ARMA MODELS, PREWHITENING, AND MINIMUM CROSS ENTROPY

The problem of spectral estimation on the basis of observations from a finite stretch of a stationary time series is considered, in connection with knowledge of a prior estimate of the spectral density. In general, the data are not exactly compatible with the prior. For example, the first p sample autocovariances might be significantly different from the first p Fourier coefficients of the prior spectral density. A reasonable ''posterior'' spectral density estimate would be the density that is closest to the prior according to some measure of divergence, while at the same time being compatible with the data. The cross entropy (relative entropy, Kullback-Leibler number) has often been proposed in the past to serve as such a measure of divergence. A connection of the original minimum cross entropy spectral analysis (MCESA) method of Shore (1981) to traditional prewhitening techniques and to ARMA models is pointed out. In view of this connection, a fast approximate solution of the minimum cross entropy problem is also proposed. The solution is in a standard multiplicative form, that is, the posterior is equal to the prior multiplied by a ''correction'' factor, and has many favorable properties, including its asymptotic consistency.