{"id":20986,"date":"2024-02-24T19:45:28","date_gmt":"2024-02-24T18:45:28","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=20986"},"modified":"2024-02-24T20:32:45","modified_gmt":"2024-02-24T19:32:45","slug":"holt-winters","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/holt-winters\/","title":{"rendered":"Holt-Winters multiplicative method"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"20986\" class=\"elementor elementor-20986\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6b2e6b1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6b2e6b1\" data-element_type=\"section\" data-e-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-33 elementor-top-column elementor-element elementor-element-d5f03d6\" data-id=\"d5f03d6\" data-element_type=\"column\" data-e-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-8ccfd39 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"8ccfd39\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Forecasting<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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-33 elementor-top-column elementor-element elementor-element-5d2b4f5\" data-id=\"5d2b4f5\" data-element_type=\"column\" data-e-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-5c01db3 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"5c01db3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Home page<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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-33 elementor-top-column elementor-element elementor-element-db839c0\" data-id=\"db839c0\" data-element_type=\"column\" data-e-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-1f5a56b elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"1f5a56b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/plat.ai\/blog\/difference-between-prediction-and-forecast\/\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-92100eb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"92100eb\" data-element_type=\"section\" data-e-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-21549ac\" data-id=\"21549ac\" data-element_type=\"column\" data-e-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-dc2fe35 elementor-widget elementor-widget-heading\" data-id=\"dc2fe35\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">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\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" 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><\/span><\/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-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/holt-winters\/#Methode-multiplicative-de-Holt-Winters\" >Holt-Winters multiplicative method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/holt-winters\/#Lissage-exponentiel-simple-Exponential-Smoothing\" >Simple exponential smoothing<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/holt-winters\/#Tendance-lineaire-de-Holt\" >Holt&#039;s linear trend<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/holt-winters\/#Methode-de-Holt-Winters-multiplicative\" >Multiplicative Holt-Winters method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/holt-winters\/#Methode-additive-Holt-Winters\" >Holt-Winters additive method<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Methode-multiplicative-de-Holt-Winters\"><\/span>Holt-Winters multiplicative method<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a26ea57 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a26ea57\" data-element_type=\"section\" data-e-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-c50fe34\" data-id=\"c50fe34\" data-element_type=\"column\" data-e-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-2149831 elementor-widget elementor-widget-text-editor\" data-id=\"2149831\" data-element_type=\"widget\" data-e-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\t\t<p>This is how the Holt-Winters multiplicative or triple smoothing method works.<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"Holt-Winters\" width=\"97\" height=\"97\" title=\"\"><\/p>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e3e9ed3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e3e9ed3\" data-element_type=\"section\" data-e-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-c46eded\" data-id=\"c46eded\" data-element_type=\"column\" data-e-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-31573e3 elementor-widget elementor-widget-heading\" data-id=\"31573e3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Lissage-exponentiel-simple-Exponential-Smoothing\"><\/span>Simple exponential smoothing<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b300c16 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b300c16\" data-element_type=\"section\" data-e-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-885b061\" data-id=\"885b061\" data-element_type=\"column\" data-e-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-30b6bd6 elementor-widget elementor-widget-text-editor\" data-id=\"30b6bd6\" data-element_type=\"widget\" data-e-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\t\t<p>In simple (aka simple) exponential smoothing, the predicted value at time i+1 is based on the value at time i and the predicted value at time i (and therefore indirectly on all previous time values). In particular, for some \u03b1 where 0 \u2264 \u03b1 \u2264 1, for all i &gt; 1, we define<\/p><p><img decoding=\"async\" class=\"alignnone size-full wp-image-20994\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters1.png\" alt=\"holt-winters\" width=\"189\" height=\"21\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters1.png 189w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters1-18x2.png 18w\" sizes=\"(max-width: 189px) 100vw, 189px\" \/><\/p><p>Note that we do not include time i = 1 in the calculations of MAE and MSE.