Hilbert Huang Transform(HHT)&

Empirical Mode Decomposition(EMD)

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)

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.

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 functionVariable amplitude and frequency along the

time axis

Two Steps of HHT:

Empirical Mode Decomposition (Sifting)

Hilbert Spectrum Analysis

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

Sifting Process Explained

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…..

Sifting……..continuedCalculate mean of the local and upper

minimaSubtract 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.

Stoppage CriteriaLimit 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.

Comparative Study

Advantages of EMD in Financial Prediction

Reduction in noise

More choices in training the neural network

Drawbacks

Less Robust System

Restricted use of time-series neural network

Longer Computational Time

Related mathematical problemsAdaptive data analysis methodology in

general

Nonlinear system identification methods

Prediction problem for nonstationary

processes

Spline problems

References

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