Information Topological Characterization of Periodically Correlated Processes by Dilation Operators

Giving process information through spectral considerations has been tackled for decades. We propose, in this paper, a new way of dealing with such an objective by giving hidden information topology of the spectral measure of non-stationary and periodically correlated processes. We used first the Kolmogorov decomposition which is a natural extension of the Naimark operator theory to obtain a sequence of rotation matrices called the dilation matrices. These matrices carry all the spectral information of the process and belong to SO(n) or SU(n) with respect to, respectively, the real or complex nature of the periodically correlated processes studied. In order to give a topological interpretation of the positioning of these matrices on the space of rotation matrices, we have applied a persistent homology technique and next exposed fundamental attributes. We showed that different types of periodically correlated processes are endowed with a point cloud structure that can be easily discriminated by topological and information features.