Empirical mode decomposition - Introduction

Empirical mode decomposition (EMD) is a data-driven decomposition method and was originally proposed by Huang et. al in 1998 [1]. Since that time the method has gained a lot of attention in the science community. EMD has been applied in a wide range of different fields, including geophysics, biomedicine, neuroscience, finance and many more.

The method is defined by an algorithm as follows:

1. Identify all local extrema (both minima and maxima) in input signal $s(t)$.
2. If the number of extrema is less or equal 2 then $s(t)$ is considered as a trend ($r(t) := s(t)$ ) --- finish with one component.
3. Estimate top (env_max) and bottom(env_min) envelopes by interpolating respectively local maxima and local minima with natural cubic splines.
4. Calculate local mean (mean of both envelopes) --- $m(t) = 0.5(env_{max} + env_{min})$.
5. Subtract the mean from the input signal $h(t) = s(t) - m(t)$ .
6. If $h_j$ fulfills the stopping criteria, then it is considered an intrinsic mode function (IMF) (a component $c_j(t)$ ) and algorithm starts again from step 1 for a signal $s(t) := m(t)$. Otherwise, it starts with $s(t) := h(t)$.

Empirical evidence is that the algorithm converges to finite number of IMFs, with which input signal can be reconstructed :

$$s(t) = \sum_n^N c_i (t) + r(t)$$

where N is the number of components.

[1] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 454, pp. 903-995, 1998.