<\/p><p>In <a href=\"https:\/\/complex-systems-ai.com\/en\/logic-math-27\/\">algebra<\/a> simple, this iteration can also be expressed as<\/p><p><img decoding=\"async\" class=\"alignnone size-full wp-image-20995\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters2.png\" alt=\"exponential smoothing\" width=\"239\" height=\"21\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters2.png 239w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters2-18x2.png 18w\" sizes=\"(max-width: 239px) 100vw, 239px\" \/><\/p><p>Let&#039;s look at the formula with Excel. The formula in cell C4 is =B4 and the formula in cell C5 is =C4+B$21*(B4-C4).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-20996 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters3.png\" alt=\"Exponential Smoothing\" width=\"859\" height=\"444\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters3.png 859w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters3-300x155.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters3-768x397.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters3-18x9.png 18w\" sizes=\"(max-width: 859px) 100vw, 859px\" \/><\/p><p>The forecast for the next value in the time series is 74.0 (cell C19), using the formula =C18+B$21*(B18-C18)<\/p><p>Excel provides the Exponential Smoothing data analysis tool to simplify the calculations described above.<\/p><p>To use this tool for example, select Data &gt; Analysis|<a href=\"https:\/\/complex-systems-ai.com\/en\/data-analysis\/\">Data analysis<\/a> and choose Exponential Smoothing from the menu that appears. A dialog box now appears. A Damping Factor field is used instead of the Interval field. If this field is left blank, its default value is 0.7.<\/p><p>The damping factor is only 1 \u2013 \u03b1. So, for example, you should use 0.6 as the damping factor.<\/p><p>The result is presented in columns D and E of the figure with the graph.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-20998 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters4.png\" alt=\"Exponential Smoothing\" width=\"746\" height=\"364\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters4.png 746w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters4-300x146.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters4-18x9.png 18w\" sizes=\"(max-width: 746px) 100vw, 746px\" \/><\/p>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7a619ad elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7a619ad\" data-element_type=\"section\" data-e-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-77a5e19\" data-id=\"77a5e19\" data-element_type=\"column\" data-e-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-eab5aab elementor-widget elementor-widget-heading\" data-id=\"eab5aab\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Tendance-lineaire-de-Holt\"><\/span>Holt&#039;s linear trend<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0140827 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0140827\" data-element_type=\"section\" data-e-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-dd3af9c\" data-id=\"dd3af9c\" data-element_type=\"column\" data-e-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-9d6c8ff elementor-widget elementor-widget-text-editor\" data-id=\"9d6c8ff\" data-element_type=\"widget\" data-e-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\t\t<p>The data in the previous figure from simple exponential smoothing (as well as previous figures on this web page) show a clear upward trend. The moving average and single exponential smoothing methods do not model this adequately, but Holt&#039;s linear trend method (aka double exponential smoothing) does. This is accomplished by adding a second unique exponential smoothing model to capture the trend (upward or downward). The model takes the following form for all i &gt; 1.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22003 size-full\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image002z.png\" alt=\"Double Exponential Smoothing\" width=\"127\" height=\"21\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22004 size-full\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image003z.png\" alt=\"Double Exponential Smoothing\" width=\"209\" height=\"21\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22005 size-full\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image004z.png\" alt=\"Double Exponential Smoothing\" width=\"209\" height=\"21\" title=\"\"><\/p><p><a href=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image005z.png\" rel=\"attachment wp-att-22006 nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22006\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image005z.png\" alt=\"image005z\" width=\"92\" height=\"21\" title=\"\"><\/a><\/p><p>where 0 \u2264 \u03b1 \u2264 1 and 0 &lt; \u03b2 \u2264 1.<\/p><p>An alternative form of these equations is<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22003\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image002z.png\" alt=\"image002z\" width=\"127\" height=\"21\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22007\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image006z.png\" alt=\"image006z\" width=\"146\" height=\"21\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22008\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image007z.png\" alt=\"image007z\" width=\"109\" height=\"21\" title=\"\"><\/p><p><a href=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image005z.png\" rel=\"attachment wp-att-22006 nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22006\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image005z.png\" alt=\"image005z\" width=\"92\" height=\"21\" title=\"\"><\/a><\/p><p>Or<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22009\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image008z.png\" alt=\"image008z\" width=\"226\" height=\"21\" title=\"\"><\/p><p>If beta=0 then we have simple exponentiation.