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

We have introduce a new vision of stochastic processes through the geometryinduced by the dilation. The dilation matrices of a given processes areobtained by a composition of rotations matrices, contain the measureinformation in a condensed way. Particularly interesting is the fact that theobtention of dilation matrices is regardless of the stationarity of theunderlying process. When the process is stationary, it coincides with theNaimark Dilation and only one rotation matrix is computed, when the process isnon-stationary, a set of rotation matrices are computed. In particular, theperiodicity of the correlation function that may appear in some classes ofsignal is transmitted to the set of dilation matrices. These rotation matrices,which can be arbitrarily close to each other depending on the sampling or therescaling of the signal are seen as a distinctive feature of the signal. Inorder to study this sequence of matrices, and guided by the possibility torescale the signal, the correct geometrical framework to use with thedilation's theoretic results is the space of curves on manifolds, that is theset of all curve that lies on a base manifold. To give a complete sight aboutthe space of curve, a metric and the derived geodesic equation are provided.The general results are adapted to the more specific case where the basemanifold is the Lie group of rotation matrices. The notion of the shape of acurve can be formalized as the set of equivalence classes of curves given bythe 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 andenable comparison between shapes.