Tutorial on Empirical Mode Decomposition: Basis Decomposition and Frequency Adaptive Graduation in Non-Stationary Time Series

Abstract

This tutorial explores the class of non-parametric time series basis decomposition methods particularly suited for nonstationary time series known as Empirical Mode Decomposition (EMD). In outlining a statistical perspective of the EMD method, it will be contrasted and combined (for the betterment of both methods) with other existing nonstationary basis decomposition methods. Some such techniques are functional Independent Component Analysis (ICA), Empirical Fourier Decomposition (EFD) (nonstationary extension of the Short-Time Fourier Transform (STFT), Empirical Wavelet Transform (EWT) (nonstationary extension of Morlet Wavelet Transform (MWT)), and Singular Spectrum Decomposition (SSD) (nonstationary extension and refinement of Singular Spectrum Analysis (SSA)). A detailed review of this time series basis decomposition approach is presented that explores 3 core aspects for a statistical audience: 1) the basis functions (Intrinsic Mode Functions (IMFs)) representation and estimation methods including robustness and optimal spline representations including smoothing and knot placements; 2) the computational and numerical robustness of various aspects of the iterative algorithmic design for EMD basis extraction, including treating carefully boundary effects; and 3) the first attempt at a population-based characterisation of EMD that provides a novel stochastic embedding of the EMD method within a stochastic model framework. Furthermore, the basis representations considered will be connected to local frequency graduation smoothing methods, demonstrating that these can be adapted to a local frequency adaptive framework within the EMD context. This will provide new practical insights into the interface between time series basis decomposition and graduation-smoothed representations.

Keywords

• Empirical mode decomposition
• fourier analysis
• graduation
• independent component analysis
• non-stationary
• signal decomposition