Download Inverse Correlations for Multiple Time Series and Gaussian Random

Inverse Correlations for Gaussian Random Fields
and Measures of its Linear Determinism
1
R.J. Bhansali and 2L. Ippoliti
Division of Statistics and Probability, Department of Mathematical Sciences
University of Liverpool, L69 7ZL, Liverpool
2
Department of Quantitative Methods and Economic Theory
University ‘G. d’Annunzio’, Viale Pindaro 42, 65127 Pescara, Italy
1
Abstract: For a discrete-time vector linear stationary process, {X(t)}, admitting forward
and backward autoregressive representations, the variance matrix of an optimal linear
interpolator of X(t), based on a knowledge of {X(t-j), j0}, is known to be given by Ri(0)1
where Ri(0) denotes the inverse variance of the process. Let A=I sRi(0)ˉ1R(0)ˉ1, where
R(0) denotes the variance matrix of {X(t)} and I s an ss, identity matrix. A measure of
linear interpolability of the process, called an index of linear determinism, may be
constructed from the determinant, Det[I s  A], of Is  A  Ri(0)1R(0)1 . An alternative
measure is constructed by relating tr[Ri(0) 1], the trace of Ri(0)-1, to tr[R(0)]. Close
parallels between the problem of interpolation for a stationary univariate two-dimensional
Gaussian random field and time series are examined and an index of linear determinism
for the latter class of processes is also defined. An application of this index for model
specification and diagnostic testing of a Gaussian Markov Random Field is investigated
together with the question of its estimation from observed data. Results are illustrated by
a simulation study.