We introduce a differential-geometric structure for spectral density functions of discrete-time random processes. This is quite analogous to the Riemannian structure of information geometry, which is used to study perturbations of probability density functions, and which is based on the Fisher information metric. Herein, we introduce an analogous Riemannian metric, which we motivate with a problem in prediction theory. It turns out that this problem also provides a prediction theoretic interpretation to the Itakura distortion measure, which relates to our metric. Geodesics and geodesic distances are characterized in closed form and, hence, the geodesic distance between two spectral density functions provides an explicit, intrinsic (pseudo)metric on the cone of density functions. Certain other distortion measures that involve generalized means of spectral density functions are shown to lead to the same Riemannian metric. Finally, an alternative Riemannian metric is introduced, which is motivated by an analogous problem involving smoothing instead of prediction.
Bibliographical noteFunding Information:
Manuscript received July 29, 2006; revised January 2, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Peter Handel. This research has been supported by the National Science Foundation and the Air Force Office of Scientific Research.
- Differential structure
- Distortion measures
- Information geometry
- Spectral density functions