Hilbert Huang Transform(HHT ) & Empirical Mode Decomposition (EMD)

1. Hilbert Huang Transform(HHT)&Empirical Mode Decomposition(EMD)

2. What is HHT??? • An algorithm for analyzing the data obtained from non-linear and non stationary systems • Decomposes signal into “Intrinsic Mode Functions” • Obtains “Instantaneous frequency” (not used in our project)

3. Hilbert Huang Transform: Need Traditional methods, e.g. Fourier Integral Transform, Fast Fourier Transform (FFT) and Wavelet Transform have a strong priori assumption that the signals being processed should be linear and/or stationary. They are actually not suitable for nonlinear and non-stationary, the signals encountered in practical engineering.

4. Intrinsic Mode Functions(IMF) Formal Definition:Any function with the same number of extrema and zero crossings, with its envelopes being symmetric with respect to zero • Counterpart to simple harmonic function • Variable amplitude and frequency along the time axis

6. Two Steps of HHT: • Empirical Mode Decomposition (Sifting) • Hilbert Spectrum Analysis

7. Empirical Mode Decomposition:Assumptions • Data consists of different simple intrinsic modes of oscillations • Each simple mode (linear or non linear) represents a simple oscillations • Oscillation will also be symmetric with respect to the local mean

8. Sifting Process Explained

15. Algorithm • Between each successive pair of zero crossings, identify a local extremum in the test data. • Connect all the local maxima by a cubic spline line as the upper envelope. • Repeat the procedure for the local minima to produce the lower envelope. Continued…..

16. Sifting……..continued • Calculate mean of the local and upper minima • Subtract this mean from the data set • Take h1 as data set and repeat above procedure till hi satisfies the criteria of IMF, say Ci • We take Ri=X(t)-Ci and repeat the above steps to find further IMF using Ri as the data set. • Finally Ri becomes monotonic function from which we no IMF can further be obtained.

17. Stoppage Criteria • Limit on SDk • S Number: The number of consecutive siftings when the numbers of zero-crossings and extrema are equal or at most differing by one.

18. Comparative Study

19. Advantages of EMD in Financial Prediction • Reduction in noise • More choices in training the neural network

20. Drawbacks • Less Robust System • Restricted use of time-series neural network • Longer Computational Time

21. Related mathematical problems • Adaptive data analysis methodology in general • Nonlinear system identification methods • Prediction problem for nonstationary processes • Spline problems

22. References • Introduction to the Hilbert Huang Transform and its related mathematical problems by Nordan E. Huang