SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Autoregressive model</title>

</head>

Autoregressive model

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of <a href="Stochastic_process" title="Stochastic process">random process</a>; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a <a href="Stochastic" title="Stochastic">stochastic</a> term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation) which should not be confused with a <a href="Differential_equation" title="Differential equation">differential equation</a>. Together with the <a href="Moving-average_model" title="Moving-average model">moving-average (MA) model</a>, it is a special case and key component of the more general <a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">autoregressive–moving-average</a> (ARMA) and <a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">autoregressive integrated moving average</a> (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable. Another important extension is the time-varying autoregressive (TVAR) model, where the autoregressive coefficients are allowed to change over time to model evolving or non-stationary processes. TVAR models are widely applied in cases where the underlying dynamics of the system are not constant, such as in sensors time series modelling<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>, finance<a href="#cite_note-3">[3]</a>, climate science<a href="#cite_note-4">[4]</a>, economics<a href="#cite_note-5">[5]</a>, signal processing<a href="#cite_note-6">[6]</a> and telecommunications<a href="#cite_note-7">[7]</a>, radar systems<a href="#cite_note-8">[8]</a>, and biological signals<a href="#cite_note-9">[9]</a>.

Unlike the moving-average (MA) model, the autoregressive model is not always stationary; non-stationarity can arise either due to the presence of a unit root or due to time-varying model parameters, as in time-varying autoregressive (TVAR) models.

<a href="Large_language_model" title="Large language model">Large language models</a> are called autoregressive, but they are not a classical autoregressive model in this sense because they are not linear.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AR(p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AR(p)}</annotation>

</semantics>

</math><img src="./656b65718fc07fb84c0d8186c6e43ce76723427b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.486ex; height:2.843ex;" alt="{\displaystyle AR(p)}" loading="lazy"> indicates an autoregressive model of order p. The AR(p) model is defined as

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}}</annotation>

</semantics>

</math><img src="./f0bec4e5efbcc95f1fb648c8c03686c4b3843b04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.257ex; height:7.009ex;" alt="{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1},\ldots ,\varphi _{p}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1},\ldots ,\varphi _{p}}</annotation>

</semantics>

</math><img src="./d4fdc81c8ea00ebabdeee779385a8e22e3c2b385.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.332ex; height:2.343ex;" alt="{\displaystyle \varphi _{1},\ldots ,\varphi _{p}}" loading="lazy"> are the parameters of the model, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> is <a href="White_noise" title="White noise">white noise</a>.<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a> This can be equivalently written using the <a href="Backshift_operator" class="mw-redirect" title="Backshift operator">backshift operator</a> B as

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}B^{i}X_{t}+\varepsilon _{t}}</annotation>

</semantics>

</math><img src="./f1df0884c60da1ead623e7b11b99fb0945441f5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.974ex; height:7.009ex;" alt="{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}B^{i}X_{t}+\varepsilon _{t}}" loading="lazy"></dd></dl>

so that, moving the summation term to the left side and using <a href="Polynomial_notation" class="mw-redirect" title="Polynomial notation">polynomial notation</a>, we have

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi [B]X_{t}=\varepsilon _{t}}</annotation>

</semantics>

</math><img src="./427fcdc777a1ef12ec7498f413bfbcb6d0932af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.201ex; height:2.843ex;" alt="{\displaystyle \phi [B]X_{t}=\varepsilon _{t}}" loading="lazy"></dd></dl>

An autoregressive model can thus be viewed as the output of an all-<a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> <a href="Infinite_impulse_response" title="Infinite impulse response">infinite impulse response</a> filter whose input is white noise.

Some parameter constraints are necessary for the model to remain <a href="Stationary_process#Weak_or_wide-sense_stationarity" title="Stationary process">weak-sense stationary</a>. For example, processes in the AR(1) model with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\varphi _{1}|\geq 1}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\varphi _{1}|\geq 1}</annotation>

</semantics>

</math><img src="./5feb9de2fe8907d308f9eaaab3ff940d54b8fbf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.129ex; height:2.843ex;" alt="{\displaystyle |\varphi _{1}|\geq 1}" loading="lazy"> are not stationary. More generally, for an AR(p) model to be weak-sense stationary, the roots of the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (z):=\textstyle 1-\sum _{i=1}^{p}\varphi _{i}z^{i}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (z):=\textstyle 1-\sum _{i=1}^{p}\varphi _{i}z^{i}}</annotation>

</semantics>

</math><img src="./28c983acd51d75155b677b0237020037000c501c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.275ex; height:3.176ex;" alt="{\displaystyle \Phi (z):=\textstyle 1-\sum _{i=1}^{p}\varphi _{i}z^{i}}" loading="lazy"> must lie outside the <a href="Unit_circle" title="Unit circle">unit circle</a>, i.e., each (complex) root <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation>

</semantics>

</math><img src="./5c6e920bac39ad09fff4efef16254595091a1025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}" loading="lazy"> must satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z_{i}|>1}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |z_{i}|>1}</annotation>

</semantics>

</math><img src="./ff75c25f97e572744f6a3c93410506ce35cd1d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:2.843ex;" alt="{\displaystyle |z_{i}|>1}" loading="lazy"> (see pages 89,92 <a href="#cite_note-12">[12]</a>).

<div class="mw-heading mw-heading2"><h2 id="Intertemporal_effect_of_shocks">Intertemporal effect of shocks</h2></div>

In an AR process, a one-time shock affects values of the evolving variable infinitely far into the future. For example, consider the AR(1) model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=\varphi _{1}X_{t-1}+\varepsilon _{t}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\varphi _{1}X_{t-1}+\varepsilon _{t}}</annotation>

</semantics>

</math><img src="./62e702d3e4a52abad2784f06fb45d2c530702773.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.024ex; height:2.676ex;" alt="{\displaystyle X_{t}=\varphi _{1}X_{t-1}+\varepsilon _{t}}" loading="lazy">. A non-zero value for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> at say time t=1 affects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>

</semantics>

</math><img src="./f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy"> by the amount <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{1}}</annotation>

</semantics>

</math><img src="./9e900f9bee793f99d10877ef108da074cbca60ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{1}}" loading="lazy">. Then by the AR equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>

</semantics>

</math><img src="./2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>

</semantics>

</math><img src="./f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy">, this affects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>

</semantics>

</math><img src="./2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"> by the amount <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}\varepsilon _{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}\varepsilon _{1}}</annotation>

</semantics>

</math><img src="./e6771fe396d6eec7394996b530cb986db71feea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.712ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}\varepsilon _{1}}" loading="lazy">. Then by the AR equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation>

</semantics>

</math><img src="./14f17eb6a51e16ea92736c904a92d8a78e73a598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}" loading="lazy"> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>

</semantics>

</math><img src="./2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy">, this affects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation>

</semantics>

</math><img src="./14f17eb6a51e16ea92736c904a92d8a78e73a598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}" loading="lazy"> by the amount <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}^{2}\varepsilon _{1}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}^{2}\varepsilon _{1}}</annotation>

</semantics>

</math><img src="./e47bea8b4509e8572bfbd52a6989c11ab084d724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.712ex; height:3.176ex;" alt="{\displaystyle \varphi _{1}^{2}\varepsilon _{1}}" loading="lazy">. Continuing this process shows that the effect of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{1}}</annotation>

</semantics>

</math><img src="./9e900f9bee793f99d10877ef108da074cbca60ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{1}}" loading="lazy"> never ends, although if the process is <a href="Stationary_process" title="Stationary process">stationary</a> then the effect diminishes toward zero in the limit.

Because each shock affects X values infinitely far into the future from when they occur, any given value Xt is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (B)X_{t}=\varepsilon _{t}\,}</annotation>

</semantics>

</math><img src="./8c07c3c7172406b06bb1b30313744788a4b99481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.104ex; height:2.843ex;" alt="{\displaystyle \phi (B)X_{t}=\varepsilon _{t}\,}" loading="lazy"></dd></dl>

(where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}={\frac {1}{\phi (B)}}\varepsilon _{t}\,.}</annotation>

</semantics>

</math><img src="./788086c2d58f2233bb3ae2d8e5f7eccc17628173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.587ex; height:6.009ex;" alt="{\displaystyle X_{t}={\frac {1}{\phi (B)}}\varepsilon _{t}\,.}" loading="lazy"></dd></dl>

When the <a href="Polynomial_long_division" title="Polynomial long division">polynomial division</a> on the right side is carried out, the polynomial in the backshift operator applied to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> has an infinite order—that is, an infinite number of lagged values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> appear on the right side of the equation.

<div class="mw-heading mw-heading2"><h2 id="Characteristic_polynomial">Characteristic polynomial</h2></div>

The <a href="Autocorrelation_function" class="mw-redirect" title="Autocorrelation function">autocorrelation function</a> of an AR(p) process can be expressed as

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (\tau )=\sum _{k=1}^{p}a_{k}y_{k}^{-|\tau |},}</annotation>

</semantics>

</math><img src="./979241068653d54f8ce728e1d16e45df688a69fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.455ex; height:7.009ex;" alt="{\displaystyle \rho (\tau )=\sum _{k=1}^{p}a_{k}y_{k}^{-|\tau |},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation>

</semantics>

</math><img src="./4b2ab0248723a410cc2c67ce06ad5c043dcbb933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" loading="lazy"> are the roots of the polynomial

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (B)=1-\sum _{k=1}^{p}\varphi _{k}B^{k}}</annotation>

</semantics>

</math><img src="./dfea5de760da3cb674b9aa8ea3509d86ab7e3bd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.264ex; height:7.009ex;" alt="{\displaystyle \phi (B)=1-\sum _{k=1}^{p}\varphi _{k}B^{k}}" loading="lazy"></dd></dl>

where B is the <a href="Backshift_operator" class="mw-redirect" title="Backshift operator">backshift operator</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (\cdot )}</annotation>

</semantics>

</math><img src="./7ead8078ac455302b763cfc6a02b75282077140b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.842ex; height:2.843ex;" alt="{\displaystyle \phi (\cdot )}" loading="lazy"> is the function defining the autoregression, and where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{k}}</annotation>

</semantics>

</math><img src="./dad86b92f0c76d343e12c4a90b368834329bd5d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.609ex; height:2.176ex;" alt="{\displaystyle \varphi _{k}}" loading="lazy"> are the coefficients in the autoregression. The formula is valid only if all the roots have multiplicity 1.

