Counting the Number of Different Scaling Exponents in Multivariate Scale-Free Dynamics: Clustering by Bootstrap in the Wavelet Domain

Abstract

Multivariate selfsimilarity has become a classical tool to analyze collections of time series recorded jointly on one same system. Often, it amounts to estimating as many scaling exponents as time series. However, this leaves open the important question how many such scaling exponents are actually different. Elaborating on earlier work aiming to test the hypothesis that all exponents are equal, we intend here to count the number of different scaling exponents from a single finite size multivariate time series. To this end, we devise an original clustering procedure that combines a wavelet domain block multivariate bootstrap scheme with a test strategy for a reduced set of multiple hypotheses on the pairwise equality of scaling exponents that are relevant to clustering. Monte Carlo simulations, making use of synthetic multivariate selfsimilar processes, assess the relevance and performance of the proposed procedure under different scenarios and demonstrate that the proposed method yields practically satisfactory cluster number and size estimations.

Hurst multimodality detection based on large wavelet random matrices

“…Future work includes the construction of methodology for the identification of the distribution π(dH) once unimodality has been rejected (cf. [21] in the multivariate context). Studying high-dimensional limits is not a purely mathematical issue; instead, it is absolutely crucial for practical use in applications.…”

“…Examples of the proposed techniques include principal component analysis, factor analysis and sparse graphical Gaussian models [19]. Nevertheless, there has been a paucity of estimation methodologies for both high-dimensional and scale invariant signals; see a contrario [20] or [21], proposing a bootstrap-based method for counting the number of distinct Hurst exponents based on wavelet eigenanalysis, yet in a multivariate context with modest dimension. A key related difficulty is the study of random matrices under dependence, as they emerge in wavelet eigenanalysis, which is still a very active area of research [22], [23].…”

Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series

“…In (33), the ensemble average E is practically computed as the mean across independent realizations. Samples of the random variable (33) are plotted against a χ 2 distribution with M degrees of freedom, since this distribution is expected to hold under the exact joint normality of ĤM,bc . Fig.…”

“…These empirical observations and analytical calculations reveal an important advantage of the mutivariate ĤM,bc (and ĤM ) against the univariate ĤU , even for nonmixing data: namely, that the former consist of asymptotically Gaussian, weakly dependent random vectors. This is a major practical feature, e.g., in the design of tests for the equality of selfsimilarity parameters, which is of significant interest in many applications [27], [33].…”