2017/01/16 by Herwig Wendt, Gustavo Didier, Sébastien Combrexelle +1 · 24 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #Algorithm #Applied mathematics #Artificial intelligence #Brownian motion #Complex Systems and Time Series Analysis #Computer science #Estimator #Financial Risk and Volatility Modeling #Fractal #Fractional Brownian motion #Geometry #Hadamard transform #Hurst exponent #Mathematical analysis #Mathematics #Monte Carlo method #Multivariate statistics #Scaling #Statistical hypothesis testing #Statistical physics #Statistics #Time Series Analysis and Forecasting #Wavelet #math.ST #stat.TH
paper · pdf · doi:10.1016/j.physd.2017.07.001
published in Physica D Nonlinear Phenomena 356-357, 1-36 (Elsevier BV)
arxiv created 2017/01/16 · openalex publication_date 2017/07/16 · arxiv updated 2017/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
While scale invariance is commonly observed in each component of real world multivariate signals, it is also often the case that the inter-component correlation structure is not fractally connected, i.e., its scaling behavior is not determined by that of the individual components. To model this situation in a versatile manner, we introduce a class of multivariate Gaussian stochastic processes called Hadamard fractional Brownian motion (HfBm). Its theoretical study sheds light on the issues raised by the joint requirement of entry-wise scaling and departures from fractal connectivity. An asymptotically normal wavelet-based estimator for its scaling parameter, called the Hurst matrix, is proposed, as well as asymptotically valid confidence intervals. The latter are accompanied by original finite sample procedures for computing confidence intervals and testing fractal connectivity from one single and finite size observation. Monte Carlo simulation studies are used to assess the estimation performance as a function of the (finite) sample size, and to quantify the impact of omitting wavelet cross-correlation terms. The simulation studies are shown to validate the use of approximate confidence intervals, together with the significance level and power of the fractal connectivity test. The test performance and properties are further studied as functions of the HfBm parameters.