The autocorrelation function of an AR(p) process is a sum of decaying exponentials.

<ul><li>Each real root contributes a component to the autocorrelation function that decays exponentially.</li>

<li>Similarly, each pair of complex conjugate roots contributes an exponentially damped oscillation.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Graphs_of_AR(p)_processes">Graphs of AR(p) processes</h2></div>

The simplest AR process is AR(0), which has no dependence between the terms. Only the error/innovation/noise term contributes to the output of the process, so in the figure, AR(0) corresponds to white noise.

For an AR(1) process with a positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy">, only the previous term in the process and the noise term contribute to the output. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is close to 0, then the process still looks like white noise, but as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> approaches 1, the output gets a larger contribution from the previous term relative to the noise. This results in a "smoothing" or integration of the output, similar to a <a href="Low_pass_filter" class="mw-redirect" title="Low pass filter">low pass filter</a>.

For an AR(2) process, the previous two terms and the noise term contribute to the output. If both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}}</annotation>

</semantics>

</math><img src="./d7daf493c8f6ef669c04c7b9715532fc35d12d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{2}}</annotation>

</semantics>

</math><img src="./c08631714273b6c8edaa9573ef3d8c548314a930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{2}}" loading="lazy"> are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}}</annotation>

</semantics>

</math><img src="./d7daf493c8f6ef669c04c7b9715532fc35d12d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}}" loading="lazy"> is positive while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{2}}</annotation>

</semantics>

</math><img src="./c08631714273b6c8edaa9573ef3d8c548314a930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{2}}" loading="lazy"> is negative, then the process favors changes in sign between terms of the process. The output oscillates. This can be linked to edge detection or detection of change in direction.

<div class="mw-heading mw-heading2"><h2 id="Example:_An_AR(1)_process">Example: An AR(1) process</h2></div>

An AR(1) process is given by:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=\varphi X_{t-1}+\varepsilon _{t}\,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\varphi X_{t-1}+\varepsilon _{t}\,}</annotation>

</semantics>

</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> is a white noise process with zero mean and constant variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}}</annotation>

</semantics>

</math><img src="./eec379b86e73255492d3266c76f6e17acfdfabd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.843ex;" alt="{\displaystyle \sigma _{\varepsilon }^{2}}" loading="lazy">.

(Note: The subscript on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}}</annotation>

</semantics>

</math><img src="./d7daf493c8f6ef669c04c7b9715532fc35d12d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}}" loading="lazy"> has been dropped.) The process is <a href="Stationary_process#Weak_or_wide-sense_stationarity" title="Stationary process">weak-sense stationary</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\varphi |<1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\varphi |<1}</annotation>

</semantics>

</math><img src="./7ac88cf3e11f7577b756abedbee667cce5069563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.075ex; height:2.843ex;" alt="{\displaystyle |\varphi |<1}" loading="lazy"> since it is obtained as the output of a stable filter whose input is white noise. (If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi =1}</annotation>

</semantics>

</math><img src="./a4d2b07a5e6f6058e04da966ccdc8506fe8ffeb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi =1}" loading="lazy"> then the variance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>

</semantics>

</math><img src="./82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"> depends on time lag t, so that the variance of the series diverges to infinity as t goes to infinity, and is therefore not weak-sense stationary.) Assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\varphi |<1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\varphi |<1}</annotation>

</semantics>

</math><img src="./7ac88cf3e11f7577b756abedbee667cce5069563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.075ex; height:2.843ex;" alt="{\displaystyle |\varphi |<1}" loading="lazy">, the mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X_{t})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X_{t})}</annotation>

</semantics>

</math><img src="./be42d3e2b3cfe283a64cf222c06972633e8ded66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.142ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (X_{t})}" loading="lazy"> is identical for all values of t by definition of weak sense stationarity. If the mean is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">, it follows from<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X_{t})=\varphi \operatorname {E} (X_{t-1})+\operatorname {E} (\varepsilon _{t}),}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X_{t})=\varphi \operatorname {E} (X_{t-1})+\operatorname {E} (\varepsilon _{t}),}</annotation>

</semantics>

</math>that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\varphi \mu +0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\varphi \mu +0,}</annotation>

</semantics>

</math>and hence

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =0.}</annotation>

</semantics>

</math><img src="./6c1fbd9b60e51f99639d432b9b86c1f1f486e1b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.309ex; height:2.676ex;" alt="{\displaystyle \mu =0.}" loading="lazy"></dd></dl>

The <a href="Variance" title="Variance">variance</a> is

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textrm {var}}(X_{t})=\operatorname {E} (X_{t}^{2})-\mu ^{2}={\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}},}</annotation>

</semantics>

</math><img src="./4f1742b6092c63b09414e86c9e208b2da49c8dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.858ex; height:6.343ex;" alt="{\displaystyle {\textrm {var}}(X_{t})=\operatorname {E} (X_{t}^{2})-\mu ^{2}={\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }}</annotation>

</semantics>

</math><img src="./04852f481494a445c9f5b9082df1ead002c098a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle \sigma _{\varepsilon }}" loading="lazy"> is the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy">. This can be shown by noting that

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textrm {var}}(X_{t})=\varphi ^{2}{\textrm {var}}(X_{t-1})+\sigma _{\varepsilon }^{2},}</annotation>

</semantics>

</math><img src="./c5594a5bf1cd9c13fb15eb23763328c684b41778.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.367ex; height:3.176ex;" alt="{\displaystyle {\textrm {var}}(X_{t})=\varphi ^{2}{\textrm {var}}(X_{t-1})+\sigma _{\varepsilon }^{2},}" loading="lazy"></dd></dl>

and then by noticing that the quantity above is a stable fixed point of this relation.

The <a href="Autocovariance" title="Autocovariance">autocovariance</a> is given by

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}=\operatorname {E} (X_{t+n}X_{t})-\mu ^{2}={\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,\,\varphi ^{|n|}.}</annotation>

</semantics>

</math><img src="./8437c88291c59fdbb8055f7630becfb993b1fb17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.121ex; height:6.343ex;" alt="{\displaystyle B_{n}=\operatorname {E} (X_{t+n}X_{t})-\mu ^{2}={\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,\,\varphi ^{|n|}.}" loading="lazy"></dd></dl>

It can be seen that the autocovariance function decays with a decay time (also called <a href="Time_constant" title="Time constant">time constant</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =1/(1-\varphi )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =1/(1-\varphi )}</annotation>

</semantics>

</math><img src="./85fb1ec990c3aae74a04441e046c7689f9180342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.958ex; height:2.843ex;" alt="{\displaystyle \tau =1/(1-\varphi )}" loading="lazy">.<a href="#cite_note-13">[13]</a>

The <a href="Spectral_density" title="Spectral density">spectral density</a> function is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform:

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (\omega )={\frac {1}{\sqrt {2\pi }}}\,\sum _{n=-\infty }^{\infty }B_{n}e^{-i\omega n}={\frac {1}{\sqrt {2\pi }}}\,\left({\frac {\sigma _{\varepsilon }^{2}}{1+\varphi ^{2}-2\varphi \cos(\omega )}}\right).}</annotation>

</semantics>

</math><img src="./7fd707a70fc78fabba4b103d6099d2a75f109d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.082ex; height:6.843ex;" alt="{\displaystyle \Phi (\omega )={\frac {1}{\sqrt {2\pi }}}\,\sum _{n=-\infty }^{\infty }B_{n}e^{-i\omega n}={\frac {1}{\sqrt {2\pi }}}\,\left({\frac {\sigma _{\varepsilon }^{2}}{1+\varphi ^{2}-2\varphi \cos(\omega )}}\right).}" loading="lazy"></dd></dl>

This expression is periodic due to the discrete nature of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation>

</semantics>

</math><img src="./ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}" loading="lazy">, which is manifested as the cosine term in the denominator. If we assume that the sampling time (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t=1}</annotation>

</semantics>

</math><img src="./26978f1c5f3ac3161ce7635cbde2a207c7519aab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.036ex; height:2.176ex;" alt="{\displaystyle \Delta t=1}" loading="lazy">) is much smaller than the decay time (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">), then we can use a continuum approximation to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy">:

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(t)\approx {\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,\,\varphi ^{|t|}}</annotation>

</semantics>

</math><img src="./23ab1061089249aaff9c9522ab7ba1da4296f38b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.96ex; height:6.343ex;" alt="{\displaystyle B(t)\approx {\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,\,\varphi ^{|t|}}" loading="lazy"></dd></dl>

which yields a <a href="Cauchy_distribution" title="Cauchy distribution">Lorentzian profile</a> for the spectral density:

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (\omega )={\frac {1}{\sqrt {2\pi }}}\,{\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,{\frac {\gamma }{\pi (\gamma ^{2}+\omega ^{2})}}}</annotation>

</semantics>

</math><img src="./09c4d8d2e045f649f287fc359418e5e65e69f042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.138ex; height:6.676ex;" alt="{\displaystyle \Phi (\omega )={\frac {1}{\sqrt {2\pi }}}\,{\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,{\frac {\gamma }{\pi (\gamma ^{2}+\omega ^{2})}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1/\tau }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =1/\tau }</annotation>

</semantics>

</math><img src="./64537ae1c00fe728d4fc64b69a72daec33b8e99c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.888ex; height:2.843ex;" alt="{\displaystyle \gamma =1/\tau }" loading="lazy"> is the angular frequency associated with the decay time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">.

