# Nonstationary and Nonlinear Time Series Analysis

### Pages to are hidden for

"Nonstationary and Nonlinear Time Series Analysis"

```					THE EMPIRICAL MODE DECOMPOSITION 1
By Manuel D. Ortigueira and Raul T. Rato

UNINOVA

INTRODUCTION
The Empirical Mode Decomposition (EMD) is an alternative approach to the analysis of non stationary and non linear signals. The EMD is the base of a method called “Hilbert-Huang Transform (HHT)” proposed by Norden Huang and al [1,2]. The HHT is a two step algorithm composed of a) Empirical Mode Decomposition b) Hilbert Spectral Analysis The first decomposes the original signal on a set of elemental signals called “intrinsic mode functions (IMF)” and the second perform a spectral analysis using the Hilbert Transform followed by an instant frequency computation. We will describe both steps and make a brief analysis of their advantages and drawbacks. I.1 IMF computation The EMD procedure allows us to obtain a decomposition of the original signal into a set of amplitude+frequency modulated sinusoids, the Intrinsic Mode Functions: ϕi(t)=Ai(t).cos[θi(t)] . Each IMF is a signal such that: a) the number of extrema and the number of zero-crossings are either equal or differ by one b) at any point the mean value of the envelopes defined by the maxima and minima is zero. The IMF’s are computed by an iterative algorithm called sifting. We are going to describe it.

1

Biopattern NoE sub-projects 16 and 21 meeting, Chania, Crete, June 25 to 27, 2004.

Let s(t) be a given signal. The sifting evolves according to the steps [1,2]: a) Find the all maxima and minima. b) Compute the corresponding interpolating signals M(t) and m(t) c) Let e(t) = M(t)+m(t) 2

d) Subtract e(t) from s(t): s(t)-e(t) ⇒ s(t) e) Return to a) – stop when e(t) remains nearly unchanged. Once we obtain a IMF, ϕi(t), we remove it from the signal s(t) - ϕi(t) ⇒ s(t) and return to a), if s(t) is not a constant or a trend. Otherwise, stop. In figure 1, we can see an ECG beat and in figure 2 a step in the IMF computation.

10

5

0

-5

-10

-15

90

100

110

120

130

140

150

160

170

fig.1- a set in sifting: signal envelopes and mean

The set of the IMF, if properly computed, represents a base of the space. In fact, if the computation is well done, the IMF’s are linearly independent. On the other hand, the procedure lasts while there is a signal that is not a constant or a trend. This can be detected by counting the number of maxima and minima. If there is only one maximum and a minimum (eventually, equal) we stop the algorithm. So, the signal can be written as

s(t) = ∑ Ai(t).cos[θi(t)] + f(t)
i=1

N

where f(t) is the trend. We can conclude that this base is complete. On the other hand, the EMD can be used in segments of the signal (even with overlap), we obtain a local decomposition and we can consider the algorithm as adaptive. We must remark the IMF’s are not necessarily orthogonal (in general, they are not), but this is not very important. To finish we must remark that the algorithm • is simple • appears naturally • does not assume anything about the signal, mainly stationarity • can be applied to a wide class of signals • allows a clear separation of mean and/or trend component and eventually noise but it has some drawbacks: Envelope computations: They are obtained through the use of a spline of order 3 In some situations the envelopes “cut” the signal Stopping criterion: somehow arbitrary

I.2 Examples In the following figures we present the decomposition of an ECG beat (left). On the right, we present the synthesis of the signal from the IMF’s. We present results for high and low signal to noise ratia. It is interesting to observe that the noise affects essentially the highest frequency IMF.

fig.2- one beat – high SNR

fig.3- one beat – low SNR

In figure 4, we present the results of using 2 beats to see if there is a “visible” similarity between the IMF’s obtained here with those obtained in figure 2. As it can be seen, this similarity is clear. This may be important in a sequential adaptive implementation.

fig.4- two beats – high SNR

fig.5- previous example with changed scales In figure 5, we present the example of figure 4 with a change in the scale of the plot to observe better. In the following figures we present the decompositions of the EP’s corresponding to channels 1 to 3 of trial 31. As it is clear, there is a clear similarity between the IMF’s obtained for channels 2 and 3.

5 0 -5 20 0 -2 50 0 -5 50 0 -5 20 0 -2 20 0 -2 0 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400

1 0 -1 20 0 -2 20 0 -2 10 0 -1 10 0 -1 20 0 -2 0 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400

fig.6 – IMF’s of channel 1 in trial 31

100 50 0 50 0 0 -50 50 0 0 -50 50 0 0 -50 20 0 0 -20 0 200 400 200 400 200 400 200 400 200 400

20 0 -20 10 0 0 -10 10 0 0 -10 50 0 -5 0 200 400 200 400 200 400 200 400

fig.7 – IMF’s of channel 2 in trial 31

100 50 0 20 0 0 -20 20 0 0 -20 10 0 0 -10 20 0 0 -20 0 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400

20 0 -20 10 0 0 -10 20 0 0 -20 10 0 0 -10 0 100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400

fig.8 – IMF’s of channel 3 in trial 31

I.3 Hilbert Spectral Analysis ^ 1 - Computation of the Hilbert Transform of each IMF ϕi(t) obtaining the analytical signal: ^ z(t) = ϕi(t) + jϕi(t) jθi(t) = Ai(t).e 2 – Determination of the instantaneous frequency and amplitude: dθi(t) ωi(t) = dt and ^ ϕi(t)   with θi(t) = arctan  ϕi(t) 

Ai(t) =
• •

^ [ϕi(t)]2+ [ϕi(t) ]2
The Hilbert Transform computed over a very small segment of a signal does not have any special meaning We do not guarantee a positive instantaneous frequency, since we are working with a small segment of the signal.

This algorithm is not very useful since:

To avoid these problems we are implementing a two step algorithm:

a) Demodulate each IMF producing amplitude and phase signals b) Create a model (eg ARMA) for each one or perform a high resolution sequential spectral estimation. The modelling is interesting since it allows us to establish relations among IMF’s from different channels.

FINAL CONSIDERATIONS
The algorithm we just presented allows us to decompose a given segment of a signal into simple components: the IMF, that are modulated sinusoids. We proposed to demodulate them and to study the amplitude and phase separately, instead of the original Hilbert transform computation. We are devising algorithms based on sequential spectral estimation or modelling. We must refer that this algorithm must not be considered as an alternative to ICA or similar. Rather, it must be considered as a complement. In fact it may be suitable for studying each signal source after ICA decomposition.

References [1] Huang NE, Shen Z, Long SR, Wu MC, Shih EH, Zheng Q, Tung CC, Liu HH, “The empirical mode decomposition and the Hilbert Spectrum for nonlinear and non-stationary time series analysis”, Proceedings of the Royal Society London 1998; A454:903–995. [2] Huang,NE, Wu MC, Qu W, Long, SR, Shen SSP and Zhang JE, “Applications of Hilbert–Huang transform to non-stationary financial time series analysis”, Applied Stochastic Models in Business and Industry, Appl. Stochastic Models Bus. Ind., 2003; 19:361

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 101 posted: 1/7/2010 language: Galician pages: 9