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

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.