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.
Bibliographical noteFunding Information:
Manuscript received December 08, 2007; revised September 06, 2008. First published November 25, 2008; current version published February 13, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Antonio Napolitano. This work was supported in part by the Swedish Research Council, Göran Gustafsson Foundation, National Science Foundation, and by the Air Force Office of Scientific Research.
- Geometry of spectral measures
- Power spectra
- Spectral distances