An alternative expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>

</semantics>

</math><img src="./82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"> can be derived by first substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi X_{t-2}+\varepsilon _{t-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi X_{t-2}+\varepsilon _{t-1}}</annotation>

</semantics>

</math><img src="./0817a0d3b28f992b815dd0f145ebafded3a438c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.221ex; height:2.676ex;" alt="{\displaystyle \varphi X_{t-2}+\varepsilon _{t-1}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t-1}}</annotation>

</semantics>

</math><img src="./7cdeab0b9b98bc0b52250537bc0234d7d3de6b60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.851ex; height:2.509ex;" alt="{\displaystyle X_{t-1}}" loading="lazy"> in the defining equation. Continuing this process N times yields

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\varphi ^{N}X_{t-N}+\sum _{k=0}^{N-1}\varphi ^{k}\varepsilon _{t-k}.}</annotation>

</semantics>

</math><img src="./07c4b3b8a831a9ef5c83ea225c43d179e00d5fce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.636ex; height:7.509ex;" alt="{\displaystyle X_{t}=\varphi ^{N}X_{t-N}+\sum _{k=0}^{N-1}\varphi ^{k}\varepsilon _{t-k}.}" loading="lazy"></dd></dl>

For N approaching infinity, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{N}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{N}}</annotation>

</semantics>

</math><img src="./a3c7d06db11de79871f23fce1a87a85a61e19f12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.212ex; height:3.176ex;" alt="{\displaystyle \varphi ^{N}}" loading="lazy"> will approach zero and:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{k=0}^{\infty }\varphi ^{k}\varepsilon _{t-k}.}</annotation>

</semantics>

</math><img src="./5cfa9e0b02bb065b02cf44a0f01480ac9219e358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.891ex; height:7.009ex;" alt="{\displaystyle X_{t}=\sum _{k=0}^{\infty }\varphi ^{k}\varepsilon _{t-k}.}" loading="lazy"></dd></dl>

It is seen that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>

</semantics>

</math><img src="./82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"> is white noise convolved with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{k}}</annotation>

</semantics>

</math><img src="./718a443a1669e482cf76903e77f49dc3f9a286d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.609ex; height:3.176ex;" alt="{\displaystyle \varphi ^{k}}" loading="lazy"> kernel plus the constant mean. If the white noise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> is a <a href="Gaussian_process" title="Gaussian process">Gaussian process</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>

</semantics>

</math><img src="./82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"> is also a Gaussian process. In other cases, the <a href="Central_limit_theorem" title="Central limit theorem">central limit theorem</a> indicates that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>

</semantics>

</math><img src="./82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"> will be approximately normally distributed when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is close to one.

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}=0}</annotation>

</semantics>

</math><img src="./274d9a088f12eef9b2bc0b4c6c797a637dd69e23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.17ex; height:2.509ex;" alt="{\displaystyle \varepsilon _{t}=0}" loading="lazy">, the process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=\varphi X_{t-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\varphi X_{t-1}}</annotation>

</semantics>

</math><img src="./3c5eeeffc952b16bda581094fe1eb451bdc14f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.22ex; height:2.676ex;" alt="{\displaystyle X_{t}=\varphi X_{t-1}}" loading="lazy"> will be a <a href="Geometric_progression" title="Geometric progression">geometric progression</a> (exponential growth or decay). In this case, the solution can be found analytically: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=a\varphi ^{t}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=a\varphi ^{t}}</annotation>

</semantics>

</math><img src="./055903c03e931cb2c247112743409af7745932af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.425ex; height:3.009ex;" alt="{\displaystyle X_{t}=a\varphi ^{t}}" loading="lazy"> whereby <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is an unknown constant (<a href="Initial_condition" title="Initial condition">initial condition</a>).

<div class="mw-heading mw-heading3"><h3 id="Explicit_mean/difference_form_of_AR(1)_process">Explicit mean/difference form of AR(1) process</h3></div>

The AR(1) model is the discrete-time analogy of the continuous <a href="Ornstein-Uhlenbeck_process" class="mw-redirect" title="Ornstein-Uhlenbeck process">Ornstein-Uhlenbeck process</a>. It is therefore sometimes useful to understand the properties of the AR(1) model cast in an equivalent form. In this form, the AR(1) model, with process parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \in \mathbb {R} }</annotation>

</semantics>

</math><img src="./b31351fd5a472b6b13c8ab8228000ff1576fdbdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle \theta \in \mathbb {R} }" loading="lazy">, is given by

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t+1}=X_{t}+(1-\theta )(\mu -X_{t})+\varepsilon _{t+1}}</annotation>

</semantics>

</math><img src="./306d60853556b6a43b768eebb8308010acf85cff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.094ex; height:2.843ex;" alt="{\displaystyle X_{t+1}=X_{t}+(1-\theta )(\mu -X_{t})+\varepsilon _{t+1}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\theta |<1\,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\theta |<1\,}</annotation>

</semantics>

</math><img src="./5780668779972c655ed2ab5186f7e3074ea4530d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.032ex; height:2.843ex;" alt="{\displaystyle |\theta |<1\,}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu :=E(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu :=E(X)}</annotation>

</semantics>

</math><img src="./6a042b43dac885166129d0b40bc0277c810e742e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.712ex; height:2.843ex;" alt="{\displaystyle \mu :=E(X)}" loading="lazy"> is the model mean, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\epsilon _{t}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\epsilon _{t}\}}</annotation>

</semantics>

</math><img src="./bd983bf30e05d88a2ab48ed476f56d1c1b3dd7d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.095ex; height:2.843ex;" alt="{\displaystyle \{\epsilon _{t}\}}" loading="lazy"> is a white-noise process with zero mean and constant variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">.</dd></dl>

By rewriting this as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t+1}=\theta X_{t}+(1-\theta )\mu +\varepsilon _{t+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t+1}=\theta X_{t}+(1-\theta )\mu +\varepsilon _{t+1}}</annotation>

</semantics>

</math><img src="./4f54418bcf2b8928d86d155c73ad9b4e1615a2c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.785ex; height:2.843ex;" alt="{\displaystyle X_{t+1}=\theta X_{t}+(1-\theta )\mu +\varepsilon _{t+1}}" loading="lazy"> and then deriving (by induction) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t+n}=\theta ^{n}X_{t}+(1-\theta ^{n})\mu +\Sigma _{i=1}^{n}\left(\theta ^{n-i}\epsilon _{t+i}\right)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t+n}=\theta ^{n}X_{t}+(1-\theta ^{n})\mu +\Sigma _{i=1}^{n}\left(\theta ^{n-i}\epsilon _{t+i}\right)}</annotation>

</semantics>

</math><img src="./5336d06b43f4540856867db550712aa1499ea232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.242ex; height:3.343ex;" alt="{\displaystyle X_{t+n}=\theta ^{n}X_{t}+(1-\theta ^{n})\mu +\Sigma _{i=1}^{n}\left(\theta ^{n-i}\epsilon _{t+i}\right)}" loading="lazy">, one can show that

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X_{t+n}|X_{t})=\mu \left[1-\theta ^{n}\right]+X_{t}\theta ^{n}}</annotation>

</semantics>

</math><img src="./57e7ce545d36dfb0de17a55881ba8322e99955a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.196ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (X_{t+n}|X_{t})=\mu \left[1-\theta ^{n}\right]+X_{t}\theta ^{n}}" loading="lazy"> and</dd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X_{t+n}|X_{t})=\sigma ^{2}{\frac {1-\theta ^{2n}}{1-\theta ^{2}}}}</annotation>

</semantics>

</math><img src="./faae1ccdfe1d391ae209a92012a05e69afe032ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.492ex; height:6.176ex;" alt="{\displaystyle \operatorname {Var} (X_{t+n}|X_{t})=\sigma ^{2}{\frac {1-\theta ^{2n}}{1-\theta ^{2}}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Choosing_the_maximum_lag">Choosing the maximum lag</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Partial_autocorrelation_function" title="Partial autocorrelation function">Partial autocorrelation function</a></div>

The partial autocorrelation of an AR(p) process equals zero at lags larger than p, so the appropriate maximum lag p is the one after which the partial autocorrelations are all zero.

<div class="mw-heading mw-heading2"><h2 id="Calculation_of_the_AR_parameters">Calculation of the AR parameters</h2></div>

There are many ways to estimate the coefficients, such as the <a href="Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> procedure or <a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a> (through Yule–Walker equations).

The AR(p) model is given by the equation

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}.\,}</annotation>

</semantics>

</math><img src="./b1a017e273d484ea82c3c0effe3153b1e991a0ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.291ex; height:7.009ex;" alt="{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}.\,}" loading="lazy"></dd></dl>

It is based on parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{i}}</annotation>

</semantics>

</math><img src="./70503774fb21be77396899900d3aa1e47d8f9e10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.32ex; height:2.176ex;" alt="{\displaystyle \varphi _{i}}" loading="lazy"> where i = 1, ..., p. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the Yule–Walker equations.

<div class="mw-heading mw-heading3"><h3 id="Yule–Walker_equations">Yule–Walker equations</h3></div>

The Yule–Walker equations, named for <a href="Udny_Yule" title="Udny Yule">Udny Yule</a> and <a href="Gilbert_Walker_(physicist)" title="Gilbert Walker (physicist)">Gilbert Walker</a>,<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a> are the following set of equations.<a href="#cite_note-16">[16]</a>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{m}=\sum _{k=1}^{p}\varphi _{k}\gamma _{m-k}+\sigma _{\varepsilon }^{2}\delta _{m,0},}</annotation>

</semantics>

</math><img src="./9da60172903d9e7119978e11c9b113da40544a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.202ex; height:7.009ex;" alt="{\displaystyle \gamma _{m}=\sum _{k=1}^{p}\varphi _{k}\gamma _{m-k}+\sigma _{\varepsilon }^{2}\delta _{m,0},}" loading="lazy"></dd></dl>

where m = 0, …, p, yielding p + 1 equations. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{m}}</annotation>

</semantics>

</math><img src="./27a890b6f03d91a6f3f1033e4ffdf6b2c46c7737.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.879ex; height:2.176ex;" alt="{\displaystyle \gamma _{m}}" loading="lazy"> is the autocovariance function of Xt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }}</annotation>

</semantics>

</math><img src="./04852f481494a445c9f5b9082df1ead002c098a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle \sigma _{\varepsilon }}" loading="lazy"> is the standard deviation of the input noise process, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{m,0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{m,0}}</annotation>

</semantics>

</math><img src="./e1370316eefa8164f8231f804f30d9e9a15cc652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.987ex; height:3.009ex;" alt="{\displaystyle \delta _{m,0}}" loading="lazy"> is the <a href="Kronecker_delta_function" class="mw-redirect" title="Kronecker delta function">Kronecker delta function</a>.

