View source for Mixed autoregressive moving-average process

''autoregressive moving-average process, ARMA process''

A wide-sense stationary [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642001.png" /> with discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642002.png" /> the values of which satisfy a difference equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642004.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642007.png" /> being the Kronecker delta (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642008.png" /> is a white noise process with spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m0642009.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420011.png" /> are non-negative integers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420013.png" /> are constant coefficients. If all roots of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420014.png" /></td> </tr></table>

are of modulus distinct from 1, then the stationary autoregressive moving-average process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420015.png" /> exists and has [[Spectral density|spectral density]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420017.png" />. However, for the solution of equation (1) with given initial values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420018.png" /> to tend to the stationary process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420019.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420020.png" />, it is necessary that all roots of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420021.png" /> be situated outside the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420022.png" /> (see [[#References|[1]]] and [[#References|[2]]], for example).

The class of Gaussian autoregressive moving-average processes coincides with the class of stationary processes that have a spectral density and are one-dimensional components of multi-dimensional Markov processes (see [[#References|[3]]]). Special cases of autoregressive moving-average processes are auto-regressive processes (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420023.png" />, cf. [[Auto-regressive process|Auto-regressive process]]) and moving-average processes (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064200/m06420024.png" />, cf. [[Moving-average process|Moving-average process]]).

Generalizations of autoregressive moving-average processes are the autoregressive integrated moving-average processes introduced by G.E.P. Box and G.M. Jenkins (see [[#References|[1]]]) and often used in applied problems. These are non-stationary processes with stationary increments such that the increments of some fixed order form an autoregressive moving-average process.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.E.P. Box,   G.M. Jenkins,   "Time series analysis. Forecasting and control" , '''1–2''' , Holden-Day  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,   "The statistical analysis of time series" , Wiley  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Doob,   "The elementary Gaussian processes"  ''Ann. Math. Stat.'' , '''15'''  (1944)  pp. 229–282</TD></TR></table>



====Comments====
The class of autoregressive moving-average processes is of interest because they represent stationary processes with a rational spectral density.

The problem of representing a stationary process as an autoregressive moving-average process is known in the Western literature as the stochastic realization problem; see [[#References|[a2]]], [[#References|[a4]]] for references on this problem.

Autoregressive moving-average processes are used by statisticians [[#References|[a3]]], econometricians [[#References|[a1]]] and engineers [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Aoki,   "Notes on economic time series analysis: system theory perspectives" , ''Lect. notes in econom. and math. systems'' , '''220''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Faurre,   M. Clerget,   F. Germain,   "Opérateurs rationnels positifs" , Dunod  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.J. Hannan,   "Multiple time series" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Lindquist,   G. Picci,   "Realization theory for multivariate stationary Gaussian processes"  ''SIAM J. Control Optim.'' , '''23'''  (1985)  pp. 809–857</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Ljung,   T. Söderström,   "Theory and practice of recursive identification" , M.I.T.  (1983)</TD></TR></table>