Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis

Abstract

Abstract The Fourier transform (or spectral analysis) has become a universal tool for data analysis in many different real-world applications, notably for the characterization of temporal/spatial dynamics in data. The wavelet transform (or multiscale analysis) can be regarded as tailoring spectral estimation to classes of signals or functions defined by scale-free dynamics. The present contribution first formally reviews these connections in the context of multivariate stationary processes, and second details the ability of the wavelet transform to extend multivariate scale-free temporal dynamics analysis beyond second-order statistics (Fourier spectrum and autocovariance function) to multivariate self-similarity and multivariate multifractality. Illustrations and qualitative discussions of the relevance of scale-free dynamics for macroscopic brain activity description using MEG data are proposed.