;s method&nbsp;<span style=\"margin: 0px; padding: 0px;\">detects Point (A) as an outlier and removes it from the&nbsp;<a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\" target=\"_blank\">dataset<\/a><\/span>.<\/p>\n<p>Plotting the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#BoxPlots\">box plot<\/a> for that <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Variables\">variable<\/a> again, we can notice that the outlier has been removed. As a consequence, the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Distributions\">distribution<\/a> of the data is now much better.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/outliers-box-plot-2.webp\" alt=\"\" width=\"600\" height=\"400\"><\/p>\n<p>However, this univariate method has not detected Point (B), so we are not finished.<\/p>\n<\/section>\n<section>\n<h2>2. Multivariate method<\/h2>\n<p>Outliers do not need to be extreme values. Indeed, as we have seen with Point (B), the <a href=\"#UnivariateMethod\">univariate method<\/a> does not always work well. The multivariate approach&nbsp;<span style=\"margin: 0px; padding: 0px;\">attempts to address this by building a predictive model using all available data and removing&nbsp;<a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set#Instances\" target=\"_blank\" rel=\"noopener\">instances<\/a>&nbsp;with errors exceeding a given threshold<\/span>.<\/p>\n<p>In this case, we have trained a <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\">neural network<\/a> using all the available data (but Point (A), excluded by the univariate method). Then, we perform a <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/testing-analysis#LinearRegressionAnalysis\">linear regression analysis<\/a> to obtain the following graph. The predicted values are plotted versus the real ones. The colored line indicates the best linear fit, and the grey line indicates a perfect fit.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/linear_regression_outliers.webp\" alt=\"Linear regression analytis showing an outlier\"><\/p>\n<p>In the above chart, a point falls too far from the model. This point spoils the model, and therefore, it might be another outlier.<\/p>\n<p>To find that point quantitatively, we can calculate the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/testing-analysis#MaximalErrors\">maximum&nbsp;<\/a><span style=\"margin: 0px; padding: 0px;\"><a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/testing-analysis#MaximalErrors\" target=\"_blank\" rel=\"noopener\">error<\/a>&nbsp;between the model&#8217;s outputs<\/span>&nbsp;and the targets. The following table lists the five instances with maximum errors.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Instance<\/th>\n<th>Error<\/th>\n<th>X<\/th>\n<th>Y<\/th>\n<\/tr>\n<tr>\n<th>11<\/th>\n<td>0.430<\/td>\n<td>0.500<\/td>\n<td>0.500<\/td>\n<\/tr>\n<tr>\n<th>10<\/th>\n<td>0.069<\/td>\n<td>0.550<\/td>\n<td>0.987<\/td>\n<\/tr>\n<tr>\n<th>12<\/th>\n<td>0.067<\/td>\n<td>0.450<\/td>\n<td>0.987<\/td>\n<\/tr>\n<tr>\n<th>9<\/th>\n<td>0.064<\/td>\n<td>0.600<\/td>\n<td>0.951<\/td>\n<\/tr>\n<tr>\n<th>13<\/th>\n<td>0.058<\/td>\n<td>0.400<\/td>\n<td>0.951<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As we can notice, sample 11 has a significant error compared to the others. If we look at the linear regression chart, we see that this instance matches the point far from the model.<\/p>\n<p>By selecting 20% of maximum error, this method identifies Point B as an outlier and cleans it from the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\">data set<\/a>. We can see that by performing a <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/testing-analysis#LinearRegressionAnalysis\">linear regression analysis<\/a> again.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/linear_regression_no_outliers.webp\" alt=\"Linear regression analysis without outliers\"><\/p>\n<p>There are no more outliers in the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\">data set<\/a>, so the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\">neural network&#8217;s<\/a> generalization capabilities improve notably.<\/p>\n<\/section>\n<section>\n<h2>3. Minkowski error<\/h2>\n<p>Now, we will discuss an alternative method for handling outliers. Unlike the <a href=\"#UnivariateMethod\">univariate<\/a> and <a href=\"#MultivariateMethod\">multivariate<\/a> methods, it doesn&#8217;t detect or clean the outliers. Instead, it reduces the impact that outliers will have on the model.