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''autoregressive moving-average process, ARMA 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
+
A wide-sense stationary [[Stochastic process|stochastic process]] $  X ( t) $
 +
with discrete time $  t = 0 , \pm  1 \dots $
 +
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>
+
$$ \tag{1 }
 +
X ( t) + a _ {1} X ( t - 1 ) + \dots +
 +
a _ {p} X ( t - p ) =
 +
$$
  
<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>
+
$$
 +
= \
 +
Y ( t) + b _ {1} Y ( t - 1 ) + \dots + b _ {q} Y ( t - q ) ,
 +
$$
  
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
+
where $  {\mathsf E} Y ( t) = 0 $,  
 +
$  {\mathsf E} Y ( t) Y ( s) = \sigma  ^ {2} \delta _ {ts} $,  
 +
$  \delta _ {ts} $
 +
being the Kronecker delta (i.e. $  Y ( t) $
 +
is a white noise process with spectral density $  \sigma  ^ {2} / 2 \pi $),  
 +
$  p $
 +
and $  q $
 +
are non-negative integers, and $  a _ {1} \dots a _ {p} $,  
 +
$  b _ {1} \dots b _ {q} $
 +
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>
+
$$
 +
\phi ( z)  = 1 + a _ {1} z + \dots + a _ {p} z  ^ {p}  = 0
 +
$$
  
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]]
+
are of modulus distinct from 1, then the stationary autoregressive moving-average process $  X ( t) $
 +
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>
+
$$
 +
f ( \lambda )  =
 +
\frac{\sigma  ^ {2} }{2 \pi }
  
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).
+
\frac{| \psi ( e ^ {i \lambda } )
 +
|  ^ {2} }{| \phi ( e ^ {i \lambda } ) | ^ {2} }
 +
,
 +
$$
  
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]]).
+
where  $  \psi ( z) = 1 + b _ {1} z + \dots + b _ {q} z  ^ {q} $.
 +
However, for the solution of equation (1) with given initial values  $  X ( t _ {0} - 1 ) \dots X ( t _ {0} - p ) $
 +
to tend to the stationary process  $  X ( t) $
 +
as  $  t - t _ {0} \rightarrow \infty $,
 +
it is necessary that all roots of the equation  $  \phi ( z) = 0 $
 +
be situated outside the unit disc  $  | z | \leq  1 $(
 +
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 $  q = 0 $,  
 +
cf. [[Auto-regressive process|Auto-regressive process]]) and moving-average processes (when $  p = 0 $,  
 +
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.
 
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.
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:01, 6 June 2020


autoregressive moving-average process, ARMA process

A wide-sense stationary stochastic process $ X ( t) $ with discrete time $ t = 0 , \pm 1 \dots $ the values of which satisfy a difference equation

$$ \tag{1 } X ( t) + a _ {1} X ( t - 1 ) + \dots + a _ {p} X ( t - p ) = $$

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

where $ {\mathsf E} Y ( t) = 0 $, $ {\mathsf E} Y ( t) Y ( s) = \sigma ^ {2} \delta _ {ts} $, $ \delta _ {ts} $ being the Kronecker delta (i.e. $ Y ( t) $ is a white noise process with spectral density $ \sigma ^ {2} / 2 \pi $), $ p $ and $ q $ are non-negative integers, and $ a _ {1} \dots a _ {p} $, $ b _ {1} \dots b _ {q} $ are constant coefficients. If all roots of the equation

$$ \phi ( z) = 1 + a _ {1} z + \dots + a _ {p} z ^ {p} = 0 $$

are of modulus distinct from 1, then the stationary autoregressive moving-average process $ X ( t) $ exists and has spectral density

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

where $ \psi ( z) = 1 + b _ {1} z + \dots + b _ {q} z ^ {q} $. However, for the solution of equation (1) with given initial values $ X ( t _ {0} - 1 ) \dots X ( t _ {0} - p ) $ to tend to the stationary process $ X ( t) $ as $ t - t _ {0} \rightarrow \infty $, it is necessary that all roots of the equation $ \phi ( z) = 0 $ be situated outside the unit disc $ | z | \leq 1 $( see [1] and [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 [3]). Special cases of autoregressive moving-average processes are auto-regressive processes (when $ q = 0 $, cf. Auto-regressive process) and moving-average processes (when $ p = 0 $, cf. 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 [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

[1] G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , 1–2 , Holden-Day (1976)
[2] T.W. Anderson, "The statistical analysis of time series" , Wiley (1971)
[3] J.L. Doob, "The elementary Gaussian processes" Ann. Math. Stat. , 15 (1944) pp. 229–282

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 [a2], [a4] for references on this problem.

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

References

[a1] M. Aoki, "Notes on economic time series analysis: system theory perspectives" , Lect. notes in econom. and math. systems , 220 , Springer (1983)
[a2] P. Faurre, M. Clerget, F. Germain, "Opérateurs rationnels positifs" , Dunod (1979)
[a3] E.J. Hannan, "Multiple time series" , Wiley (1970)
[a4] A. Lindquist, G. Picci, "Realization theory for multivariate stationary Gaussian processes" SIAM J. Control Optim. , 23 (1985) pp. 809–857
[a5] L. Ljung, T. Söderström, "Theory and practice of recursive identification" , M.I.T. (1983)
How to Cite This Entry:
Mixed autoregressive moving-average process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed_autoregressive_moving-average_process&oldid=47861
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article