Metrics for power spectra: an axiomatic approach

Authors:
Tryphon T. Georgiou;Johan Karlsson;Mir Shahrouz Takyar
Affiliations:
Department of Electrical Engineering, University of Minnesota, Minneapolis, MN;Department of Mathematics, Division of Optimization and Systems Theory, Royal Institute of Technology, Stockholm, Sweden;Department of Electrical Engineering, University of Minnesota, Minneapolis, MN
Venue:
IEEE Transactions on Signal Processing
Year:
2009

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Abstract

We present an axiomatic framework for seeking distances between power spectral density functions. The axioms require that the sought metric respects the effects of additive and multiplicative noise in reducing our ability to discriminate spectra, as well as they require continuity of statistical quantities with respect to perturbations measured in the metric. We then present a particular metric which abides by these requirements. The metric is based on the Monge-Kantorovich transportation problem and is contrasted with an earlier Riemannian metric based on the minimum-variance prediction geometry of the underlying time-series. It is also being compared with the more traditional Itakura-Saito distance measure, as well as the aforementioned prediction metric, on two representative examples.