<\/p><p>The result is presented in the figure. Here, cell C4 contains the formula =B4, cell D4 contains the value 0, cell C5 contains the formula =B$21*B5+(1-B$21)*(C4+D4), cell D5 contains the formula =C$21* (C5-C4)+(1-C$21)*D4 and cell E5 contains the formula =C4+D4.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21000 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters7.png\" alt=\"Double Exponential Smoothing\" width=\"988\" height=\"448\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters7.png 988w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters7-300x136.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters7-768x348.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters7-18x8.png 18w\" sizes=\"(max-width: 988px) 100vw, 988px\" \/><\/p><p>For any value of i, the forecast at time i+h is given by the formula<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22010\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image009z.png\" alt=\"image009z\" width=\"101\" height=\"21\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21001 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters8.png\" alt=\"holt-winters\" width=\"732\" height=\"465\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters8.png 732w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters8-300x191.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters8-18x12.png 18w\" sizes=\"(max-width: 732px) 100vw, 732px\" \/><\/p>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ca80f83 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ca80f83\" data-element_type=\"section\" data-e-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-3e45f4f\" data-id=\"3e45f4f\" data-element_type=\"column\" data-e-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-efd985c elementor-widget elementor-widget-heading\" data-id=\"efd985c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Methode-de-Holt-Winters-multiplicative\"><\/span>Multiplicative Holt-Winters method<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0b0a4f7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0b0a4f7\" data-element_type=\"section\" data-e-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-99da799\" data-id=\"99da799\" data-element_type=\"column\" data-e-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-97001d1 elementor-widget elementor-widget-text-editor\" data-id=\"97001d1\" data-element_type=\"widget\" data-e-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\t\t<p>In the Holt-Winters method (aka triple exponential smoothing), we add a seasonal component to Holt&#039;s linear trend model. We explore two of these models: the multiplicative seasonality model and the additive seasonality model. We consider the first of these models on this web page. See Holt\u2013Winters additive model for the second model.<\/p><p>Let c be the duration of a seasonal cycle. Thus, c = 12 for the months of a year, c = 7 for the days of a week and c = 4 for the quarters of a year. The model takes the following recursive form for all i &gt; c<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-26562 size-full\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/03\/image159d.png\" alt=\"Holt\u2013Winter u equation\" width=\"246\" height=\"20\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22005\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/03\/image004z.png\" alt=\"image004z\" width=\"209\" height=\"21\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-26563 size-full\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/03\/image160d.png\" alt=\"Holt\u2013Winter s equation\" width=\"171\" height=\"20\" title=\"\"><\/p><p>where 0 \u2264 \u03b1 \u2264 1, 0 \u2264 \u03b2 \u2264 1 and 0 \u2264 \u03b3 \u2264 1.<\/p><p>The ui values represent the baseline, the vi values represent the trend (i.e. slope), and the si values represent the seasonal component. In the multiplicative model, for any c consecutive time periods, the sum of the si values is approximately equal to c (at least for reasonable values of \u03b1, \u03b2, \u03b3).<\/p><p>Predictions for data elements yi can be expressed as<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-28460\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2019\/03\/image005m.png\" alt=\"Predicted y values\" width=\"143\" height=\"21\" title=\"\"><\/p><p>For forecasts at future times, we use the form<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-28461\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2019\/03\/image006m.png\" alt=\"Forecasted future values\" width=\"165\" height=\"21\" title=\"\"><\/p><p>where h\u2032 =INT((h\u20131)\/c)+1.<\/p><p>Alternative forms<br \/>We can also use the following alternative version of the term seasonality:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33936\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image041.png\" sizes=\"(max-width: 258px) 100vw, 258px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2021\/06\/image041.png 334w, https:\/\/real-statistics.com\/wp-content\/uploads\/2021\/06\/image041-300x31.png 300w\" alt=\"Alternative seasonality term\" width=\"258\" height=\"27\" title=\"\"><\/p><p>Based on this version of the seasonality term, we have the following alternative form of the recursive equations:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33937\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image042.png\" alt=\"Alternative u_i term\" width=\"207\" height=\"24\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33938\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image043.png\" alt=\"Alternative v_i term\" width=\"164\" height=\"24\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33939\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image044.png\" alt=\"Alternative