{"id":6644,"date":"2018-09-15T15:19:09","date_gmt":"2018-09-15T14:19:09","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6644"},"modified":"2022-12-03T23:02:00","modified_gmt":"2022-12-03T22:02:00","slug":"chaines-de-markov-en-temps-continu","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/continuous-time-markov-chains\/","title":{"rendered":"Continuous-time Markov chains"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"6644\" class=\"elementor elementor-6644\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e8a8b3d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e8a8b3d\" 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-287e317\" data-id=\"287e317\" 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-ae2feaa elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"ae2feaa\" 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\/processus-de-markov\/\">\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\">Processus de Markov<\/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-b3fc716\" data-id=\"b3fc716\" 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-30617e2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"30617e2\" 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\/\">\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\">Page 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elementor-widget-progress\" data-id=\"281f04e\" 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-281f04e\">\n\t\t\t\tDifficult\u00e9\t\t\t<\/span>\n\t\t\n\t\t<div aria-labelledby=\"elementor-progress-bar-281f04e\" 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\">Moyen<\/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-26c917c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"26c917c\" 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-1d328412\" data-id=\"1d328412\" 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-7b9d97bb elementor-widget elementor-widget-text-editor\" data-id=\"7b9d97bb\" 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\">Contenus<\/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\/continuous-time-markov-chains\/#Chaines-de-Markov-en-temps-continu\" >Chaines de Markov en temps continu<\/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\/continuous-time-markov-chains\/#Temps-continu\" >Temps continu<\/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\/markov-process\/continuous-time-markov-chains\/#Modele-de-duree\" >Mod\u00e8le de dur\u00e9e<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Chaines-de-Markov-en-temps-continu\"><\/span>Chaines de Markov en temps continu<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Dans le cas du temps discret, nous observons les \u00e9tats sur des moments instantan\u00e9s et immuables. Dans le cadre des chaines de Markov en temps continu, les observations se dont de fa\u00e7on continu, c&rsquo;est \u00e0 dire sans interruption temporelle.<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Temps-continu\"><\/span>Temps continu<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Soient M + 1 \u00e9tats mutuellement exclusifs. L\u2019analyse d\u00e9bute au temps 0 et le temps s\u2019\u00e9coule de fa\u00e7on continue, on nomme X(t) l\u2019\u00e9tat du syst\u00e8me au temps t. Les points de changement d\u2019\u00e9tats t<sub>i<\/sub> sont des points al\u00e9atoires dans le temps (ils ne sont pas n\u00e9cessairement entiers). Il est impossible d&rsquo;avoir deux changements d&rsquo;\u00e9tats en m\u00eame temps.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6648\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage62.png\" alt=\"Chaines de Markov en temps continu\" width=\"545\" height=\"78\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage62.png 545w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage62-300x43.png 300w\" sizes=\"(max-width: 545px) 100vw, 545px\" \/><\/figure>\n<\/div>\n\n<p>Consid\u00e9rons trois points cons\u00e9cutifs dans le temps o\u00f9 il y a eu un changement d\u2019\u00e9tats r dans le pass\u00e9, s \u00e0 l\u2019instant pr\u00e9sent et s + t dans le futur. X(s) = i et X(r) = l . Un processus stochastique en temps continu \u00e0 la propri\u00e9t\u00e9 de Markov si :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6649\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage63.png\" alt=\"Chaines de Markov en temps continu\" width=\"779\" height=\"91\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage63.png 779w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage63-300x35.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/langage63-768x90.png 768w\" sizes=\"(max-width: 779px) 100vw, 779px\" \/><\/figure>\n<\/div>\n\n<p>Les probabilit\u00e9s de transition sont stationnaires puisqu&rsquo;elles sont ind\u00e9pendantes de s. On note p<sub>ij<\/sub>(t) = P(X(t) = j , X(0) = i ) la fonction de probabilit\u00e9 de transition en temps continu.<\/p>\n\n<p>Notons T<sub>i<\/sub> une variable al\u00e9atoire d\u00e9notant le temps pass\u00e9 dans l\u2019\u00e9tat i avant de se d\u00e9placer vers un autre \u00e9tat, i\u2208{0, &#8230;,M}. Supposons que le processus entre dans l\u2019\u00e9tat i au temps t&rsquo; = s. Alors pour une dur\u00e9e t &gt; 0, T<sub>i<\/sub> &gt; t \u21d4 X(t&rsquo; = i ), \u2200t&rsquo;\u2208[s, s + t].<\/p>\n\n<p>La propri\u00e9t\u00e9 de stationnarit\u00e9 des probabilit\u00e9s de transition entra\u00eene : P(T<sub>i<\/sub> &gt; s + t, T<sub>i<\/sub> &gt; s) = P(T<sub>i<\/sub> &gt; t). La distribution du temps restant d\u2019ici la prochaine sortie de i par le processus est la m\u00eame quel que soit le temps d\u00e9j\u00e0 pass\u00e9 dans l\u2019\u00e9tat i . La variable T<sub>i<\/sub> est sans m\u00e9moire. La seule distribution de variable al\u00e9atoire continue ayant cette propri\u00e9t\u00e9 est la distribution exponentielle.