On the Behavior of MEMD in Presence of Multivariate Fractional Gaussian Noise

Abstract

Multivariate empirical mode decomposition (MEMD) has been introduced to make standard EMD suitable for direct multichannel signals processing. Unlike EMD, MEMD is able to align sifted intrinsic mode functions (IMFs) from multiple data channels. The aim of this work is to analyze the behavior of MEMD under multivariate fGn (MfGn) with different Hurst exponents, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula> , and strengths of link between pairs of channels, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> , of each sifted IMF. We report results supporting the claim that, regardless of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> values and for both MfGns long-range and short-range dependent, MEMD acts as filter bank on each channel of the input multivariate signal. Whatever the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula> values, this equivalent filter bank structure is dyadic with constant-Q band-pass filters. The observed self-similar filter bank structure leads to a deeper statistical studies of the variance distribution and zero-crossings alignment in order to express this self-similarity in terms of spectral densities of multidimensional IMFs. These statistical properties generalize what was previously conducted for EMD to MEMD and estimation strategy of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula> exponent is proposed. The filter bank behavior of MEMD is illustrated on real turbulent flow data and the estimated <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula> exponent brings out the long-range-dependent nature of the turbulent flow data. An application to EEG data is also proposed.