Modeling of multichannel time series and extrapolation of matrix-valued autocorrelation sequences

This paper deals with multichannel time series, and extends the autocorrelation method for fitting the time series from the nondegenerate case to the general case. We shall show that the AR model obtained by solving the normal equations is always stable even in the degenerate case. The multichannel Levinson algorithm for solving the normal equations is extended from the nondegenerate case to the general case. In the nondegenerate case, the first n+1 values of the auto-correlation sequence of the obtained AR model of order n are identical with those of the original data. The next considered is the extrapolation problem of an autocorrelation sequence, that is, given data
{R_{0}, R_{1},..., R_{n}}
, it is to construct a model whose first n+1 values of the autocorrelation sequence are identical with these data. We shall show that, even in the degenerate case, when the given data
{R_{0}, R_{1},..., R_{n}}
are taken from the autocorrelation sequence of a purely nondeterministic time series, they are identical with the first n+1 values of the autocorrelation sequence of the AR model of order n.