An introduction to Empirical Mode Decomposition

1. An introduction to Empirical Mode Decomposition

2. The simplest model for a signal is given by circular functions of the type (Type I) Such “Fourier modes” are of particular interest in the case of stationary signals and linear systems

3. However, many physical situations are known to undergo nonstationary and/or nonlinear behaviors we can think of representing these signals in terms of amplitude and frequency modulated (AM–FM) components (Type II) The rationale for such a modelling is to compactly encode possible nonstationarities in a time variation of the amplitudes and frequencies of Fourier-likemodes.

4. More generally, signals may also be generated by nonlinear systems for which oscillations are not necessarily associated with circular functions, thus suggesting decompositions of the following form (Type III) Each of the components has to have physical and mathematical meaning.

5. Empirical Mode Decomposition (EMD) is designed primarily for obtaining representations of Type II or TypeIII in the case of signals which are oscillatory, possibly nonstationary or generated by a nonlinear system, in some automatic, fully data-driven way.

6. The starting point of EMD is to consider oscillatory signals at the level of their local oscillations and to formalize the idea that: “signal = fast oscillations superimposed to slow oscillations”

7. “signal = fast oscillations superimposed to slow oscillations” Iterate on the slow oscillations component considered as a new signal.

9. Empirical Mode Decomposition (EMD) Decomposing a complicated set of data into a finite number of Intrinsic Mode Functions (IMF), that admit well behaved Hilbert Transforms. Intrinsic Mode Functions (IMF) 1. In the whole set of data, the numbers of local extrema and the numbers of zero crossings must be equal or differ by 1 at most. 2. At any time point, the mean value of the “upper envelope” (defined by the local Maxima) and the “lower envelope” (defined by the local minima) must be zero.

11. How to find one Intrinsic Mode Functions of a signal? Sifting procedure 1. identify all extrema of x(t). 2. Interpolate the local maxima to form an upper envelope u(x). 3. Interpolate the local minima to form an lower envelope l(x). 4. Calculate the mean envelope: m(t)=[u(x)+l(x)]/2. 5. Extract the mean from the signal: h(t)=x(t)-m(t) 6. Check whether h(t) satisfies the IMF condition. YES: h(t) is an IMF, stop sifting. NO: let x(t)=h(t), keep sifting.

12. An example Refer to emd_Flandrin.ppt

13. By construction, the number of extrema decreases when going from one residual to the next, thus guaranteeing that the complete decomposition is achieved in a finite number of steps. “Residual” shows overall trend in data.

14. Another example (puts emphasis on the potentially “nonharmonic” nature of EMD) • the signal is composed of • a “high frequency” triangular waveform whose amplitude is slowly (linearly) growing. • a “middle frequency”sine wave whose amplitude is quickly (linearly) decaying • a “low frequency” triangular waveform Both linear and nonlinear oscillations are effectively identified and separated by EMD. whereas any “harmonic” analysis (Fourier, wavelets, . . . ) would end up in the same context with a much less compact and physically less meaningful decomposition.

15. Some remarks on Huang’s EMD Rationale — Intuitive, simple, local and fully data-driven. Still lacks from solid theoretical grounds No analytic definition — The decomposition is only defined as the output of an algorithm

16. Applications of EMD Acoustics, noise, and vibration: Machine vibration analysis, Speech/sound analysis Environmental: Oceanography, Earthquake engineering , Water and wind dynamics Industrial: Machine monitoring and failure prediction ……. http://tco.gsfc.nasa.gov/HHT/

17. Bibliography • P. FLANDRIN, P. GONCALVES, 2004 :  "Empirical Mode Decompositions as Data-Driven Wavelet-Like Expansions," Int. J. of Wavelets, Multires. and Info. Proc., Vol. 2, No. 4, pp. 477-496.

18. Thank you! Questions or Suggestions? Email to: qinwu@math.wvu.edu