Title: Multivariate Hadamard self-similarity: testing fractal connectivity,
Abstract: 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.