Maximum entropy spectral estimation for regular time series of degenerate rank

This paper deals with multichannel time series of degenerate rank, and extends the maximum entropy method over the degenerate rank case. In order to define the entropy of a multichannel time series of degenerate rank, we must first clarify all the deterministic relationships in the time series. This will be done for any regular time series matching given finite data of the autocorrelation sequence . A necessary and sufficient condition of the existence of a regular time series matching the data will be presented. Next, the entropy Hm(P) of a time series with its power spectrum P(ω) of rank less than equal to m is defined by where Sm[P] denotes the sum of all the principal minors of order m of the matrix P. The main purpose of this paper is to show that the maximum entropy power spectrum (i.e., the power spectrum which maximizes the entropy) is identical with the autoregressive power spectrum (i.e., the power spectrum obtained by the autoregressive fitting) even in the degenerate rank case.