View source for Moving-average process

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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  $  X ( t) $
in discrete time  $  t = 0 , \pm  1 \dots $
that is representable in the form

$$ \tag{1 }
X ( t)  =  Y ( t) +
b _ {1} Y ( t - 1 ) + \dots
+ b _ {q} Y ( t - q ) ,
$$

where  $  {\mathsf E} Y ( t) = 0 $, 
$  {\mathsf E} Y ( t) Y ( s) = \sigma  ^ {2} \delta _ {ts} $, 
with  $  \delta _ {ts} $
the Kronecker delta (so that  $  Y ( t) $
is a white noise process with spectral density  $  \sigma  ^ {2} / 2 \pi $), 
$  q $
is a positive integer, and  $  b _ {1} \dots b _ {q} $
are constant coefficients. The [[Spectral density|spectral density]]  $  f ( \lambda ) $
of such a process is given by

$$ 
f ( \lambda )  =  
\frac{\sigma  ^ {2} }{2 \pi }
 | \psi ( e ^ {i \lambda } ) |  ^ {2} ,
$$

$$ 
\psi ( z)  =  b _ {0} + b _ {1} z + \dots + b _ {q} z  ^ {q} ,\  b _ {0} = 1 ,
$$

and its correlation function  $  r ( k) = {\mathsf E} X ( t) X ( t - k ) $
has the form

$$ 
r ( k)  =  \sigma  ^ {2}
\sum _ { j= } 0 ^ { {q }  - | k | }
b _ {j} b _ {j + | k | }  \ \ 
\textrm{ if }  | k | \leq  q ,
$$

$$ 
r ( k)  =  0 \  \textrm{ if }  | k | > q .
$$

Conversely, if the correlation function  $  r ( k) $
of a stationary process  $  X ( t) $
in discrete time  $  t $
has the property that  $  r ( k) = 0 $
when  $  | k | > q $
for some positive integer  $  q $, 
then  $  X ( t) $
is a moving-average process of order  $  q $, 
that is, it has a representation of the form (1) where  $  Y ( t) $
is a white noise (see, for example, [[#References|[1]]]).

Along with the moving-average process of finite order  $  q $, 
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

$$ \tag{2 }
X ( t)  = \ 
\sum _ { j= } 0 ^  \infty  
b _ {j} Y ( t - j ) ,
$$

where  $  Y ( t) $
denotes white noise and the series on the right-hand side converges in mean-square (so that  $  \sum _ {j=} 0  ^  \infty  | b _ {j} |  ^ {2} < \infty $), 
and also more general two-sided moving-average processes, of the form

$$ \tag{3 }
X ( t)  = \ 
\sum _ {j = - \infty } ^  \infty  
b _ {j} Y ( t - j ) ,
$$

where  $  Y ( t) $
denotes white noise and  $  \sum _ {j = - \infty }   ^  \infty  | b _ {j} |  ^ {2} < \infty $. 
The class of two-sided moving-average processes coincides with that of stationary processes  $  X ( t) $
having spectral density  $  f ( \lambda ) $, 
while the class of one-sided moving-average processes coincides with that of processes having spectral density  $  f ( \lambda ) $
such that

$$ 
\int\limits _ {- \pi } ^  \pi  
 \mathop{\rm log}  f ( \lambda ) \ 
d \lambda  >  - \infty
$$

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

A continuous-time stationary process  $  X ( t) $, 
$  - \infty < t < \infty $, 
is called a one-sided or two-sided moving-average process if it has the form

$$ 
X ( t)  =  \int\limits _ { 0 } ^  \infty  
b ( s)  d Y ( t - s ) ,\ \ 
\int\limits _ { 0 } ^  \infty  
| b ( s) |  ^ {2}  d s  <  \infty ,
$$

or

$$ 
X ( t)  =  \int\limits _
{- \infty } ^  \infty  
b ( s)  d Y ( t - s ) ,\ \ 
\int\limits _ {- \infty } ^  \infty  
| b ( s) |  ^ {2}  d s  <  \infty ,
$$

respectively, where  $  {\mathsf E} [ d Y ( t) ]  ^ {2} = \sigma  ^ {2}  d t $, 
that is,  $  Y  ^  \prime  ( t) $
is a generalized white noise process. The class of two-sided moving-average processes in continuous time coincides with that of stationary processes  $  X ( t) $
having spectral density  $  f ( \lambda ) $, 
while the class of one-sided moving-average processes in continuous time coincides with that of processes having spectral density  $  f ( \lambda ) $
such that

$$ 
\int\limits _ {- \infty } ^  \infty  
 \mathop{\rm log}  f ( \lambda )
( 1 + \lambda  ^ {2} )  ^ {-} 1 \ 
d \lambda  >  - \infty
$$

(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]].