## Distances and Riemannian metrics for multivariate spectral densities## Preliminaries on multivariate predictionConsider a multivariate discrete-time, zero mean, weakly stationary stochastic process with taking values in . Let denote the sequence of matrix covariances and be the corresponding matricial power spectral measure for which We will be concerned with the case of non-deterministic process of full rank with an absolutely continuous power spectrum. Hence, with being a matrix valued power spectral density (PSD) function. Further, is assumed to be integrable throughout. ## Geometry of multivariable processDefine to be the closure of -vector-valued finite linear combinations of with respect to convergence in the mean: This space is endowed with both, a matricial inner product as well as a scalar inner product It is standard to establish the Kolmogorov isomorphism between the “temporal” space and the “spectral” space , with for . It is convenient to endow the latter space the matricidal inner product We denote for simplicity. The corresponding scalar inner product The least-variance linear prediction can be expressed in the spectral domain The minimizer coincides with that of where the minimum is sought in the positive-definite sense. Let denote the the minimizer of such a problem, then the minimal matrix of the above problem, denoted by , is ## Spectral factorization and optimal predictionFor a non-deterministic process of full rank, the determinant of the error variance is non-zero, and this the is well-known Szeg-Kolmogorov formula. We consider only non-deterministic processes of full rank and hence we assume that with and , where , and denotes the Hardy space of functions which are analytic in the unit disk with square-integrable radial limits. The spectral factorization presents an expression of the optimal prediction error in the form ## Comparison of PSD's## Prediction errors and innovations processesConsider two processes with and the corresponding PSD's and the optimal prediction filters, respectively. Let for be an innovations process. Then, from stationarity, whereas ## The color of innovations and PSD mismatchWe normalize the innovation processes as follows: then the power spectral density of the process is The mismatch between the two power spectra , can be quantified by the distance of to the identity. To this end, we define The following choices of divergence measures lead to the same Riemannian structure as the above one: These are the Frobenius distance, the generalized Hellinger distance, the multivariate Itakuta-Saito distance, the log-spectral deviation between and , respectively. ## Suboptimal prediction and PSD mismatchAs was given in , . The above equality holds if and only if . Thus a mismatch between the two spectral densities can be quantified by the strength of the above inequality. Since , we consider as a “divergence measure” between the two PSD's. The following options lead to the same Riemannian structure: ## Riemannian structure on multivariate spectraConsider the following class of PSD's Let be a class of admissible perturbations of for with It was shown in [1] that for PSD's in and in the distance induces the following Riemannian metric In particular, for and . Then for sufficiently small The geodesic path connecting two spectra is The geodesic distance is For the distance , the corresponding Riemannian metic is The associtated geodesic path for is still unknown. ## Examples## A scalar exampleConsider the two power spectral densities where The two power spectra, and , are shown in the following figure. We evaluate , the corresponding spectra are shown in the following figure. ## A multivariable exampleConsider the two matrix-valued power spectral densities The following two figures show the two spectra where the block is the and the block is , for . We compute the geodesic connecting and as The geodesic is shown in the following figure. ## Reference[1] X. Jiang, L. Ning and T.T. Georgiou, “Distances and Riemannian metrics for multivariate spectral densities,” IEEE |