# Distances and Riemannian metrics for multivariate spectral densities

## Preliminaries on multivariate prediction

Consider 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 process

Define 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 prediction

For 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

In this case, admits a unique factorization

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 processes

Consider 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 mismatch

We 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 mismatch

As 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 spectra

Consider 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 example

Consider 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 example

Consider 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

code1

[1] X. Jiang, L. Ning and T.T. Georgiou, “Distances and Riemannian metrics for multivariate spectral densities,” IEEE Trans. on Signal Processing, to appear 2012. http://arxiv.org/abs/1107.1345