Because the last part of an individual equation is non-zero only if m = 0, the set of equations can be solved by representing the equations for m > 0 in matrix form, thus getting the equation

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\gamma _{1}\\\gamma _{2}\\\gamma _{3}\\\vdots \\\gamma _{p}\\\end{bmatrix}}={\begin{bmatrix}\gamma _{0}&\gamma _{-1}&\gamma _{-2}&\cdots \\\gamma _{1}&\gamma _{0}&\gamma _{-1}&\cdots \\\gamma _{2}&\gamma _{1}&\gamma _{0}&\cdots \\\vdots &\vdots &\vdots &\ddots \\\gamma _{p-1}&\gamma _{p-2}&\gamma _{p-3}&\cdots \\\end{bmatrix}}{\begin{bmatrix}\varphi _{1}\\\varphi _{2}\\\varphi _{3}\\\vdots \\\varphi _{p}\\\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./3a564a8ed89ab0e811cec86e6381723792950663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.082ex; margin-bottom: -0.256ex; width:42.536ex; height:17.843ex;" alt="{\displaystyle {\begin{bmatrix}\gamma _{1}\\\gamma _{2}\\\gamma _{3}\\\vdots \\\gamma _{p}\\\end{bmatrix}}={\begin{bmatrix}\gamma _{0}&\gamma _{-1}&\gamma _{-2}&\cdots \\\gamma _{1}&\gamma _{0}&\gamma _{-1}&\cdots \\\gamma _{2}&\gamma _{1}&\gamma _{0}&\cdots \\\vdots &\vdots &\vdots &\ddots \\\gamma _{p-1}&\gamma _{p-2}&\gamma _{p-3}&\cdots \\\end{bmatrix}}{\begin{bmatrix}\varphi _{1}\\\varphi _{2}\\\varphi _{3}\\\vdots \\\varphi _{p}\\\end{bmatrix}}}" loading="lazy"></dd></dl>

which can be solved for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}.}</annotation>

</semantics>

</math><img src="./a852eb40a2580da26a00424bfa85c14c077bc73e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.046ex; height:2.843ex;" alt="{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}.}" loading="lazy"> The remaining equation for m = 0 is

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{0}=\sum _{k=1}^{p}\varphi _{k}\gamma _{-k}+\sigma _{\varepsilon }^{2},}</annotation>

</semantics>

</math><img src="./9b026ab9a1802d7f6365a76f22b966a46f39f3f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.151ex; height:7.009ex;" alt="{\displaystyle \gamma _{0}=\sum _{k=1}^{p}\varphi _{k}\gamma _{-k}+\sigma _{\varepsilon }^{2},}" loading="lazy"></dd></dl>

which, once <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}}</annotation>

</semantics>

</math><img src="./11b162c79b064fc3de8d868c0a5210897b051f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.399ex; height:2.843ex;" alt="{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}}" loading="lazy"> are known, can be solved for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }^{2}.}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}.}</annotation>

</semantics>

</math><img src="./aa5d61c2a70c33444023a77f42f7502a2abd7c18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.032ex; height:2.843ex;" alt="{\displaystyle \sigma _{\varepsilon }^{2}.}" loading="lazy">

An alternative formulation is in terms of the <a href="Autocorrelation_function" class="mw-redirect" title="Autocorrelation function">autocorrelation function</a>. The AR parameters are determined by the first p+1 elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (\tau )}</annotation>

</semantics>

</math><img src="./7f1b4786ec22c80e8b5d81429869524e2fadca5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.213ex; height:2.843ex;" alt="{\displaystyle \rho (\tau )}" loading="lazy"> of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (\tau )=\sum _{k=1}^{p}\varphi _{k}\rho (k-\tau )}</annotation>

</semantics>

</math><img src="./b7510506c4854869e268ae4a004ffdeaf07a0713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.927ex; height:7.009ex;" alt="{\displaystyle \rho (\tau )=\sum _{k=1}^{p}\varphi _{k}\rho (k-\tau )}" loading="lazy"></dd></dl>

Examples for some Low-order AR(p) processes

<ul><li>p=1

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{1}=\varphi _{1}\gamma _{0}}</annotation>

</semantics>

</math><img src="./7e8bf2eecd26d68040d14b4395e6e6ee608d80b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.19ex; height:2.176ex;" alt="{\displaystyle \gamma _{1}=\varphi _{1}\gamma _{0}}" loading="lazy"></li>

<li>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}=\varphi _{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}=\varphi _{1}}</annotation>

</semantics>

</math><img src="./b497cf1e359dd28af10fadeb373548eb34656b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.707ex; height:2.843ex;" alt="{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}=\varphi _{1}}" loading="lazy"></li></ul></li>

<li>p=2

<ul><li>The Yule–Walker equations for an AR(2) process are

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{1}=\varphi _{1}\gamma _{0}+\varphi _{2}\gamma _{-1}}</annotation>

</semantics>

</math><img src="./d8a22c7a06835e594cec72e5c83a633b812064b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.142ex; height:2.509ex;" alt="{\displaystyle \gamma _{1}=\varphi _{1}\gamma _{0}+\varphi _{2}\gamma _{-1}}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{2}=\varphi _{1}\gamma _{1}+\varphi _{2}\gamma _{0}}</annotation>

</semantics>

</math><img src="./cf33ac37c5209d496139c69ee8a0fb2fd9fa4bdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.863ex; height:2.509ex;" alt="{\displaystyle \gamma _{2}=\varphi _{1}\gamma _{1}+\varphi _{2}\gamma _{0}}" loading="lazy"></dd></dl>

<ul><li>Remember that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{-k}=\gamma _{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{-k}=\gamma _{k}}</annotation>

</semantics>

</math><img src="./8ce33584a8a0c4817bba5a476b369ab6d1ba76fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.963ex; height:2.176ex;" alt="{\displaystyle \gamma _{-k}=\gamma _{k}}" loading="lazy"></li>

<li>Using the first equation yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}={\frac {\varphi _{1}}{1-\varphi _{2}}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}={\frac {\varphi _{1}}{1-\varphi _{2}}}}</annotation>

</semantics>

</math><img src="./37eeb70989f5c088e165c97a2be5dbbaacb2c291.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.546ex; height:5.343ex;" alt="{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}={\frac {\varphi _{1}}{1-\varphi _{2}}}}" loading="lazy"></li>

<li>Using the recursion formula yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}=\gamma _{2}/\gamma _{0}={\frac {\varphi _{1}^{2}-\varphi _{2}^{2}+\varphi _{2}}{1-\varphi _{2}}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{2}=\gamma _{2}/\gamma _{0}={\frac {\varphi _{1}^{2}-\varphi _{2}^{2}+\varphi _{2}}{1-\varphi _{2}}}}</annotation>

</semantics>

</math><img src="./6e7156ced938798e237ac752692eb1e07195dbc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.373ex; height:6.343ex;" alt="{\displaystyle \rho _{2}=\gamma _{2}/\gamma _{0}={\frac {\varphi _{1}^{2}-\varphi _{2}^{2}+\varphi _{2}}{1-\varphi _{2}}}}" loading="lazy"></li></ul></li></ul></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Estimation_of_AR_parameters">Estimation of AR parameters</h3></div>

The above equations (the Yule–Walker equations) provide several routes to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values.<a href="#cite_note-18">[18]</a> Some of these variants can be described as follows:

<ul><li>Estimation of autocovariances or autocorrelations. Here each of these terms is estimated separately, using conventional estimates. There are different ways of doing this and the choice between these affects the properties of the estimation scheme. For example, negative estimates of the variance can be produced by some choices.</li>

<li>Formulation as a <a href="Least_squares_regression" class="mw-redirect" title="Least squares regression">least squares regression</a> problem in which an ordinary least squares prediction problem is constructed, basing prediction of values of Xt on the p previous values of the same series. This can be thought of as a forward-prediction scheme. The <a href="Normal_equations" class="mw-redirect" title="Normal equations">normal equations</a> for this problem can be seen to correspond to an approximation of the matrix form of the Yule–Walker equations in which each appearance of an autocovariance of the same lag is replaced by a slightly different estimate.</li>

<li>Formulation as an extended form of ordinary least squares prediction problem. Here two sets of prediction equations are combined into a single estimation scheme and a single set of normal equations. One set is the set of forward-prediction equations and the other is a corresponding set of backward prediction equations, relating to the backward representation of the AR model:</li></ul>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t+i}+\varepsilon _{t}^{*}\,.}</annotation>

</semantics>

</math><img src="./c77daca7a31182f4b78f428aafd3bcebb460b400.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.519ex; height:7.009ex;" alt="{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t+i}+\varepsilon _{t}^{*}\,.}" loading="lazy"></dd></dl></dd>

<dd>Here predicted values of Xt would be based on the p future values of the same series. This way of estimating the AR parameters is due to John Parker Burg,<a href="#cite_note-Burg-19">[19]</a> and is called the Burg method:<a href="#cite_note-Brockwell-20">[20]</a> Burg and later authors called these particular estimates "maximum entropy estimates",<a href="#cite_note-Burg1-21">[21]</a> but the reasoning behind this applies to the use of any set of estimated AR parameters. Compared to the estimation scheme using only the forward prediction equations, different estimates of the autocovariances are produced, and the estimates have different stability properties. Burg estimates are particularly associated with <a href="Maximum_entropy_spectral_estimation" title="Maximum entropy spectral estimation">maximum entropy spectral estimation</a>.<a href="#cite_note-Bos-22">[22]</a></dd></dl>

Other possible approaches to estimation include <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a>. Two distinct variants of maximum likelihood are available: in one (broadly equivalent to the forward prediction least squares scheme) the likelihood function considered is that corresponding to the conditional distribution of later values in the series given the initial p values in the series; in the second, the likelihood function considered is that corresponding to the unconditional joint distribution of all the values in the observed series. Substantial differences in the results of these approaches can occur if the observed series is short, or if the process is close to non-stationarity.

