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 non-stationary 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 non-stationary basis decomposition methods. Some such techniques are functional Independent Component Analysis (ICA), Empirical Fourier Decomposition (EFD) (non-stationary extension of the Short-Time Fourier Transform (STFT)), Empirical Wavelet Transform (EWT) (non-stationary extension of Morlet Wavelet Transform (MWT)), and Singular Spectrum Decomposition (SSD) (non-stationary 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: Time Series Analysis, Empirical Mode Decomposition, Fourier Analysis, Wavelet Analysis, Independent Component Analysis, X11, Non-Stationary, Graduation, Signal Decomposition

JEL Classification: C02, C14, C22, C32, C38