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Wavelet-Based Detection and Estimation of Fractional Lévy Signals in High Dimensions

In the modern world, systems are routinely monitored by multiple sensors, generating “Big Data” in the form of a large collection of time series. However, dynamic signals are often low-dimensional and characterized by joint scale-free dynamics (self-similarity) and non-Gaussianity. In this paper, we put forward a statistical methodology for identifying the number of multivariate self-similar, Lévy-driven components immersed in high-dimensional noise, as well as for estimating the underlying scaling exponents. It relies on the analysis of the evolution over scales of the eigenvalues of random wavelet matrices. Monte Carlo simulations show that the proposed methodology is accurate for realistic sample sizes. This holds even at low signal-to-noise ratios and for a large number of observed mixed and noisy time series. The mathematical framework further allows us to analyze the impact of the tails of the Lévy noise marginal distribution on the estimation performance.