<div class="mw-heading mw-heading2"><h2 id="Spectrum">Spectrum</h2></div>

The <a href="Spectral_density#Power_spectral_density" title="Spectral density">power spectral density</a> (PSD) of an AR(p) process with noise variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Var} (Z_{t})=\sigma _{Z}^{2}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Var} (Z_{t})=\sigma _{Z}^{2}}</annotation>

</semantics>

</math><img src="./fcf6634824713d3cfd153951776fe39e92578b21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.886ex; height:3.176ex;" alt="{\displaystyle \mathrm {Var} (Z_{t})=\sigma _{Z}^{2}}" loading="lazy"> is<a href="#cite_note-Storch-17">[17]</a>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{|1-\sum _{k=1}^{p}\varphi _{k}e^{-i2\pi fk}|^{2}}}.}</annotation>

</semantics>

</math><img src="./42f50b7ac757b2de256f1e9ff1571e969884c257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.844ex; height:7.509ex;" alt="{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{|1-\sum _{k=1}^{p}\varphi _{k}e^{-i2\pi fk}|^{2}}}.}" loading="lazy"></dd></dl>

For white noise (AR(0))

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(f)=\sigma _{Z}^{2}.}</annotation>

</semantics>

</math><img src="./06468b780d3e82778416cc74ff45f2bb6298547f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.08ex; height:3.176ex;" alt="{\displaystyle S(f)=\sigma _{Z}^{2}.}" loading="lazy"></dd></dl>

For AR(1)

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{|1-\varphi _{1}e^{-2\pi if}|^{2}}}={\frac {\sigma _{Z}^{2}}{1+\varphi _{1}^{2}-2\varphi _{1}\cos 2\pi f}}}</annotation>

</semantics>

</math><img src="./fe899e2c744e152309b413956cb163d64218246b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.024ex; height:7.176ex;" alt="{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{|1-\varphi _{1}e^{-2\pi if}|^{2}}}={\frac {\sigma _{Z}^{2}}{1+\varphi _{1}^{2}-2\varphi _{1}\cos 2\pi f}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}>0}</annotation>

</semantics>

</math><img src="./abe87447932492440bc26bb36dd9106b3a00b94e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.835ex; height:2.676ex;" alt="{\displaystyle \varphi _{1}>0}" loading="lazy"> there is a single spectral peak at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=0}</annotation>

</semantics>

</math><img src="./1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy">, often referred to as <a href="Red_noise" class="mw-redirect" title="Red noise">red noise</a>. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}}</annotation>

</semantics>

</math><img src="./d7daf493c8f6ef669c04c7b9715532fc35d12d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}}" loading="lazy"> becomes nearer 1, there is stronger power at low frequencies, i.e. larger time lags. This is then a low-pass filter, when applied to full spectrum light, everything except for the red light will be filtered.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}<0}</annotation>

</semantics>

</math><img src="./df3bd2845ccc8efc15e3eb61ca848ac2af258802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.835ex; height:2.676ex;" alt="{\displaystyle \varphi _{1}<0}" loading="lazy"> there is a minimum at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=0}</annotation>

</semantics>

</math><img src="./1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy">, often referred to as <a href="Blue_noise" class="mw-redirect" title="Blue noise">blue noise</a>. This similarly acts as a high-pass filter, everything except for blue light will be filtered.</li></ul>

The behavior of an AR(2) process is determined entirely by the roots of it <a href="Characteristic_equation_(calculus)" title="Characteristic equation (calculus)"> characteristic equation</a>, which is expressed in terms of the <a href="Lag_operator" title="Lag operator">lag operator</a> as:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-\varphi _{1}B-\varphi _{2}B^{2}=0,}</annotation>

</semantics>

</math><img src="./7af3e38e0138f8bcb67856401cdb332e075def5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.482ex; height:3.176ex;" alt="{\displaystyle 1-\varphi _{1}B-\varphi _{2}B^{2}=0,}" loading="lazy"></dd></dl>

or equivalently by the poles of its <a href="Transfer_function" title="Transfer function">transfer function</a>, which is defined in the <a href="Z-transform" title="Z-transform">Z domain</a> by:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{z}=(1-\varphi _{1}z^{-1}-\varphi _{2}z^{-2})^{-1}.}</annotation>

</semantics>

</math><img src="./42a3d758905d4bf4af1b22f12d47e6499b3e00b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.659ex; height:3.176ex;" alt="{\displaystyle H_{z}=(1-\varphi _{1}z^{-1}-\varphi _{2}z^{-2})^{-1}.}" loading="lazy"></dd></dl>

It follows that the poles are values of z satisfying:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-\varphi _{1}z^{-1}-\varphi _{2}z^{-2}=0}</annotation>

</semantics>

</math><img src="./5791fcc18c955dcc9d51e650a78bcef77e052c02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.099ex; height:3.176ex;" alt="{\displaystyle 1-\varphi _{1}z^{-1}-\varphi _{2}z^{-2}=0}" loading="lazy">,</dd></dl>

which yields:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{1},z_{2}={\frac {1}{2\varphi _{2}}}\left(\varphi _{1}\pm {\sqrt {\varphi _{1}^{2}+4\varphi _{2}}}\right)}</annotation>

</semantics>

</math><img src="./e580697a2dc39f1304e0a34757188b4ab71b8b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.029ex; height:5.676ex;" alt="{\displaystyle z_{1},z_{2}={\frac {1}{2\varphi _{2}}}\left(\varphi _{1}\pm {\sqrt {\varphi _{1}^{2}+4\varphi _{2}}}\right)}" loading="lazy">.</dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{1}}</annotation>

</semantics>

</math><img src="./c3621e468231ab352b7caa30bcf0ce9b452241a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{2}}</annotation>

</semantics>

</math><img src="./5abf655fa14f7ea44ad0ca781b59ff59c5f49117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{2}}" loading="lazy"> are the reciprocals of the characteristic roots, as well as the eigenvalues of the temporal update matrix:

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varphi _{1}&\varphi _{2}\\1&0\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./9f329235be209f01d37f96c2c92341066d0524da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.678ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\varphi _{1}&\varphi _{2}\\1&0\end{bmatrix}}}" loading="lazy"></dd></dl>

AR(2) processes can be split into three groups depending on the characteristics of their roots/poles:

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}^{2}+4\varphi _{2}<0}</annotation>

</semantics>

</math><img src="./0458ca96d21393755f25e547fb527f92a5b25272.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.413ex; height:3.176ex;" alt="{\displaystyle \varphi _{1}^{2}+4\varphi _{2}<0}" loading="lazy">, the process has a pair of complex-conjugate poles, creating a mid-frequency peak at:</li></ul>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}={\frac {1}{2\pi }}\cos ^{-1}\left({\frac {\varphi _{1}}{2{\sqrt {-\varphi _{2}}}}}\right),}</annotation>

</semantics>

</math><img src="./57755a084516b69b752f9c6b507e9055769b7b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.02ex; height:6.509ex;" alt="{\displaystyle f^{*}={\frac {1}{2\pi }}\cos ^{-1}\left({\frac {\varphi _{1}}{2{\sqrt {-\varphi _{2}}}}}\right),}" loading="lazy"></dd></dl>

with bandwidth about the peak inversely proportional to the moduli of the poles:

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |z_{1}|=|z_{2}|={\sqrt {-\varphi _{2}}}.}</annotation>

</semantics>

</math><img src="./ad6806931fd224663b11643262068e6814881447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.408ex; height:3.509ex;" alt="{\displaystyle |z_{1}|=|z_{2}|={\sqrt {-\varphi _{2}}}.}" loading="lazy"></dd></dl>

The terms involving square roots are all real in the case of complex poles since they exist only when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}<0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{2}<0}</annotation>

</semantics>

</math><img src="./fc7620690c960582c0f901f83633ccd7e06e3f8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.835ex; height:2.676ex;" alt="{\displaystyle \varphi _{2}<0}" loading="lazy">.

Otherwise the process has real roots, and:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}>0}</annotation>

</semantics>

</math><img src="./abe87447932492440bc26bb36dd9106b3a00b94e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.835ex; height:2.676ex;" alt="{\displaystyle \varphi _{1}>0}" loading="lazy"> it acts as a low-pass filter on the white noise with a spectral peak at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=0}</annotation>

</semantics>

</math><img src="./1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy"></li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}<0}</annotation>

</semantics>

</math><img src="./df3bd2845ccc8efc15e3eb61ca848ac2af258802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.835ex; height:2.676ex;" alt="{\displaystyle \varphi _{1}<0}" loading="lazy"> it acts as a high-pass filter on the white noise with a spectral peak at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=1/2}</annotation>

</semantics>

</math><img src="./09c9d48b46b004abc15e441ada41e3ac9b8d3e40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.864ex; height:2.843ex;" alt="{\displaystyle f=1/2}" loading="lazy">.</li></ul>

The process is non-stationary when the poles are on or outside the unit circle, or equivalently when the characteristic roots are on or inside the unit circle.

The process is stable when the poles are strictly within the unit circle (roots strictly outside the unit circle), or equivalently when the coefficients are in the triangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq \varphi _{2}\leq 1-|\varphi _{1}|}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1\leq \varphi _{2}\leq 1-|\varphi _{1}|}</annotation>

</semantics>

</math><img src="./174a4fea8bdb965bfce378f7c22e1333141e85ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.613ex; height:2.843ex;" alt="{\displaystyle -1\leq \varphi _{2}\leq 1-|\varphi _{1}|}" loading="lazy">.

The full PSD function can be expressed in real form as:

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{1+\varphi _{1}^{2}+\varphi _{2}^{2}-2\varphi _{1}(1-\varphi _{2})\cos(2\pi f)-2\varphi _{2}\cos(4\pi f)}}}</annotation>

</semantics>

</math><img src="./a013a00f38c6fa6be16d25f4b30ed3842b88a04e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:60.216ex; height:7.009ex;" alt="{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{1+\varphi _{1}^{2}+\varphi _{2}^{2}-2\varphi _{1}(1-\varphi _{2})\cos(2\pi f)-2\varphi _{2}\cos(4\pi f)}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Implementations_in_statistics_packages">Implementations in statistics packages</h2></div>

<ul><li><a href="R_(programming_language)" title="R (programming language)">R</a> – the stats package includes ar function;<a href="#cite_note-23">[23]</a> the astsa package includes sarima function to fit various models including AR.<a href="#cite_note-:0-24">[24]</a></li>

<li><a href="MATLAB" title="MATLAB">MATLAB</a> – the Econometrics Toolbox<a href="#cite_note-25">[25]</a> and System Identification Toolbox<a href="#cite_note-26">[26]</a> include AR models.<a href="#cite_note-27">[27]</a></li>

<li><a href="MATLAB" title="MATLAB">MATLAB</a> and <a href="GNU_Octave" title="GNU Octave">Octave</a> – the TSA toolbox contains several estimation functions for uni-variate, <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a>, and adaptive AR models.<a href="#cite_note-28">[28]</a></li>

<li><a href="PyMC" title="PyMC">PyMC</a>3 – the Bayesian statistics and probabilistic programming framework supports AR modes with p lags.</li>

<li>bayesloop – supports parameter inference and model selection for the AR-1 process with time-varying parameters.<a href="#cite_note-29">[29]</a></li>

<li><a href="Python_(programming_language)" title="Python (programming language)">Python</a> – statsmodels.org hosts an AR model.<a href="#cite_note-30">[30]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Impulse_response">Impulse response</h2></div>

The <a href="Impulse_response" title="Impulse response">impulse response</a> of a system is the change in an evolving variable in response to a change in the value of a shock term k periods earlier, as a function of k. Since the AR model is a special case of the vector autoregressive model, the computation of the impulse response in <a href="Vector_autoregression#Impulse_response" title="Vector autoregression">vector autoregression#impulse response</a> applies here.

