Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds

TL;DR

This paper introduces a geometric framework using dilation operators and shape space manifolds to analyze non-stationary stochastic processes, capturing their features through rotation matrices and curve analysis on manifolds.

Contribution

It develops a novel geometric approach to characterize non-stationary processes via dilation matrices and shape spaces, extending classical methods to non-stationary signals.

Findings

Dilation matrices encode process information regardless of stationarity.

The shape space framework allows comparison of process features through curves on manifolds.

The approach captures periodicity and other signal features in the geometry of dilation matrices.

Abstract

We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, it coincides with the Naimark Dilation and only one rotation matrix is computed, when the process is non-stationary, a set of rotation matrices are computed. In particular, the periodicity of the correlation function that may appear in some classes of signal is transmitted to the set of dilation matrices. These rotation matrices, which can be arbitrarily close to each other depending on the sampling or the rescaling of the signal are seen as a distinctive feature of the signal. In order to study this sequence of matrices, and guided by the possibility to rescale the signal, the correct geometrical framework to use with the dilation's theoretic results is the space of curves on manifolds, that is the set of all curve that lies on a base manifold. To give a complete sight about the space of curve, a metric and the derived geodesic equation are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices. The notion of the shape of a curve can be formalized as the set of equivalence classes of curves given by the quotient space of the space of curves and the increasing diffeomorphisms. The metric in the space of curve naturally extent to the space of shapes and enable comparison between shapes.