{"id":21033,"date":"2024-02-24T21:00:41","date_gmt":"2024-02-24T20:00:41","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=21033"},"modified":"2024-02-24T21:08:17","modified_gmt":"2024-02-24T20:08:17","slug":"dickey-fuller","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/dickey-fuller\/","title":{"rendered":"Dickey-Fuller test"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"21033\" class=\"elementor elementor-21033\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8d97884 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8d97884\" 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 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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-dd707af elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"dd707af\" 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-44fd586\" data-id=\"44fd586\" 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-f14add1 elementor-widget elementor-widget-heading\" data-id=\"f14add1\" 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\/dickey-fuller\/#Test-de-Dickey-Fuller\" >Dickey-Fuller test<\/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\/dickey-fuller\/#Bases\" >Bases<\/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\/dickey-fuller\/#Exemple\" >Example<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Test-de-Dickey-Fuller\"><\/span>Dickey-Fuller test<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-6776b2a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6776b2a\" 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-271b970\" data-id=\"271b970\" 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-6b089cb elementor-widget elementor-widget-text-editor\" data-id=\"6b089cb\" 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 Dickey-Fuller test is a way to determine if the process (time series) has a unit root.<\/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=\"Dickey-Fuller\" 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-8a94f31 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8a94f31\" 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-2db382a\" data-id=\"2db382a\" 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-158abbd elementor-widget elementor-widget-heading\" data-id=\"158abbd\" 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=\"Bases\"><\/span>Bases<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-c585c5b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c585c5b\" 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-ed17b44\" data-id=\"ed17b44\" 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-c036da7 elementor-widget elementor-widget-text-editor\" data-id=\"c036da7\" 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>We consider the stochastic process of form<\/p><p><img decoding=\"async\" class=\"aligncenter size-full wp-image-22653\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image268z.png\" alt=\"image268z\" width=\"100\" height=\"21\" title=\"\"><\/p><p>where |\u03c6| \u2264 1 and \u03b5i is white noise. If |\u03c6| = 1, we have what we call a unit root. In particular, if \u03c6 = 1, we have a random walk (without drift), which is not stationary. In fact, if |\u03c6| = 1, the process is not stationary, whereas if |\u03c6| &lt; 1, the process is stationary. We will not consider the case where |\u03c6| &gt; 1 more since in this case the process is said to be explosive and increases over time.<\/p><p>This process is a first-order autoregressive process, AR(1), which we study in more detail in Autoregressive Processes. We will also see why such processes without a unit root are stationary and why the term &quot;root&quot; is used.<\/p><p>The Dickey-Fuller test is a way to determine whether the above process has a unit root. The approach used is quite simple. First calculate the first difference, i.e.<\/p><p><img decoding=\"async\" class=\"aligncenter size-full wp-image-22654\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image269z.png\" alt=\"image269z\" width=\"190\" height=\"21\" title=\"\"><\/p><p>that&#039;s to say<\/p><p><a href=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image270z.png\" rel=\"attachment wp-att-22656 nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22656\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image270z.png\" alt=\"image270z\" width=\"185\" height=\"21\" title=\"\"><\/a><\/p><p>If we use the delta operator, defined by \u0394yi = yi \u2013 yi-1 and define \u03b2 = \u03c6 \u2013 1, then the equation becomes the equation of <a href=\"https:\/\/complex-systems-ai.com\/en\/correlation-and-regressions\/\">regression<\/a> linear<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22655\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image271z.png\" alt=\"image271z\" width=\"108\" height=\"21\" title=\"\"><\/p><p>where \u03b2 \u2264 0 and therefore the test for \u03c6 is transformed into a test according to which the slope parameter \u03b2 = 0. Thus, we have a one-sided test (since \u03b2 cannot be positive) where<\/p><p>H0: \u03b2 = 0 (equivalent to \u03c6 = 1)<\/p><p>H1: \u03b2 &lt; 0 (equivalent to \u03c6 &lt; 1)<\/p><p>Under the alternative hypothesis, if b is the ordinary least squares (OLS) estimate of \u03b2, and therefore \u03c6-bar = 1 + b is the ordinary least squares (OLS) estimate of \u03c6, then for sufficiently large n<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22657\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image272z.png\" sizes=\"(max-width: 156px) 100vw, 156px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image272z.png 156w, https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image272z-150x24.png 150w\" alt=\"image272z\" width=\"156\" height=\"24\" title=\"\"><\/p><p>or<\/p><p><a href=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image273z.png\" rel=\"attachment wp-att-22658 nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22658\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image273z.png\" alt=\"image273z\" width=\"102\" height=\"25\" title=\"\"><\/a><\/p><p>We can use the usual linear regression approach, except that when the null hypothesis is true, the t coefficient does not follow a normal distribution and therefore we cannot use the usual t test. Instead, this coefficient follows a tau distribution, and so our test is whether the tau statistic \u03c4 (which is equivalent to the usual t statistic) is less than \u03c4crit based on a table of critical tau statistic values shown in the Dickey-Fuller table. .