s_i term\" width=\"218\" height=\"24\" title=\"\"><\/p><p><a href=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image045.png\" rel=\"nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33940\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image045.png\" alt=\"Alternative y_i\" width=\"201\" height=\"24\" title=\"\"><\/a><\/p><p>The initial values of the model, i.e. where 1 \u2264 i \u2264 c, are<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-28462\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2019\/03\/image007m.png\" alt=\"Initial values Holt-Winters\" width=\"255\" height=\"42\" title=\"\"><\/p><p>Alternatively, we can set the initial trend value using the average slope of the first two years, namely:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-28465\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2019\/03\/image008m.png\" sizes=\"(max-width: 333px) 100vw, 333px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2019\/03\/image008m.png 333w, https:\/\/real-statistics.com\/wp-content\/uploads\/2019\/03\/image008m-300x40.png 300w\" alt=\"Initial trend\" width=\"333\" height=\"44\" title=\"\"><\/p><p>Note that if \u03b3 = 0, then the Holt-Winters model is equivalent to the Holt linear trend model, and if \u03b2 = 0 and \u03b3 = 0, then the Holt-Winters model is equivalent to the simple exponential smoothing model.<\/p><p>Calculate the predicted values of the time series shown in the range C4:C19 of the figure using the Holt-Winter method with \u03b1 = 0.5, \u03b2 = 0.5, and \u03b3 = 0.5.<\/p><p>The result is presented in the figure. First, we calculate s1, s2, s3, s4, where c = 4, as shown in the range F4:F7. We do this by inserting the formula =C4\/AVERAGE(C$4:C$7) into cell F4, highlighting the range F4:F7, and pressing Ctrl-D.<\/p><p>Next, we calculate uc and vc by placing the formula =C7\/F7 in cell D7 and the value 0 in cell E7.<\/p><p>We now insert the formula =C$22*C8\/F4+(1-C$22)*(D7+E7) in cell D8, the formula =D$22*(D8-D7)+(1-D$22)*E7 in cell E8, =E$22*(C8\/D8)+(1-E$22)*F4 in cell F8 and the formula =(D7+E7)*F4 in cell G8, then highlight the range D8:F19 and press Ctrl- D.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21002 size-large\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters9-1024x444.png\" alt=\"Multiplicative Holt-Winters\" width=\"1024\" height=\"444\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters9-1024x444.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters9-300x130.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters9-768x333.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters9-18x8.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters9.png 1075w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p><p>We can use Solver to determine which values of alpha, beta and <a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/\">gamma<\/a> give the best Holt-Winters fit for the example data.<\/p><p>The optimization approach using Excel Solver is, however, likely to find a local minimum instead of a global minimum. For this reason, the optimized values for alpha, beta, and gamma are sensitive to the initial values used. You can ask Solver to try different initial values to find the parameter values that reduce the MAE value.<\/p><p>To do this, select Data &gt; Analysis | Solver and press the Options button in the Solver dialog box. The Solver dialog box automatically contains the values collected by the Basic Forecasting data analysis tool. Now choose the Multistart option from the GRG Nonlinear tab of the Options dialog box. The solver will now run multiple times using different starting values, selecting the values that produce the best result. The cost of this option is slower execution times.<\/p>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8101990 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8101990\" data-element_type=\"section\" data-e-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-37bdb27\" data-id=\"37bdb27\" data-element_type=\"column\" data-e-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-e010b14 elementor-widget elementor-widget-heading\" data-id=\"e010b14\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Methode-additive-Holt-Winters\"><\/span>Holt-Winters additive method<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4bccbb7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4bccbb7\" data-element_type=\"section\" data-e-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-d125351\" data-id=\"d125351\" data-element_type=\"column\" data-e-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-0501e80 elementor-widget elementor-widget-text-editor\" data-id=\"0501e80\" data-element_type=\"widget\" data-e-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\t\t<p>The Holt-Winters additive model is identical to the multiplicative model, except that seasonality is considered additive. This means that the predicted value for each data element is the sum of the baseline, trend, and seasonal components. The sum of the seasonality components for c consecutive periods is approximately 1.<\/p><p>The recursive approach to the additive model is<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-31445 size-medium\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image189-300x23.png\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image189-300x23.png 300w, https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image189.png 397w\" alt=\"Additive level\" width=\"300\" height=\"23\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-31446\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image190-300x30.png\" sizes=\"(max-width: 240px) 100vw, 240px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image190-300x30.png 300w, https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image190.png 314w\" alt=\"Additive trend\" width=\"240\" height=\"24\" title=\"\"><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-38170\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2022\/08\/image014.png\" alt=\"\" width=\"233\" height=\"24\" title=\"\"><\/p><p>where 0 \u2264 \u03b1 \u2264 1, 0 \u2264 \u03b2 \u2264 1 and 0 \u2264 \u03b3 \u2264 1.