Identification of the Multivariate Fractional Brownian Motion

Abstract

This paper deals with the identification of the multivariate fractional Brownian motion, a recently developed extension of the fractional Brownian motion to the multivariate case. This process is a pmultivariate self-similar Gaussian process parameterized by p different Hurst exponents Hi, p scaling coefficients σi (of each component) and also by p(p − 1) coefficients ρij, ηij (for i, j = 1, . . . , p with j > i) allowing two components to be more or less strongly correlated and allowing the process to be time reversible or not. We investigate the use of discrete filtering techniques to estimate jointly or separately the different parameters and prove the efficiency of the methodology with a simulation study and the derivation of asymptotic results.

Interplay between functional connectivity and scale-free dynamics in intrinsic fMRI networks

“…However, to assess functional connectivity, it is needed that a collection of time series each associated with a different region of interest, are studied jointly (or simultaneously), to measure for instance how they correlate one to another. It is thus natural to make use of a model inspired from the multivariate extension of fGn (mfGn), proposed e.g., in [69, 70]. In essence, this model assumes joint Gaussianity for the time series and power-law behaviors both for the auto- and cross-spectra, across a large range of frequencies.…”

Mixed-correlated ARFIMA processes for power-law cross-correlations

“…The construction of the multivariate ARFIMA process implies that the bivariate Hurst exponent is the average of the separate Hurst exponents [35]. The same property holds for the fractional Brownian motion [36]. The long-range cross-correlations thus simply arise from the specification of these processes.…”

Multifractional Vector Brownian Motions, Their Decompositions, and Generalizations

“…Recently, the fractional Brownian motion is extended to a multivariate framework, with the necessary condition of the corresponding covariance matrix structure being derived in [34]. Basic properties of the multivariate fractional Brownian motion are reviewed and extended in [3,15], its identification is investigated in [2], and its wavelet analysis is performed in [16]. A couple of examples of fractional vector Brownian motions in R d are given in [22], with a general format based on a scalar variogram (structure function).…”

“…The multifractional vector Brownian motion we propose has the direct and cross covariances of the form (5), and enjoys an orthogonal decomposition like (2). More precisely, we introduce three types of covariance matrix structures for Gaussian or elliptically contoured vector random fields in space and/or time, which are the generalized forms of (multi-, bi-, tri-)fractional vector Brownian motions and related stochastic vector processes.…”