Effective Solutions of Linear Approximation Problems for Multivariate Stationary Processes with a Rational Spectrum

Abstract

We consider a class of multivariate stationary random processes ${\bf \xi} (t) = \{ \xi _1 (t), \cdots ,\xi _k (t)\} $ having the nonsingular spectral density matrix $||f_{jk} (\lambda )||$, where all $f_{jk} (\lambda )$ are rational functions of $\lambda $. The following linear approximation problems for the processes are studied: 1) the simplest extrapolation problem of determining a linear least-square estimate of $\xi _k (t + \tau ),\tau > 0$, by known values of $\xi _j (t'), j = 1, \cdots ,n,t' \leqq t$; 2) the finite interval extrapolation problem of a linear least-square estimation of $\xi _k (t + \tau )$ by $\xi _j (t'),j = 1, \cdots ,n,t - T \leqq t' \leqq t$; 3) the interpolation problem of a least-square estimation of $\xi _k (t + \tau ),0 < \tau < T$ by $\xi _j (t'),j = 1, \cdots ,n,t' \leqq t$ or $t' \geqq t + T$; 4) the filtration problem of a least-square estimation of the value of some random variable $\Xi $ (such that the functions $f_{\Xi k} (\lambda ),k = 1, \cdots ,n$, from equations (3.1) – (3.2) have the form (3.4), where all $q_{rk} (\lambda )$ are rational) by the values of $\xi _j (t'),j = 1, \cdots ,n,t' \leqq t$ or $t - T \leqq t' \leqq t$. In all cases the method used in previous papers [11] and [12] enables the explicit extrapolation, interpolation or filtration formulae to be derived by merely solving the algebraical equation $D(\lambda ) = \det ||f_{jk} (\lambda )|| = 0$ and afterwards a simple system of linear algebraical equations. The same method can also be applied to the case when we wish to find a least-square estimate of $\xi _k (t + \tau )$ or $\Xi $ by the values of $\xi _j (t'),j = 1, \cdots ,n$, on any set of closed intervals on the time axis. Some other generalizations of extrapolation, interpolation and filtration problems may be solved by the same method; they are given in the last section of the paper.