Moving-average process
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 in discrete time that is representable in the form
(1) |
where , , with the Kronecker delta (so that is a white noise process with spectral density ), is a positive integer, and are constant coefficients. The spectral density of such a process is given by
and its correlation function has the form
Conversely, if the correlation function of a stationary process in discrete time has the property that when for some positive integer , then is a moving-average process of order , that is, it has a representation of the form (1) where is a white noise (see, for example, [1]).
Along with the moving-average process of finite order , 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
(2) |
where denotes white noise and the series on the right-hand side converges in mean-square (so that ), and also more general two-sided moving-average processes, of the form
(3) |
where denotes white noise and . The class of two-sided moving-average processes coincides with that of stationary processes having spectral density , while the class of one-sided moving-average processes coincides with that of processes having spectral density such that
A continuous-time stationary process , , is called a one-sided or two-sided moving-average process if it has the form
or
respectively, where , that is, is a generalized white noise process. The class of two-sided moving-average processes in continuous time coincides with that of stationary processes having spectral density , while the class of one-sided moving-average processes in continuous time coincides with that of processes having spectral density such that
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.
Moving-average process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moving-average_process&oldid=12234