<\/p>\n<p>The <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MinkowskiError\">Minkowski error<\/a> is a <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#LossIndex\">loss index<\/a>&nbsp;that is less sensitive to outliers than the standard <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MeanSquaredError\">mean squared error<\/a>.<\/p>\n<p>The <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MeanSquaredError\">mean squared error<\/a> raises each instance error to the square, making a too-big contribution of outliers to the total error,<\/p>\n<p>$$mean\\_squared\\_error = \\frac{\\sum \\left(outputs &#8211; targets\\right)^2}{instances\\_number}$$<\/p>\n<p>The <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MinkowskiError\">Minkowski error<\/a>&nbsp;addresses this by raising each instance error to a power smaller than 2. This number is called the <em>Minkowski parameter<\/em>&nbsp;and reduces the contribution of outliers to the total error,<\/p>\n<p>$$minkowski\\_error = \\frac{\\sum\\left(outputs &#8211; targets\\right)^{minkowski\\_parameter}}{instances\\_number}$$<\/p>\n<p>A typical value for the Minkowski parameter is 1.5.<\/p>\n<p>For instance, if an outlier has an error of 10, the squared error for that instance is $(10^2=100)$, while the Minkowski error is $10^{1.5} = 31.62$.<\/p>\n<p>To illustrate this method, we build two different <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\">neural networks<\/a>&nbsp;<span style=\"margin: 0px; padding: 0px;\">using our&nbsp;<a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/data-set\" target=\"_blank\">dataset<\/a>, which contains<\/span>&nbsp;two outliers (A and B). The architecture selected for this network is 1:24:1. The loss index for the first neural network is the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MeanSquaredError\">mean squared error<\/a>. The loss index for the second neural network is the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MinkowskiError\">Minkowski error<\/a>.<\/p>\n<p>The <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\">neural network<\/a> trained with the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MeanSquaredError\">mean squared error<\/a> is plotted in the following figure. As we can see, two outliers are spoiling the model.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/model_trained.webp\" alt=\"Model trained with the normalized squared error, sensitive to outliers\"><\/p>\n<p>Now, we train the same <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-network\">neural network<\/a> with the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MinkowskiError\">Minkowski error<\/a>. The following figure depicts the resulting model.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/images\/trained_minkowski.webp\" alt=\"Model trained with the normalized squared error, sensitive to outliers\"><\/p>\n<p>As a result, the <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/training-strategy#MinkowskiError\">Minkowski error<\/a> has made the training process less sensitive to outliers and has improved the quality of our model.<\/p>\n<\/section>\n<section>\n<h2>Conclusions<\/h2>\n<p>We have observed that outliers are a significant issue when building a predictive model. Indeed, they cause data scientists to achieve more unsatisfactory results than they could. We need practical methods to deal with those spurious points and remove them to resolve this issue.<\/p>\n<p>In this article, we have seen 3 different methods for dealing with outliers: the <a href=\"#UnivariateMethod\">univariate method<\/a>, the <a href=\"#MultivariateMethod\">multivariate method<\/a>, and the <a href=\"#MinkowskiError\">Minkowski error<\/a>. These methods are complementary, and we may need to try them all if our dataset contains many severe outliers.<\/p>\n<\/section>\n<section>\n<h2>Related posts<\/h2>\n<\/section>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"author":20,"featured_media":1767,"template":"","categories":[],"tags":[36],"class_list":["post-3367","blog","type-blog","status-publish","has-post-thumbnail","hentry","tag-tutorials"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>3 methods to treat outliers in machine learning<\/title>\n<meta name=\"description\" content=\"Discover three methods to treat outliers in machine learning, ensuring more accurate models and better results.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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