<div class="mw-heading mw-heading2"><h2 id="n-step-ahead_forecasting">n-step-ahead forecasting</h2></div>

Once the parameters of the autoregression

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}\,}</annotation>

</semantics>

</math><img src="./989ee564796d9883fe00ba3d1faf373a731c0065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.644ex; height:7.009ex;" alt="{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}\,}" loading="lazy"></dd></dl>

have been estimated, the autoregression can be used to forecast an arbitrary number of periods into the future. First use t to refer to the first period for which data is not yet available; substitute the known preceding values Xt-i for i=1, ..., p into the autoregressive equation while setting the error term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}}</annotation>

</semantics>

</math><img src="./7c1ff8b8945e6a4fccf6071f806b9ef232492b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}}" loading="lazy"> equal to zero (because we forecast Xt to equal its expected value, and the expected value of the unobserved error term is zero). The output of the autoregressive equation is the forecast for the first unobserved period. Next, use t to refer to the next period for which data is not yet available; again the autoregressive equation is used to make the forecast, with one difference: the value of X one period prior to the one now being forecast is not known, so its expected value—the predicted value arising from the previous forecasting step—is used instead. Then for future periods the same procedure is used, each time using one more forecast value on the right side of the predictive equation until, after p predictions, all p right-side values are predicted values from preceding steps.

There are four sources of uncertainty regarding predictions obtained in this manner: (1) uncertainty as to whether the autoregressive model is the correct model; (2) uncertainty about the accuracy of the forecasted values that are used as lagged values in the right side of the autoregressive equation; (3) uncertainty about the true values of the autoregressive coefficients; and (4) uncertainty about the value of the error term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{t}\,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{t}\,}</annotation>

</semantics>

</math><img src="./0dddd630dd121297ad959b33a2adf8d58c3101ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.297ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{t}\,}" loading="lazy"> for the period being predicted. Each of the last three can be quantified and combined to give a <a href="Confidence_interval" title="Confidence interval">confidence interval</a> for the n-step-ahead predictions; the confidence interval will become wider as n increases because of the use of an increasing number of estimated values for the right-side variables.

<ul><li><a href="Moving_average_model" class="mw-redirect" title="Moving average model">Moving average model</a></li>

<li><a href="Linear_difference_equation" class="mw-redirect" title="Linear difference equation">Linear difference equation</a></li>

<li><a href="Predictive_analytics" title="Predictive analytics">Predictive analytics</a></li>

<li><a href="Linear_predictive_coding" title="Linear predictive coding">Linear predictive coding</a></li>

<li><a href="Resonance" title="Resonance">Resonance</a></li>

<li><a href="Levinson_recursion" title="Levinson recursion">Levinson recursion</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Infinite_impulse_response" title="Infinite impulse response">Infinite impulse response</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFSouzaLeao2023" class="citation journal cs1">Souza, Douglas Baptista de; Leao, Bruno Paes (26 October 2023). "Data Augmentation of Sensor Time Series using Time-varying Autoregressive Processes". Annual Conference of the PHM Society. 15 (1). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.36001%2Fphmconf.2023.v15i1.3565">10.36001/phmconf.2023.v15i1.3565</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFSouzaLeao2024" class="citation journal cs1">Souza, Douglas Baptista de; Leao, Bruno Paes (5 November 2024). "Data Augmentation of Multivariate Sensor Time Series using Autoregressive Models and Application to Failure Prognostics". Annual Conference of the PHM Society. 16 (1). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.36001%2Fphmconf.2024.v16i1.4145">10.36001/phmconf.2024.v16i1.4145</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFJiaLiJiangLiu2025" class="citation journal cs1">Jia, Zhixuan; Li, Wang; Jiang, Yunlong; Liu, Xingshen (9 July 2025). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath13142230">"The Use of Minimization Solvers for Optimizing Time-Varying Autoregressive Models and Their Applications in Finance"</a>. Mathematics. 13 (14): 2230. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath13142230">10.3390/math13142230</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFDiodatoDi_SalvoBellocchi2025" class="citation journal cs1">Diodato, Nazzareno; Di Salvo, Cristina; Bellocchi, Gianni (18 March 2025). "Climate driven generative time-varying model for improved decadal storm power predictions in the Mediterranean". Communications Earth & Environment. 6 (1): 212. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2025ComEE...6..212D">2025ComEE...6..212D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs43247-025-02196-2">10.1038/s43247-025-02196-2</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFInayatiIriawan2024" class="citation journal cs1">Inayati, Syarifah; Iriawan, Nur (31 December 2024). "Time-Varying Autoregressive Models for Economic Forecasting". Matematika: 131–142. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.11113%2Fmatematika.v40.n3.1654">10.11113/matematika.v40.n3.1654</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBaptista_de_SouzaKuhnSeara2019" class="citation journal cs1">Baptista de Souza, Douglas; Kuhn, Eduardo Vinicius; Seara, Rui (January 2019). "A Time-Varying Autoregressive Model for Characterizing Nonstationary Processes". IEEE Signal Processing Letters. 26 (1): 134–138. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019ISPL...26..134B">2019ISPL...26..134B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLSP.2018.2880086">10.1109/LSP.2018.2880086</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFWangChenWang2023" class="citation journal cs1">Wang, Shihan; Chen, Tao; Wang, Hongjian (17 March 2023). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10052024">"IDBD-Based Beamforming Algorithm for Improving the Performance of Phased Array Radar in Nonstationary Environments"</a>. Sensors. 23 (6): 3211. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023Senso..23.3211W">2023Senso..23.3211W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fs23063211">10.3390/s23063211</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10052024">10052024</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/36991922">36991922</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFAbramovichSpencerTurley2007" class="citation journal cs1">Abramovich, Yuri I.; Spencer, Nicholas K.; Turley, Michael D. E. (April 2007). "Time-Varying Autoregressive (TVAR) Models for Multiple Radar Observations". IEEE Transactions on Signal Processing. 55 (4): 1298–1311. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007ITSP...55.1298A">2007ITSP...55.1298A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTSP.2006.888064">10.1109/TSP.2006.888064</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFGutierrezSalazar-Varas2011" class="citation book cs1">Gutierrez, D.; Salazar-Varas, R. (August 2011). "EEG signal classification using time-varying autoregressive models and common spatial patterns". 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society. pp. 6585–6588. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEMBS.2011.6091624">10.1109/IEMBS.2011.6091624</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4577-1589-1</bdi>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22255848">22255848</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFBox1994" class="citation book cs1">Box, George E. P. (1994). Time series analysis : forecasting and control. Gwilym M. Jenkins, Gregory C. Reinsel (3rd ed.). Englewood Cliffs, N.J.: Prentice Hall. p. 54. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-060774-6</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/28888762">28888762</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFShumway2000" class="citation book cs1">Shumway, Robert H. (2000). Time series analysis and its applications. David S. Stoffer. New York: Springer. pp. 90–91. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98950-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42392178">42392178</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFShumwayStoffer2010" class="citation book cs1">Shumway, Robert H.; Stoffer, David (2010). Time series analysis and its applications : with R examples (3rd ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1441978646</bdi>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> Lai, Dihui; and Lu, Bingfeng; <a rel="nofollow" class="external text" href="https://www.soa.org/globalassets/assets/library/newsletters/predictive-analytics-and-futurism/2017/june/2017-predictive-analytics-iss15-lai-lu.pdf">"Understanding Autoregressive Model for Time Series as a Deterministic Dynamic System"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230324041726/https://www.soa.org/globalassets/assets/library/newsletters/predictive-analytics-and-futurism/2017/june/2017-predictive-analytics-iss15-lai-lu.pdf">Archived</a> 2023-03-24 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, in Predictive Analytics and Futurism, June 2017, number 15, June 2017, pages 7-9

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> Yule, G. Udny (1927) <a rel="nofollow" class="external text" href="http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56031">"On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110514094546/http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56031">Archived</a> 2011-05-14 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="Philosophical_Transactions_of_the_Royal_Society" title="Philosophical Transactions of the Royal Society">Philosophical Transactions of the Royal Society</a> of London, Ser. A, Vol. 226, 267–298.]

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Walker, Gilbert (1931) <a rel="nofollow" class="external text" href="http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56224">"On Periodicity in Series of Related Terms"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110607170511/http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56224">Archived</a> 2011-06-07 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="Proceedings_of_the_Royal_Society" title="Proceedings of the Royal Society">Proceedings of the Royal Society</a> of London, Ser. A, Vol. 131, 518–532.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFTheodoridis2015" class="citation book cs1">Theodoridis, Sergios (2015-04-10). "Chapter 1. Probability and Stochastic Processes". Machine Learning: A Bayesian and Optimization Perspective. Academic Press, 2015. pp. 9–51. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-801522-3</bdi>.</cite>

</li>

<li id="cite_note-Storch-17">^ <a href="#cite_ref-Storch_17-0">a</a> <a href="#cite_ref-Storch_17-1">b</a> <cite id="CITEREFVon_StorchZwiers2001" class="citation book cs1">Von Storch, Hans; Zwiers, Francis W. (2001). Statistical analysis in climate research. Cambridge University Press. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511612336">10.1017/CBO9780511612336</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-01230-9</bdi>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFEshel" class="citation web cs1">Eshel, Gidon. <a rel="nofollow" class="external text" href="http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/YWSourceFiles/YW-Eshel.pdf">"The Yule Walker Equations for the AR Coefficients"</a> (PDF). stat.wharton.upenn.edu. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180713135223/http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/YWSourceFiles/YW-Eshel.pdf">Archived</a> (PDF) from the original on 2018-07-13. Retrieved 2019-01-27.</cite>

</li>

<li id="cite_note-Burg-19"><a href="#cite_ref-Burg_19-0">^</a> Burg, John Parker (1968); "A new analysis technique for time series data", in Modern Spectrum Analysis (Edited by D. G. Childers), NATO Advanced Study Institute of Signal Processing with emphasis on Underwater Acoustics. IEEE Press, New York.

