Auto-regressive process
A stochastic process whose values satisfy an auto-regression equation with certain constants
:
![]() | (*) |
where is some positive number and where the variables
are usually assumed to be uncorrelated and identically distributed around their average value 0 with a variance
. If all the zeros of the function
of a complex variable
lie inside the unit circle, then equation (*) has the solution
![]() |
where the are connected with the
by the relation
![]() |
For example, let be a white noise process with spectral density
; in such a case the only kind of auto-regressive process satisfying equation (*) will be a process
with spectral density
![]() |
which is stationary in the wide sense if has no real zeros. The autocovariances (cf. Autocovariance)
of the process satisfy the recurrence relation
![]() |
and, in terms of the , have the form
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The parameters of the auto-regression are connected with the auto-correlation coefficients
of the process by the matrix relation
![]() |
where ,
and
is the matrix of auto-correlation coefficients (the Yule–Walker equation).
References
[1] | U. Grenander, M. Rosenblatt, "Statistical analysis of stationary time series" , Wiley (1957) |
[2] | E.J. Hannan, "Time series analysis" , Methuen , London (1960) |
Auto-regressive process. A.V. Prokhorov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Auto-regressive_process&oldid=15186