# Multivariate Hadamard self-similarity: testing fractal connectivity

## TL;DR

This paper introduces Hadamard fractional Brownian motion to model multivariate signals with scale invariance but varying inter-component fractal connectivity, proposing wavelet-based estimators and tests validated through simulations.

## Contribution

It presents a new class of multivariate Gaussian processes and develops wavelet-based methods for estimating and testing fractal connectivity in multivariate signals.

## Key findings

- Wavelet-based estimator for Hurst matrix is asymptotically normal.
- Finite sample procedures for confidence intervals and testing are effective.
- Simulation validates the estimation and testing methods across parameters.

## 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.