<\/p>\n\n<p>La distribution exponentielle T<sub>i<\/sub> poss\u00e8de un seul param\u00e8tre\u00a0q<sub>i<\/sub> et sa moyenne (esp\u00e9rance math\u00e9matique) est R[T<sub>i<\/sub> ] = 1\/q<sub>i<\/sub>. Ce r\u00e9sultat nous permet de d\u00e9crire une <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">cha\u00eene de Markov<\/a> en temps continu d\u2019une fa\u00e7on \u00e9quivalente comme suit :<\/p>\n\n<ul class=\"wp-block-list\">\n<li>La variable al\u00e9atoire\u00a0T<sub>i<\/sub> a une distribution exponentielle de param\u00e8tre \u03bb<\/li>\n<li>quand le processus quitte l&rsquo;\u00e9tat i, il passe \u00e0 l&rsquo;\u00e9tat j avec\u00a0 une probabilit\u00e9 p<sub>ij\u00a0<\/sub>telle que (similaire \u00e0 une cha\u00eene de Markov en temps discret) :\u00a0<img decoding=\"async\" class=\"alignnone wp-image-6653\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba361.png\" alt=\"Chaines de Markov en temps continu\" width=\"362\" height=\"86\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba361.png 459w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba361-300x71.png 300w\" sizes=\"(max-width: 362px) 100vw, 362px\" \/><\/li>\n<li>le prochain \u00e9tat visit\u00e9 apr\u00e8s i est ind\u00e9pendant du temps pass\u00e9 dans l&rsquo;\u00e9tat i.<\/li>\n<li>La cha\u00eene de Markov en temps continu poss\u00e8de les m\u00eames propri\u00e9t\u00e9s de Classe et d\u2019Irr\u00e9ductibilit\u00e9 que les cha\u00eenes en temps discret.<\/li>\n<\/ul>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6663\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba41.png\" alt=\"Chaines de Markov en temps continu\" width=\"317\" height=\"214\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba41.png 317w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba41-300x203.png 300w\" sizes=\"(max-width: 317px) 100vw, 317px\" \/><\/figure>\n<\/div>\n\n<p>Voici quelques propri\u00e9t\u00e9s de la loi exponentielle :<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6664\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba42.png\" alt=\"Chaines de Markov en temps continu\" width=\"435\" height=\"292\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba42.png 435w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba42-300x201.png 300w\" sizes=\"(max-width: 435px) 100vw, 435px\" \/><\/figure>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Modele-de-duree\"><\/span>Mod\u00e8le de dur\u00e9e<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Ainsi, si on consid\u00e8re\u00a0\u03bc<sub>i<\/sub> le param\u00e8tre de la variable al\u00e9atoire exponentielle associ\u00e9 \u00e0 l&rsquo;\u00e9tat i, nous pouvons repr\u00e9senter la cha\u00eene de Markov \u00e0 temps continue comme suit :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6665\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba43.png\" alt=\"Chaines de Markov en temps continu\" width=\"435\" height=\"214\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba43.png 435w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba43-300x148.png 300w\" sizes=\"(max-width: 435px) 100vw, 435px\" \/><\/figure>\n<\/div>\n\n<p>Nous voyons bien la cha\u00eene de Markov \u00e0 temps discret incluse, d&rsquo;o\u00f9 la possibilit\u00e9 d&rsquo;effectuer une \u00e9tude du mod\u00e8le discret. A noter qu&rsquo;il n&rsquo;existe pas de notion de p\u00e9riodicit\u00e9 dans ce cadre.<\/p>\n\n<p>Si l&rsquo;on consid\u00e8re qu&rsquo;on se d\u00e9place de l&rsquo;\u00e9tat i \u00e0 l&rsquo;\u00e9tat j apr\u00e8s un temps\u00a0T<sub>ij<\/sub> et que l&rsquo;on consid\u00e8re ce temps comme une variable al\u00e9atoire exponentielle de taux \u03bc<sub>ij<\/sub>, alors il est possible d&rsquo;\u00e9crire la cha\u00eene de Markov en temps continu sous un mod\u00e8le de dur\u00e9e :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6666\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba44.png\" alt=\"Chaines de Markov en temps continu\" width=\"569\" height=\"249\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba44.png 569w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba44-300x131.png 300w\" sizes=\"(max-width: 569px) 100vw, 569px\" \/><\/figure>\n<\/div>\n\n<p>Attention, il existe un grand changement entre le c\u00f4t\u00e9 stochastique du mouvement d&rsquo;un \u00e9tat \u00e0 un autre et le c\u00f4t\u00e9 continue dans le temps. Il est important de comprendre que la matrice de transition d&rsquo;une cha\u00eene de Markov en temps continue est toujours un mod\u00e8le de dur\u00e9e.<\/p>\n\n<p>La matrice de transition d&rsquo;un mod\u00e8le de dur\u00e9e \u00e0 les propri\u00e9t\u00e9s suivantes :<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6667\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba45.png\" alt=\"Chaines de Markov en temps continu\" width=\"219\" height=\"83\" title=\"\"><\/figure>\n\n<p>Cette matrice est appel\u00e9 le g\u00e9n\u00e9rateur infinit\u00e9simale.<\/p>\n\n<p>Ainsi, \u00e0 partir du <a href=\"https:\/\/complex-systems-ai.com\/en\/graph-theory-2\/\">graphe<\/a> de Markov discret suivant (la loi exponentielle est la m\u00eame dans les trois \u00e9tats) :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6655\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba37.png\" alt=\"Chaines de Markov en temps continu\" width=\"221\" height=\"197\" title=\"\"><\/figure>\n<\/div>\n\n<p>Il est possible d&rsquo;obtenir le mod\u00e8le de temps suivant :<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6656\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba38.png\" alt=\"Chaines de Markov en temps continu\" width=\"246\" height=\"82\" title=\"\"><\/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>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Markov process Home page Wiki Difficulty Medium 50% Continuous-time Markov chains In the case of discrete time, we observe the states\u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":5007,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-6644","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6644","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=6644"}],"version-history":[{"count":11,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6644\/revisions"}],"predecessor-version":[{"id":18670,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6644\/revisions\/18670"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=6644"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}