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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 <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
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<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>
+
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 +
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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
+
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
  
<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>
+
$$ \tag{1 }
 +
X ( t)  = Y ( t) +
 +
b _ {1} Y ( t - 1 ) + \dots
 +
+ b _ {q} Y ( t - q ) ,
 +
$$
  
<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>
+
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
  
and its correlation function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065080/m06508014.png" /> has the form
+
$$
 +
f ( \lambda )  =
 +
\frac{\sigma  ^ {2} }{2 \pi }
 +
| \psi ( e ^ {i \lambda } ) |  ^ {2} ,
 +
$$
  
<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>
+
$$
 +
\psi ( z)  = b _ {0} + b _ {1} z + \dots + b _ {q} z  ^ {q} ,\  b _ {0} = 1 ,
 +
$$
  
<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>
+
and its correlation function  $  r ( k) = {\mathsf E} X ( t) X ( t - k ) $
 +
has the form
  
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]]]).
+
$$
 +
r ( k) = \sigma  ^ {2}
 +
\sum _ { j= } 0 ^ { {q }  - | k | }
 +
b _ {j} b _ {j + | k | }  \ \
 +
\textrm{ if }  | k | \leq  q ,
 +
$$
  
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
+
$$
 +
r ( k)  = 0 \  \textrm{ if }  | k | > q .
 +
$$
  
<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>
+
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]]]).
  
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
+
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
  
<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>
+
$$ \tag{2 }
 +
X ( t)  = \
 +
\sum _ { j= } 0 ^  \infty 
 +
b _ {j} Y ( t - j ) ,
 +
$$
  
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
+
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
  
<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>
+
$$ \tag{3 }
 +
X ( t)  = \
 +
\sum _ {j = - \infty } ^  \infty 
 +
b _ {j} Y ( t - j ) ,
 +
$$
 +
 
 +
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]]]).
 
(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
+
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
  
<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>
+
$$
 +
X ( t)  = \int\limits _ { 0 } ^  \infty 
 +
b ( s)  d Y ( t - s ) ,\ \
 +
\int\limits _ { 0 } ^  \infty 
 +
| b ( s) |  ^ {2}  d s  < \infty ,
 +
$$
  
 
or
 
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>
+
$$
 +
X ( t)  = \int\limits _
 +
{- \infty } ^  \infty 
 +
b ( s)  d Y ( t - s ) ,\ \
 +
\int\limits _ {- \infty } ^  \infty 
 +
| b ( s) |  ^ {2}  d s  < \infty ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
\int\limits _ {- \infty } ^  \infty 
 +
\mathop{\rm log}  f ( \lambda )
 +
( 1 + \lambda  ^ {2} )  ^ {-} 1 \
 +
d \lambda  > - \infty
 +
$$
  
 
(see [[#References|[4]]], [[#References|[3]]], [[#References|[5]]]).
 
(see [[#References|[4]]], [[#References|[3]]], [[#References|[5]]]).
Line 47: Line 139:
 
====References====
 
====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>
 
<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====
 
====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]].
 
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]].

Revision as of 08:01, 6 June 2020


A 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 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 $ 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, [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 [2], [1], [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 [4], [3], [5]).

References

[1] T.M. Anderson, "The statistical analysis of time series" , Wiley (1971)
[2] 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 & Ross (1977) pp. 66–89 (Translated from Russian)
[3] J.L. Doob, "Stochastic processes" , Wiley (1953)
[4] K. Karhunun, "Ueber lineare Methoden in der Wahrscheinlichkeitsrechnung" Ann. Acad. Sci. Fennicae Ser. A. Math. Phys. , 37 (1947)
[5] Yu.A. Rozanov, "Stationary random processes" , Holden-Day (1967) (Translated from Russian)

Comments

Both auto-regressive processes (cf. 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), which are of great importance in the study of time series.

How to Cite This Entry:
Moving-average process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moving-average_process&oldid=47910
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article