Hurst multimodality detection based on large wavelet random matrices

Abstract

In the modern world, systems are routinely monitored by multiple sensors, generating "Big Data" in the form of a large collection of time series. In this paper, we put forward a statistical methodology for detecting multimodality in the distribution of Hurst exponents in high-dimensional fractal systems. The methodology relies on the analysis of the distribution of the log-eigenvalues of large wavelet random matrices. Depending on the presence of a single or many Hurst exponents, we show that the wavelet empirical log-spectral distribution displays one or many modes, respectively, in the threefold limit as dimension, sample size and scale go to infinity. This allows for the construction of a unimodality test for the Hurst exponent distribution. Monte Carlo simulations show that the proposed methodology attains satisfactory power for realistic sample sizes.