View source for Moving-average process

A [[Stochastic process|stochastic process]] which is stationary in the wide sense and which can be obtained by applying some linear transformation to a process with non-correlated values (that is, to a [[White noise|white noise]] process). The term is often applied to the more special case of a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650801.png" /> in discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650802.png" /> that is representable in the form

<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/m065/m065080/m0650803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650805.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650806.png" /> the Kronecker delta (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650807.png" /> is a white noise process with spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650808.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m0650809.png" /> is a positive integer, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508010.png" /> are constant coefficients. The [[Spectral density|spectral density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508011.png" /> of such a process is given by

<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/m065/m065080/m06508012.png" /></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/m065/m065080/m06508013.png" /></td> </tr></table>

and its correlation function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508014.png" /> has the form

<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/m065/m065080/m06508015.png" /></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/m065/m065080/m06508016.png" /></td> </tr></table>

Conversely, if the correlation function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508017.png" /> of a stationary process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508018.png" /> in discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508019.png" /> has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508020.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508021.png" /> for some positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508023.png" /> is a moving-average process of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508025.png" />, that is, it has a representation of the form (1) where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508026.png" /> is a white noise (see, for example, [[#References|[1]]]).

Along with the moving-average process of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508027.png" />, which is representable in the form (1), there are two types of moving-average processes in discrete time of infinite order, namely: one-sided moving-average processes, having a representation of the form

<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/m065/m065080/m06508028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508029.png" /> denotes white noise and the series on the right-hand side converges in mean-square (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508030.png" />), and also more general two-sided moving-average processes, of the form

<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/m065/m065080/m06508031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508032.png" /> denotes white noise and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508033.png" />. The class of two-sided moving-average processes coincides with that of stationary processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508034.png" /> having spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508035.png" />, while the class of one-sided moving-average processes coincides with that of processes having spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508036.png" /> such that

<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/m065/m065080/m06508037.png" /></td> </tr></table>

(see [[#References|[2]]], [[#References|[1]]], [[#References|[3]]]).

A continuous-time stationary process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508039.png" />, is called a one-sided or two-sided moving-average process if it has the form

<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/m065/m065080/m06508040.png" /></td> </tr></table>

or

<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/m065/m065080/m06508041.png" /></td> </tr></table>

respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508042.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508043.png" /> is a generalized white noise process. The class of two-sided moving-average processes in continuous time coincides with that of stationary processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508044.png" /> having spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508045.png" />, while the class of one-sided moving-average processes in continuous time coincides with that of processes having spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508046.png" /> such that

<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/m065/m065080/m06508047.png" /></td> </tr></table>

(see [[#References|[4]]], [[#References|[3]]], [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.M. Anderson,   "The statistical analysis of time series" , Wiley  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   "Stationary sequences in Hilbert space"  T. Kailath (ed.) , ''Linear Least-Squares Estimation'' , ''Benchmark Papers in Electric Engin. Computer Sci.'' , '''17''' , Dowden, Hutchington &amp; Ross  (1977)  pp. 66–89  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Doob,   "Stochastic processes" , Wiley  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Karhunun,   "Ueber lineare Methoden in der Wahrscheinlichkeitsrechnung"  ''Ann. Acad. Sci. Fennicae Ser. A. Math. Phys.'' , '''37'''  (1947)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Yu.A. Rozanov,   "Stationary random processes" , Holden-Day  (1967)  (Translated from Russian)</TD></TR></table>



====Comments====
Both auto-regressive processes (cf. [[Auto-regressive process|Auto-regressive process]]) and moving-average processes are special cases of so-called ARMA processes, i.e. auto-regressive moving-average processes (cf. [[Mixed autoregressive moving-average process|Mixed autoregressive moving-average process]]), which are of great importance in the study of [[Time series|time series]].