</li>

<li id="cite_note-Brockwell-20"><a href="#cite_ref-Brockwell_20-0">^</a> <cite id="CITEREFBrockwellDahlhausTrindade2005" class="citation journal cs1">Brockwell, Peter J.; Dahlhaus, Rainer; Trindade, A. Alexandre (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121021015413/http://www3.stat.sinica.edu.tw/statistica/oldpdf/A15n112.pdf">"Modified Burg Algorithms for Multivariate Subset Autoregression"</a> (PDF). Statistica Sinica. 15: 197–213. Archived from <a rel="nofollow" class="external text" href="http://www3.stat.sinica.edu.tw/statistica/oldpdf/A15n112.pdf">the original</a> (PDF) on 2012-10-21.</cite>

</li>

<li id="cite_note-Burg1-21"><a href="#cite_ref-Burg1_21-0">^</a> Burg, John Parker (1967) "Maximum Entropy Spectral Analysis", Proceedings of the 37th Meeting of the Society of Exploration Geophysicists, Oklahoma City, Oklahoma.

</li>

<li id="cite_note-Bos-22"><a href="#cite_ref-Bos_22-0">^</a> <cite id="CITEREFBosDe_WaeleBroersen2002" class="citation journal cs1">Bos, Robert; De Waele, Stijn; Broersen, Piet M. T. (2002). <a rel="nofollow" class="external text" href="http://resolver.tudelft.nl/uuid:870559f7-f1e9-4968-83da-edd19485eaaf">"Autoregressive spectral estimation by application of the Burg algorithm to irregularly sampled data"</a>. IEEE Transactions on Instrumentation and Measurement. 51 (6): 1289. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002ITIM...51.1289B">2002ITIM...51.1289B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIM.2002.808031">10.1109/TIM.2002.808031</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230416160911/https://repository.tudelft.nl/islandora/object/uuid:870559f7-f1e9-4968-83da-edd19485eaaf?collection=research">Archived</a> from the original on 2023-04-16. Retrieved 2019-12-11.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <a rel="nofollow" class="external text" href="http://finzi.psych.upenn.edu/R/library/stats/html/ar.html">"Fit Autoregressive Models to Time Series"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160128234632/http://finzi.psych.upenn.edu/R/library/stats/html/ar.html">Archived</a> 2016-01-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (in R)

</li>

<li id="cite_note-:0-24"><a href="#cite_ref-:0_24-0">^</a> <cite id="CITEREFStofferPoison2023" class="citation web cs1">Stoffer, David; Poison, Nicky (2023-01-09). <a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/astsa/">"astsa: Applied Statistical Time Series Analysis"</a>. Retrieved 2023-08-20.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/products/econometrics.html">"Econometrics Toolbox"</a>. www.mathworks.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230416160907/https://www.mathworks.com/products/econometrics.html">Archived</a> from the original on 2023-04-16. Retrieved 2022-02-16.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/products/sysid.html">"System Identification Toolbox"</a>. www.mathworks.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220216063519/https://www.mathworks.com/products/sysid.html">Archived</a> from the original on 2022-02-16. Retrieved 2022-02-16.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/help/econ/autoregressive-model.html">"Autoregressive Model - MATLAB & Simulink"</a>. www.mathworks.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220216063648/https://www.mathworks.com/help/econ/autoregressive-model.html">Archived</a> from the original on 2022-02-16. Retrieved 2022-02-16.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://pub.ist.ac.at/~schloegl/matlab/tsa/">"The Time Series Analysis (TSA) toolbox for Octave and MATLAB"</a>. pub.ist.ac.at. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120511144225/http://pub.ist.ac.at/~schloegl/matlab/tsa/">Archived</a> from the original on 2012-05-11. Retrieved 2012-04-03.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/christophmark/bayesloop">"christophmark/bayesloop"</a>. December 7, 2021. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200928085417/https://github.com/christophmark/bayesloop">Archived</a> from the original on September 28, 2020. Retrieved September 4, 2018 – via GitHub.</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.statsmodels.org/stable/generated/statsmodels.tsa.ar_model.AutoReg.html">"statsmodels.tsa.ar_model.AutoReg — statsmodels 0.12.2 documentation"</a>. www.statsmodels.org. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210228123354/https://www.statsmodels.org/stable/generated/statsmodels.tsa.ar_model.AutoReg.html">Archived</a> from the original on 2021-02-28. Retrieved 2021-04-29.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFMills1990" class="citation book cs1">Mills, Terence C. (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/timeseriestechni0000mill">Time Series Techniques for Economists</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521343398</bdi>.</cite></li>

<li><cite id="CITEREFPercivalWalden1993" class="citation book cs1">Percival, Donald B.; Walden, Andrew T. (1993). Spectral Analysis for Physical Applications. Cambridge University Press. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993sapa.book.....P">1993sapa.book.....P</a>.</cite></li>

<li><cite id="CITEREFPanditWu1983" class="citation book cs1">Pandit, Sudhakar M.; Wu, Shien-Ming (1983). Time Series and System Analysis with Applications. John Wiley & Sons.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://paulbourke.net/miscellaneous/ar/">AutoRegression Analysis (AR)</a> by Paul Bourke</li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=b8yslhlIsA0&list=PLD15D38DC7AA3B737&index=8#t=34m25s">Econometrics lecture (topic: Autoregressive models)</a> on <a href="YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> by <a href="Mark_Thoma" title="Mark Thoma">Mark Thoma</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Stochastic_processes496" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="Random_graph" title="Random graph">Random graph</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Artificial_intelligence_(AI)426" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Artificial_intelligence_(AI)426" style="font-size:114%;margin:0 4em"><a href="Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a> (AI)</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_artificial_intelligence" title="History of artificial intelligence">History</a>

<ul><li><a href="Timeline_of_artificial_intelligence" title="Timeline of artificial intelligence">timeline</a></li></ul></li>

<li><a href="List_of_artificial_intelligence_companies" title="List of artificial intelligence companies">Companies</a></li>

<li><a href="List_of_artificial_intelligence_projects" title="List of artificial intelligence projects">Projects</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Parameter" title="Parameter">Parameter</a>

<ul><li><a href="Hyperparameter_(machine_learning)" title="Hyperparameter (machine learning)">Hyperparameter</a></li></ul></li>

<li><a href="Loss_functions_for_classification" title="Loss functions for classification">Loss functions</a></li>

<li><a href="Regression_analysis" title="Regression analysis">Regression</a>

<ul><li><a href="Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li>

<li><a href="Double_descent" title="Double descent">Double descent</a></li>

<li><a href="Overfitting" title="Overfitting">Overfitting</a></li></ul></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Clustering</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient descent</a>

<li><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton method</a></li>

<li><a href="Conjugate_gradient_method" title="Conjugate gradient method">Conjugate gradient method</a></li></ul></li>

<li><a href="Backpropagation" title="Backpropagation">Backpropagation</a></li>

<li><a href="Attention_(machine_learning)" title="Attention (machine learning)">Attention</a></li>

<li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Normalization_(machine_learning)" title="Normalization (machine learning)">Normalization</a>

<ul><li><a href="Batch_normalization" title="Batch normalization">Batchnorm</a></li></ul></li>

<li><a href="Activation_function" title="Activation function">Activation</a>

<ul><li><a href="Softmax_function" title="Softmax function">Softmax</a></li>

<li><a href="Sigmoid_function" title="Sigmoid function">Sigmoid</a></li>

<li><a href="Rectifier_(neural_networks)" title="Rectifier (neural networks)">Rectifier</a></li></ul></li>

<li><a href="Gating_mechanism" title="Gating mechanism">Gating</a></li>

<li><a href="Weight_initialization" title="Weight initialization">Weight initialization</a></li>

<li><a href="Regularization_(mathematics)" title="Regularization (mathematics)">Regularization</a></li>

<li><a href="Training%2C_validation%2C_and_test_data_sets" title="Training, validation, and test data sets">Datasets</a>

<ul><li><a href="Data_augmentation" title="Data augmentation">Augmentation</a></li></ul></li>

<li><a href="Prompt_engineering" title="Prompt engineering">Prompt engineering</a></li>

<li><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a>

<ul><li><a href="Q-learning" title="Q-learning">Q-learning</a></li>

<li><a href="State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li>

<li><a href="Imitation_learning" title="Imitation learning">Imitation</a></li>

<li><a href="Policy_gradient_method" title="Policy gradient method">Policy gradient</a></li></ul></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion</a></li>

<li><a href="Latent_diffusion_model" title="Latent diffusion model">Latent diffusion model</a></li>

<li><a href="Adversarial_machine_learning" title="Adversarial machine learning">Adversary</a></li>

<li><a href="Uncanny_valley" title="Uncanny valley">Uncanny valley</a></li>

<li><a href="Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li>

<li><a href="Reflection_(artificial_intelligence)" class="mw-redirect" title="Reflection (artificial intelligence)">Reflection</a></li>

<li><a href="Recursive_self-improvement" title="Recursive self-improvement">Recursive self-improvement</a></li>

<li><a href="Hallucination_(artificial_intelligence)" title="Hallucination (artificial intelligence)">Hallucination</a></li>

<li><a href="Word_embedding" title="Word embedding">Word embedding</a></li>

<li><a href="Vibe_coding" title="Vibe coding">Vibe coding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Machine_learning" title="Machine learning">Machine learning</a>

<ul><li><a href="Prompt_engineering#In-context_learning" title="Prompt engineering">In-context learning</a></li></ul></li>

<li><a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Artificial neural network</a>

<ul><li><a href="Deep_learning" title="Deep learning">Deep learning</a></li></ul></li>

<li><a href="Language_model" title="Language model">Language model</a>

<ul><li><a href="Large_language_model" title="Large language model">Large language model</a></li>

<li><a href="Reasoning_language_model" title="Reasoning language model">Reasoning language model</a></li>

<li><a href="Model_Context_Protocol" title="Model Context Protocol">Model Context Protocol</a></li>

<li><a href="Intelligent_agent" title="Intelligent agent">Intelligent agent</a></li>

<li><a href="Artificial_human_companion" title="Artificial human companion">Artificial human companion</a></li>

<li><a href="Humanity's_Last_Exam" title="Humanity's Last Exam">Humanity's Last Exam</a></li>

<li><a href="Artificial_general_intelligence" title="Artificial general intelligence">Artificial general intelligence (AGI)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Audio–visual</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="AlexNet" title="AlexNet">AlexNet</a></li>

