2 research outputs found
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 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.Comment: 19 pages, draft pape