The purpose of this thesis is to study the geometry of power spectra and develop geometric methods for spectral analysis, modeling, and filtering of time series. We first study notions of distance suitable to quantify mismatch between probability distributions, power spectra, and covariance matrices. To this end, we review and relate relevant concepts from statistical inference, information theory, signal processing, and quantum mechanics. We introduce several alternative metrics for scalar as well as multivariate power spectra, and explore concepts from Riemannian geometry, in particular geodesics and means, to model slowly varying time series, and to interpolate and fuse statistics from different sources.
We first study mismatch between power spectra in the context of linear prediction. We introduce two alternative metrics. The first quantifies the degradation of predictive error variance when one power spectrum is used to design an optimal filter which is then applied to a process corresponding to another power spectrum. The second metric reflects the flatness of the innovations process. For this second metric, we provide closed-form expressions for the induced geodesics and geodesic distances. The theory we develop applies to multivariate power spectra. The metrics based on the prediction paradigm lack the important property of weak continuity. To this end we explore an alternative framework based on the transportation problem. In this, energy content has a physical significance and metrics quantify transportation cost between frequencies.
Throughout we explore geometric methods on the Riemannian manifold of power spectra, in a way analogous to the familiar Euclidean geometry. In particular, we use the notion of a geodesic to model the evolution of power spectral densities for slowly varying time series. More specifically, we develop the idea of geodesic path-fitting as an analogue of the least squares line-fitting in the Euclidean space. Further, we investigate means or averages of distributions and of positive semi-definite matrices arising in signal processing, computer vision and pattern recognition applications. The transportation mean, as well as the median, for normalized scalar power spectra are provided analytically. For multivariate Gaussian distributions, we show that the transportation mean requires solving a linear matrix inequality problem, which is computationally tractable. Furthermore, linear structural constraints on the means, based on prior knowledge, can be easily incorporated and solved efficiently using the linear matrix inequality formulation. We highlight the relevance of the geometric framework with several applications in signal processing, such as spectral tracking, speech morphing, filtering, and spectral averaging.