<\/p><p>The predictions for data elements yi are given by<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-31448\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image192.png\" alt=\"Additive forecast\" width=\"194\" height=\"27\" title=\"\"><\/p><p>For forecasts at future times, we use the form<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-31449\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/08\/image193.png\" alt=\"Future additive forecast\" width=\"220\" height=\"27\" title=\"\"><\/p><p>\u00a0where h\u2032 =INT((h\u20131)\/c)+1.An alternative version of the seasonality term is:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33930\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image036.png\" sizes=\"(max-width: 290px) 100vw, 290px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2021\/06\/image036.png 350w, https:\/\/real-statistics.com\/wp-content\/uploads\/2021\/06\/image036-300x30.png 300w\" alt=\"Alternative seasonality term\" width=\"290\" height=\"29\" title=\"\"><\/p><p>Based on this version of the seasonality term, we have the following alternative form of the recursive equations:<\/p><p><a href=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image037.png\" rel=\"nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33931\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image037.png\" alt=\"Alternative u_i term\" width=\"191\" height=\"27\" title=\"\"><\/a><\/p><p><a href=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image038.png\" rel=\"nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33932\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image038.png\" alt=\"Alternative v_i term\" width=\"142\" height=\"27\" title=\"\"><\/a><\/p><p><a href=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image039.png\" rel=\"nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33933\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image039.png\" alt=\"Alternative s_i term\" width=\"124\" height=\"27\" title=\"\"><\/a><\/p><p><a href=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image040.png\" rel=\"nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-33934\" src=\"https:\/\/www.real-statistics.com\/wp-content\/uploads\/2021\/06\/image040.png\" alt=\"y_i term\" width=\"233\" height=\"27\" title=\"\"><\/a><\/p><p>Forecast y values for 2014 (i.e. the next four quarters) using the Holt-Winter additive method based on data from the E4:F19 range of the figure using alpha values, beta and gamma which minimize the MSE statistic.<\/p><p>This is the same problem as the Holt-Winters multiplicative method example, except now we need to apply the additive method.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21003 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters10.png\" alt=\"Holt-Winters&#039; Additive Forecast\" width=\"964\" height=\"513\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters10.png 964w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters10-300x160.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters10-768x409.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/holt-winters10-18x10.png 18w\" sizes=\"(max-width: 964px) 100vw, 964px\" \/><\/p><p>We can calculate the standard error and prediction intervals for Holt-Winters additive model forecasts as described in Exponential Smoothing Confidence Interval, except now the standard error of k steps ahead is<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-31263\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/07\/image184.png\" alt=\"Holt-Winters standard error\" width=\"220\" height=\"67\" title=\"\"><\/p><p>or<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-31264 size-medium\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/07\/image185-300x44.png\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/07\/image185-300x44.png 300w, https:\/\/real-statistics.com\/wp-content\/uploads\/2020\/07\/image185.png 403w\" alt=\"Standard error formulas\" width=\"300\" height=\"44\" title=\"\"><\/p><p>The standard errors and prediction intervals in the figure are calculated exactly as for the Holt linear trend (see Holt linear trend confidence interval) with the exception of cell J26 since it is a entry where c|i-1. The formula in cell J26 is<\/p><p>=22 K$*SQRT((K25\/K$22)^2-1+(F$30*(1+(ROW(J26)-ROW(J$22))*G$30+(1-F$30)*H$30) )^2+1)<\/p><p>The term (1-F$30)*H$30 in this formula is not found in other standard error formulas (although such a term is included in the 5 step lead, 9 step lead, 13 step d advance, etc. standard errors).<\/p>\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<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Forecasting Homepage Wiki Multiplicative method of Holt-Winters This is how the multiplicative method of Holt-Winters or triple smoothing works. Simple exponential smoothing \/ Exponential Smoothing \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":20753,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-20986","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/20986","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=20986"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/20986\/revisions"}],"predecessor-version":[{"id":21007,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/20986\/revisions\/21007"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/20753"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=20986"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}