Ordering Principal Components of Multivariate Fractional Brownian Motion for Solving Inverse Problems

The problem of recovering information about a generating time-dependent dynamical system from measurement data is investigated as an inverse problem. Fractional Brownian motion is often used for modelling real-world applications that deal with specific properties such as long-range dependence or self-similarity. The self-similarity parameter, also known as Hurst parameter H, is directly related to the distribution of ordinal patterns in fractional Brownian motion. Thus the corresponding information entropy measure, known as permutation entropy, can extract significant information about the generating system. As real-world applications often involve multivariate correlated measurements, multivariate variants of permutation entropy have to be considered. While pooled permutation entropy is able to estimate H, it fails to detect cross-correlations between variables. In this work, we use multivariate permutation entropy based on principal component analysis (MPE-PCA) to investigate self-similarity and cross-correlations. We examine the variation of permutation entropy of principal components with variations of
$H$
and cross-correlations and find those principal components behave equally to their origin under certain conditions, i.e., as
$H$
increases, the MPE-PCA of the principal components decreases. Furthermore, MPE-PCA discovers cross-correlations in multivariate fractional Brownian motion.