Methods of Analysis

Lesson

Introduction

Empirical Modal Decomposition (EMD) is an analysis method for non-stationary and non-linear time series which was proposed by Huang et al, [1998]^[1]. It is applied in different domains: acoustics, economics, cardiology, biology, turbulence...etc. The method principle is to decompose the time series into a sum of various "modes" or Intrinsic Mode Functions (IMFs). The EMD is frequently associated to the Hilbert Transform in order to calculate the spectral density for each separated IMF. Before starting with the EMD principle, we introduce the Hilbert spectra analysis. The Hilbert transform of function :

$\hat{I}(t) = \frac{1}{\pi} \int \frac{I(t^{'})}{t-t^{'}} dt^{'}$

Then, for each mode one can reconstructs the signal such as:

$I(t) = A(t) e^{i \theta (t)}$

Where and represent the amplitude and the phase function respectively, they are defined as:

$\left \{ \begin{array}{l} A(t)= \sqrt{\tilde{I}^{2}(t) + I^{2}(t)}\\ \theta(t)= tan^{-1}(\dfrac{\tilde{I}(t)}{I(t)}) \end{array} \right.$

The time series can be reconstructed using the sum of all IMFs:

$x(t) = real \sum \limits^{N}_{j=1} A_j (t) e^{i \theta_j (t)} + r(t)$

[Huang et al, 1998]
Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., ... & Liu, H. H. (1998, March). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 454, No. 1971, pp. 903-995). The Royal Society.

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