<li><a href="WaveNet" title="WaveNet">WaveNet</a></li>

<li><a href="Human_image_synthesis" title="Human image synthesis">Human image synthesis</a></li>

<li><a href="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Deep_learning_speech_synthesis" title="Deep learning speech synthesis">Speech synthesis</a>

<li><a href="ElevenLabs" title="ElevenLabs">ElevenLabs</a></li></ul></li>

<li><a href="Speech_recognition" title="Speech recognition">Speech recognition</a>

<ul><li><a href="Whisper_(speech_recognition_system)" title="Whisper (speech recognition system)">Whisper</a></li></ul></li>

<li><a href="Facial_recognition_system" title="Facial recognition system">Facial recognition</a></li>

<li><a href="AlphaFold" title="AlphaFold">AlphaFold</a></li>

<li><a href="Text-to-image_model" title="Text-to-image model">Text-to-image models</a>

<ul><li><a href="Aurora_(text-to-image_model)" class="mw-redirect" title="Aurora (text-to-image model)">Aurora</a></li>

<li><a href="Adobe_Firefly" title="Adobe Firefly">Firefly</a></li>

<li><a href="Ideogram_(text-to-image_model)" title="Ideogram (text-to-image model)">Ideogram</a></li>

<li><a href="Imagen_(text-to-image_model)" title="Imagen (text-to-image model)">Imagen</a></li>

<li><a href="Midjourney" title="Midjourney">Midjourney</a></li>

<li><a href="Recraft" title="Recraft">Recraft</a></li>

<li><a href="Stable_Diffusion" title="Stable Diffusion">Stable Diffusion</a></li></ul></li>

<li><a href="Text-to-video_model" title="Text-to-video model">Text-to-video models</a>

<ul><li><a href="Dream_Machine_(text-to-video_model)" title="Dream Machine (text-to-video model)">Dream Machine</a></li>

<li><a href="Runway_(company)#Services_and_technologies" title="Runway (company)">Runway Gen</a></li>

<li><a href="MiniMax_(company)#Hailuo_AI" title="MiniMax (company)">Hailuo AI</a></li>

<li><a href="Kling_(text-to-video_model)" class="mw-redirect" title="Kling (text-to-video model)">Kling</a></li>

<li><a href="Music_and_artificial_intelligence" title="Music and artificial intelligence">Music generation</a>

<ul><li><a href="Riffusion" title="Riffusion">Riffusion</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Text</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="GloVe" title="GloVe">GloVe</a></li>

<li><a href="Llama_(language_model)" title="Llama (language model)">Llama</a></li>

<li><a href="Chinchilla_(language_model)" title="Chinchilla (language model)">Chinchilla AI</a></li>

<li><a href="ChatGPT" title="ChatGPT">ChatGPT</a></li>

<li><a href="Claude_(language_model)" title="Claude (language model)">Claude</a></li>

<li><a href="Gemini_(language_model)" title="Gemini (language model)">Gemini</a>

<ul><li><a href="Gemini_(chatbot)" title="Gemini (chatbot)">chatbot</a></li></ul></li>

<li><a href="LaMDA" title="LaMDA">LaMDA</a></li>

<li><a href="BLOOM_(language_model)" title="BLOOM (language model)">BLOOM</a></li>

<li><a href="Project_Debater" title="Project Debater">Project Debater</a></li>

<li><a href="IBM_Watson" title="IBM Watson">IBM Watson</a></li>

<li><a href="IBM_Watsonx" title="IBM Watsonx">IBM Watsonx</a></li>

<li><a href="IBM_Granite" title="IBM Granite">Granite</a></li>

<li><a href="Huawei_PanGu" title="Huawei PanGu">PanGu-Σ</a></li>

<li><a href="DeepSeek_(chatbot)" title="DeepSeek (chatbot)">DeepSeek</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decisional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="AlphaGo" title="AlphaGo">AlphaGo</a></li>

<li><a href="AlphaZero" title="AlphaZero">AlphaZero</a></li>

<li><a href="OpenAI_Five" title="OpenAI Five">OpenAI Five</a></li>

<li><a href="Self-driving_car" title="Self-driving car">Self-driving car</a></li>

<li><a href="MuZero" title="MuZero">MuZero</a></li>

<li><a href="Action_selection" title="Action selection">Action selection</a>

<ul><li><a href="AutoGPT" title="AutoGPT">AutoGPT</a></li></ul></li>

<li><a href="Robot_control" title="Robot control">Robot control</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alan_Turing" title="Alan Turing">Alan Turing</a></li>

<li><a href="Warren_Sturgis_McCulloch" title="Warren Sturgis McCulloch">Warren Sturgis McCulloch</a></li>

<li><a href="Walter_Pitts" title="Walter Pitts">Walter Pitts</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li>

<li><a href="Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li>

<li><a href="Shun'ichi_Amari" title="Shun'ichi Amari">Shun'ichi Amari</a></li>

<li><a href="Kunihiko_Fukushima" title="Kunihiko Fukushima">Kunihiko Fukushima</a></li>

<li><a href="Takeo_Kanade" title="Takeo Kanade">Takeo Kanade</a></li>

<li><a href="Marvin_Minsky" title="Marvin Minsky">Marvin Minsky</a></li>

<li><a href="John_McCarthy_(computer_scientist)" title="John McCarthy (computer scientist)">John McCarthy</a></li>

<li><a href="Nathaniel_Rochester_(computer_scientist)" title="Nathaniel Rochester (computer scientist)">Nathaniel Rochester</a></li>

<li><a href="Allen_Newell" title="Allen Newell">Allen Newell</a></li>

<li><a href="Cliff_Shaw" title="Cliff Shaw">Cliff Shaw</a></li>

<li><a href="Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a></li>

<li><a href="Oliver_Selfridge" title="Oliver Selfridge">Oliver Selfridge</a></li>

<li><a href="Frank_Rosenblatt" title="Frank Rosenblatt">Frank Rosenblatt</a></li>

<li><a href="Bernard_Widrow" title="Bernard Widrow">Bernard Widrow</a></li>

<li><a href="Joseph_Weizenbaum" title="Joseph Weizenbaum">Joseph Weizenbaum</a></li>

<li><a href="Seymour_Papert" title="Seymour Papert">Seymour Papert</a></li>

<li><a href="Seppo_Linnainmaa" title="Seppo Linnainmaa">Seppo Linnainmaa</a></li>

<li><a href="Paul_Werbos" title="Paul Werbos">Paul Werbos</a></li>

<li><a href="Geoffrey_Hinton" title="Geoffrey Hinton">Geoffrey Hinton</a></li>

<li><a href="John_Hopfield" title="John Hopfield">John Hopfield</a></li>

<li><a href="J%C3%BCrgen_Schmidhuber" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a></li>

<li><a href="Yann_LeCun" title="Yann LeCun">Yann LeCun</a></li>

<li><a href="Yoshua_Bengio" title="Yoshua Bengio">Yoshua Bengio</a></li>

<li><a href="Lotfi_A._Zadeh" title="Lotfi A. Zadeh">Lotfi A. Zadeh</a></li>

<li><a href="Stephen_Grossberg" title="Stephen Grossberg">Stephen Grossberg</a></li>

<li><a href="Alex_Graves_(computer_scientist)" title="Alex Graves (computer scientist)">Alex Graves</a></li>

<li><a href="James_Goodnight" title="James Goodnight">James Goodnight</a></li>

<li><a href="Andrew_Ng" title="Andrew Ng">Andrew Ng</a></li>

<li><a href="Ilya_Sutskever" title="Ilya Sutskever">Ilya Sutskever</a></li>

<li><a href="Alex_Krizhevsky" title="Alex Krizhevsky">Alex Krizhevsky</a></li>

<li><a href="Ian_Goodfellow" title="Ian Goodfellow">Ian Goodfellow</a></li>

<li><a href="Demis_Hassabis" title="Demis Hassabis">Demis Hassabis</a></li>

<li><a href="David_Silver_(computer_scientist)" title="David Silver (computer scientist)">David Silver</a></li>

<li><a href="Andrej_Karpathy" title="Andrej Karpathy">Andrej Karpathy</a></li>

<li><a href="Ashish_Vaswani" title="Ashish Vaswani">Ashish Vaswani</a></li>

<li><a href="Noam_Shazeer" title="Noam Shazeer">Noam Shazeer</a></li>

<li><a href="Aidan_Gomez" title="Aidan Gomez">Aidan Gomez</a></li>

<li><a href="Mustafa_Suleyman" title="Mustafa Suleyman">Mustafa Suleyman</a></li>

<li><a href="Fran%C3%A7ois_Chollet" title="François Chollet">François Chollet</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Architectures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Neural_Turing_machine" title="Neural Turing machine">Neural Turing machine</a></li>

<li><a href="Differentiable_neural_computer" title="Differentiable neural computer">Differentiable neural computer</a></li>

<li><a href="Transformer_(deep_learning_architecture)" title="Transformer (deep learning architecture)">Transformer</a>

<ul><li><a href="Vision_transformer" title="Vision transformer">Vision transformer (ViT)</a></li></ul></li>

<li><a href="Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network (RNN)</a></li>

<li><a href="Long_short-term_memory" title="Long short-term memory">Long short-term memory (LSTM)</a></li>

<li><a href="Gated_recurrent_unit" title="Gated recurrent unit">Gated recurrent unit (GRU)</a></li>

<li><a href="Echo_state_network" title="Echo state network">Echo state network</a></li>

<li><a href="Multilayer_perceptron" title="Multilayer perceptron">Multilayer perceptron (MLP)</a></li>

<li><a href="Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network (CNN)</a></li>

<li><a href="Residual_neural_network" title="Residual neural network">Residual neural network (RNN)</a></li>

<li><a href="Highway_network" title="Highway network">Highway network</a></li>

<li><a href="Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li>

<li><a href="Autoencoder" title="Autoencoder">Autoencoder</a></li>

<li><a href="Variational_autoencoder" title="Variational autoencoder">Variational autoencoder (VAE)</a></li>

<li><a href="Generative_adversarial_network" title="Generative adversarial network">Generative adversarial network (GAN)</a></li>

<li><a href="Graph_neural_network" title="Graph neural network">Graph neural network (GNN)</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-01" href="https://en.wikipedia.org/wiki/?title=Autoregressive_model&oldid=1303736624">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>