Editing Autoregressive model (section)

===Yule–Walker equations===
<!-- this heading is linked from other articles -->

The Yule–Walker equations, named for [[Udny Yule]] and [[Gilbert Walker (physicist)|Gilbert Walker]],<ref>Yule, G. Udny (1927) [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"] {{Webarchive|url=https://web.archive.org/web/20110514094546/http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56031 |date=2011-05-14 }}, ''[[Philosophical Transactions of the Royal Society]] of London'', Ser. A, Vol. 226, 267–298.]</ref><ref>Walker, Gilbert  (1931) [http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56224  "On Periodicity in Series of Related Terms"] {{Webarchive|url=https://web.archive.org/web/20110607170511/http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56224 |date=2011-06-07 }}, ''[[Proceedings of the Royal Society]] of London'', Ser. A, Vol. 131,  518–532.</ref> are the following set of equations.<ref>{{cite book |last=Theodoridis |first=Sergios |title=Machine Learning: A Bayesian and Optimization Perspective |publisher=Academic Press, 2015 |chapter=Chapter 1. Probability and Stochastic Processes |pages=9–51 |isbn=978-0-12-801522-3 |date=2015-04-10 }}</ref>

:<math>\gamma_m = \sum_{k=1}^p \varphi_k \gamma_{m-k} + \sigma_\varepsilon^2\delta_{m,0},</math>

where {{nowrap|1=''m'' = 0, …, ''p''}}, yielding {{nowrap|''p'' + 1}} equations. Here <math>\gamma_m</math> is the autocovariance function of X<sub>t</sub>, <math>\sigma_\varepsilon</math> is the standard deviation of the input noise process, and <math>\delta_{m,0}</math> is the [[Kronecker delta function]].

Because the last part of an individual equation is non-zero only if {{nowrap|1=''m'' = 0}}, the set of equations can be solved by representing the equations for {{nowrap|''m'' > 0}} in matrix form, thus getting the equation

:<math>\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}

</math>

which can be solved for all <math>\{\varphi_m; m=1,2, \dots ,p\}.</math> The remaining equation for ''m'' = 0 is

:<math>\gamma_0 = \sum_{k=1}^p \varphi_k \gamma_{-k} + \sigma_\varepsilon^2 ,</math>

which, once  <math>\{\varphi_m ; m=1,2, \dots ,p \}</math> are known, can be solved for <math>\sigma_\varepsilon^2 .</math>

An alternative formulation is in terms of the [[autocorrelation function]]. The AR parameters are determined by the first ''p''+1 elements <math>\rho(\tau)</math> of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating
<ref name=Storch>{{Cite book
| publisher = Cambridge University Press
| isbn = 0-521-01230-9
| last = Von Storch
| first = Hans
| first2=Francis W. |last2=Zwiers
| title = Statistical analysis in climate research
| year = 2001
| doi = 10.1017/CBO9780511612336
}}{{Page needed|date=March 2011}}</ref>
: <math>\rho(\tau) = \sum_{k=1}^p \varphi_k \rho(k-\tau)</math>
Examples for some Low-order AR(''p'') processes
* ''p''=1
** <math>\gamma_1 = \varphi_1 \gamma_0</math>
** Hence <math>\rho_1 = \gamma_1 / \gamma_0 = \varphi_1</math>
* ''p''=2
** The Yule–Walker equations for an AR(2) process are
**: <math>\gamma_1 = \varphi_1 \gamma_0 + \varphi_2 \gamma_{-1}</math>
**: <math>\gamma_2 = \varphi_1 \gamma_1 + \varphi_2 \gamma_0</math>
*** Remember that <math>\gamma_{-k} = \gamma_k</math>
*** Using the first equation yields <math>\rho_1 = \gamma_1 / \gamma_0 = \frac{\varphi_1}{1-\varphi_2}</math>
*** Using the recursion formula yields <math>\rho_2 = \gamma_2 / \gamma_0 = \frac{\varphi_1^2 - \varphi_2^2 + \varphi_2}{1 - \varphi_2}</math>