<\/p><p>If the calculated tau value is less than the critical value in the critical value table, then we have a significant result; otherwise, we accept the null hypothesis that there is a unit root and that the time series is not stationary.<\/p><p>There are the following three versions of the Dickey-Fuller test:<\/p><p>Type 0 No constant, no trend \u0394yi = \u03b21 yi-1 + \u03b5i<br \/>Type 1 Constant, no trend \u0394yi = \u03b20 + \u03b21 yi-1 + \u03b5i<br \/>Type 2 Constant and trend \u0394yi = \u03b20 + \u03b21 yi-1 + \u03b22 i+ \u03b5i<\/p><p>Each version of the test uses a different set of critical values, as shown in the Dickey-Fuller table. It is important to select the correct version of the test for the time series being analyzed. Note that the type 2 test assumes that there is a constant term (which can be significantly equal to zero).<\/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-ea03074 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ea03074\" 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-4f01ec5\" data-id=\"4f01ec5\" 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-e8eb5e1 elementor-widget elementor-widget-heading\" data-id=\"e8eb5e1\" 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=\"Exemple\"><\/span>Example<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-d44c0b5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d44c0b5\" 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-208480c\" data-id=\"208480c\" 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-39420f8 elementor-widget elementor-widget-text-editor\" data-id=\"39420f8\" 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>A small player&#039;s net daily winnings are listed in column B of Figure 1. Use the Dickey-Fuller test to determine whether the time series is stationary.<\/p><p>We start by assuming that the correct model is type 1, i.e. constant but without trend.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21038 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller1.png\" alt=\"dickey-fuller\" width=\"740\" height=\"497\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller1.png 740w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller1-300x201.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller1-18x12.png 18w\" sizes=\"(max-width: 740px) 100vw, 740px\" \/><\/p><p>Since we use the regression model<\/p><p><a href=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image274z.png\" rel=\"attachment wp-att-22659 nofollow noopener\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22659\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2016\/04\/image274z.png\" alt=\"image274z\" width=\"147\" height=\"21\" title=\"\"><\/a><\/p><p>(constant, no trend), we use the tool<a href=\"https:\/\/complex-systems-ai.com\/en\/data-analysis\/\">data analysis<\/a> Real Statistics linear regression using range B4:B27 and data range , highlighting the range D5:D28 and pressing Ctrl-D.<\/p><p>The result of the regression analysis is shown on the right side of the figure. In particular, we see that the t statistic (cell I20) for the coefficient \u03b21 is -1.91613. This is the tau statistic. We now look in the Dickey-Fuller table and see that the critical value of tau for a Type 1 test is -2.986 when n = 25 and \u03b1 = 0.05. Since \u03c4crit = -2.986 &lt; \u2013 1.91613 = \u03c4, we cannot reject the null hypothesis that the time series is not stationary.<\/p><p>Note that the coefficient \u03b21 (cell G20) is negative as expected. If, on the contrary, the coefficient were positive, then we would know that this type of Dickey-Fuller test is inappropriate since \u03b21 = \u03c6 \u2013 1 \u2264 0.<\/p><p>We now show in the figure a plot of the time series values from the previous figure.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21039 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller2.png\" alt=\"dickey-fuller\" width=\"538\" height=\"295\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller2.png 538w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller2-300x164.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller2-18x10.png 18w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/><\/p><p>We see an apparent downward trend towards the end of the 25-day period and so it is not surprising that the time series is not stationary. In fact, this leads us to choose the Dickey-Fuller type 2 test (with constant and trend). The result of this test is shown in the figure.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21040 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller3.png\" alt=\"dickey-fuller\" width=\"529\" height=\"413\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller3.png 529w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller3-300x234.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/dickey-fuller3-15x12.png 15w\" sizes=\"(max-width: 529px) 100vw, 529px\" \/><\/p><p>Since we use the regression model<\/p><p>\u0394yi = \u03b20 + \u03b21i + \u03b22yi-1 + \u03b5i<\/p><p>this time we use A4:B27 from Figure 1 as the X data range and D5:D28 as the Y data range. We see in Figure 3 that the t statistic (cell I21) for the coefficient \u03b22 is -2, 91345. We now look in the Dickey-Fuller table and see that the critical value of tau is -3.60269 for a Type 2 test when n = 25 and \u03b1 = 0.05. Since \u03c4crit = -3.60269 &lt; -2.91345 = \u03c4, we cannot reject the null hypothesis that the time series is not stationary.<\/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 Home Page Wiki Dickey-Fuller Test The Dickey-Fuller test is a way to determine whether the process (time series) has a unit root. \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-21033","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/21033","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=21033"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/21033\/revisions"}],"predecessor-version":[{"id":21043,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/21033\/revisions\/21043"}],"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=21033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}