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Inverse Correlations for Multiple Time Series and Gaussian Random Fields and Measures of Their Linear Determinism

Author(s): R.J. Bhansali | L. Ippoliti

Journal: Journal of Mathematics and Statistics
ISSN 1549-3644

Volume: 1;
Issue: 4;
Start page: 287;
Date: 2005;
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Keywords: Spatial series analysis | time series analysis | gaussian random fields | gaussian markov random fields | inverse correlation function | linear predictor

For a discrete-time vector linear stationary process, {X(t)}, admitting forward and backwardautoregressive representations, the variance matrix of an optimal linear interpolator of X(t), based on aknowledge of {X(t-j), j¹0}, is known to be given by Ri(0) -1 where Ri(0) denotes the inverse varianceof the process. Let A=Is-Ri(0) 1 R(0) 1 , where R(0) denotes the variance matrix of {X(t)} and s I ans´s, identity matrix. A measure of linear interpolability of the process, called an index of lineardeterminism, may be constructed from the determinant, A], Det[I s - of s R(0) 0) Ri( A I -1 -1 = - . Analternative measure is constructed by relating ], 0) tr[Ri( -1 the trace of Ri(0) -1 , to tr[R(0)]. Therelationship between the matrix A and the corresponding matrix, P, obtained by considering only anoptimal one-step linear predictor of X(t) from a knowledge of its infinite past, {X(t-j),j>0}, is alsodiscussed. The possible role the inverse correlation function may have for model specification of avector ARMA model is explored. Close parallels between the problem of interpolation for a stationaryunivariate two-dimensional Gaussian random field and time series are examined and an index of lineardeterminism for the latter class of processes is also defined. An application of this index for modelspecification and diagnostic testing of a Gaussian Markov Random Field is investigated together withthe question of its estimation from observed data. Results are illustrated by a simulation study.

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