{"id":6590,"date":"2018-09-07T12:50:31","date_gmt":"2018-09-07T11:50:31","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6590"},"modified":"2022-12-03T23:01:58","modified_gmt":"2022-12-03T22:01:58","slug":"probabilite-dabsorption-dun-etat","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/probability-of-state-absorption\/","title":{"rendered":"Probability of absorption of a state"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"6590\" class=\"elementor elementor-6590\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3b20339 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3b20339\" 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-b06ded8\" data-id=\"b06ded8\" 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-c046905 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"c046905\" 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\/markov-process\/\">\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\">Markov process<\/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-d8a1db1\" data-id=\"d8a1db1\" 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-5c012d8 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"5c012d8\" 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 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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-bb5dbb9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bb5dbb9\" 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-4af1abf\" data-id=\"4af1abf\" 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-dc6b2a3 elementor-widget elementor-widget-progress\" data-id=\"dc6b2a3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"progress.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<span class=\"elementor-title\" id=\"elementor-progress-bar-dc6b2a3\">\n\t\t\t\tDifficulty\t\t\t<\/span>\n\t\t\n\t\t<div aria-labelledby=\"elementor-progress-bar-dc6b2a3\" class=\"elementor-progress-wrapper\" role=\"progressbar\" aria-valuemin=\"0\" aria-valuemax=\"100\" aria-valuenow=\"50\" aria-valuetext=\"50% (Moyen)\">\n\t\t\t<div class=\"elementor-progress-bar\" data-max=\"50\">\n\t\t\t\t<span class=\"elementor-progress-text\">Average<\/span>\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-progress-percentage\">50%<\/span>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2295c3d2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2295c3d2\" 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-5f88cbea\" data-id=\"5f88cbea\" 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-10bc68dc elementor-widget elementor-widget-text-editor\" data-id=\"10bc68dc\" 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\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 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\/markov-process\/probability-of-state-absorption\/#Absorption-dun-etat\" >Absorption of a state<\/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\/markov-process\/probability-of-state-absorption\/#Equations-lineaires\" >Linear equations<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Absorption-dun-etat\"><\/span>Absorption of a state<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">markov chain<\/a> is absorbing (absorption of a state) if and only if: there is at least one absorbing state, from any non-absorbing state, one can reach an absorbing state. For any absorbing Markov chain and for any starting state, the probability of being in an absorbing state at time t tends to 1 when t tends to infinity.<\/p>\n\n<p>When dealing with an absorbing Markov chain, we are generally interested in the following two questions:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>How long will it take on average to arrive in an absorbent state, given its initial state?<\/li>\n<li>If there are several absorbing states, what is the probability of falling into a given absorbing state?<\/li>\n<\/ul>\n\n<p>If a Markov chain is absorbent, we will place the absorbing states at the beginning; we<br \/>will then have a transition matrix of the following form (I is a unit matrix and 0<br \/>a matrix of 0):<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6606\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba24.png\" alt=\"Absorption of a state\" width=\"302\" height=\"177\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba24.png 302w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba24-300x176.png 300w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><\/figure>\n<\/div>\n\n<p>The matrix N = (IQ)<sup>-1<\/sup> is called the fundamental matrix of the absorbent chain. Consider the following stochastic matrix:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6607\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba25.png\" alt=\"Absorption of a state\" width=\"208\" height=\"193\" title=\"\"><\/figure>\n<\/div>\n\n<p>We then have to calculate N:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6608\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba26.png\" alt=\"Absorption of a state\" width=\"549\" height=\"361\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba26.png 549w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba26-300x197.png 300w\" sizes=\"(max-width: 549px) 100vw, 549px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The average number e<sub>ij<\/sub> of changes to state j (non-absorbent) before absorption when starting from state i (non-absorbent) is given by e<sub>ij<\/sub> = (N)<sub>ij<\/sub>.<br \/>The average number of steps before absorption knowing that we start from state i (not<br \/>absorbent) is the sum of the terms of the i-th row of N.<\/div>\n\n<p>In the previous example, the average number of steps before absorption is taken from the first line, starting from state 1: 320\/37 + 160\/37 + 100\/37 = 15.67.<\/p>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">In an absorbing Markov chain with P placed under <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-canonical-form-and-standard-form\/\">canonic form<\/a>, the term b<sub>ij<\/sub> of the matrix B = NR is the probability of absorption by the absorbing state j knowing that we start from state i.<\/div>\n\n<p>In the same example:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6609\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba27.png\" alt=\"Absorption of a state\" width=\"528\" height=\"169\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba27.png 528w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba27-300x96.png 300w\" sizes=\"(max-width: 528px) 100vw, 528px\" \/><\/figure>\n<\/div>\n\n<p>The probability of being absorbed by the unique absorbing state is 1 whatever the initial state!<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equations-lineaires\"><\/span>Linear equations<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>From a linear equation point of view, the vector of absorption probabilities is the smallest positive solution of the system:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6613\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba28.png\" alt=\"Absorption of a state\" width=\"534\" height=\"79\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba28.png 534w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba28-300x44.png 300w\" sizes=\"(max-width: 534px) 100vw, 534px\" \/><\/figure>\n\n<p>The vector of <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/probability-of-a-state\/\">mean time to reach<\/a> is the smallest positive solution of the system:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6614\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba29.png\" alt=\"Absorption of a state\" width=\"782\" height=\"83\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba29.png 782w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba29-300x32.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba29-768x82.png 768w\" sizes=\"(max-width: 782px) 100vw, 782px\" \/><\/figure>\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<\/div>","protected":false},"excerpt":{"rendered":"<p>Markov Process Wiki Home Difficulty Medium 50% Absorption of a state A Markov chain is absorbing (absorbing of a state) if and only if \u2026 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