Feature extraction

In order to carry out classification of the time signals, we need a feauture vector characterizing each signal. It goes without saying, that a feature vector should have considerably less elements than the number of the samples in time domain. Wavelet transformation is proved to be a succesful candidate for this data compression process Goswami and Chan [1999]. Figure 1. shows a typical time signal:

Figure 1.  A typical time signal

Employing Daubechies filter of order two for the (2^n) samples of the time signal, discrete wavelet transformation (DWT) decomposes the signal into n resolution levels, into one approximation subband and n-1 wavelet subbands. In our case n = 16, see the phase space plot of DWT of the time signal, see Figure 2:

Figure 2. The phase space plot of the DWT of the time signal

In the phase space plot, each rectangle represents a wavelet (vertical direction) and the darker a rectangle, the larger the absolute value of the corresponding wavelet coefficient.The residual trend is plotted in the bottom row, the finest detail, in the top row.
Preliminary investigations proved that  the average energy content of the coefficients at higher resolutions is negledgable, see  Figure 2, therefore the two coefficients of the approximation subband were considered as feature vector components. In addition the different values of the wavlet coefficients (represented by different gray levels ), i.e. at the resolution level six, show that the frequency is time dependent, revealing the certain instacionar nature of the signal.

Verification of the feature representation

In order to verify that the choosen feature vector is an acceptable representation of  the time signals  unsupervised classification via neural network was carried out for the 20 signals resulted form the measurements of right side of the brain,  belonging to the two different classes (normal state and state effected by drug injection).

Figure 3. Unsupervised classification of signals of right brain side

Only two signals being  very close to the separation surface, see Figure 3. were misclassified. The same classification was carried out for the signals of the left side of the brain, see Figure 4.

Figure 4. Unsupervised classification of signals of left brain side

Now, again two signals - clearly seen in neigbourhood  of the separation surface- were misclassified. In addition it turned out, that  signals of the right and left side of the brain do not differ from each other considerably, only a narrow band contains misclassified signals, see Figure 5.:

Figure 5. Unsupervised classification of signals of right and left brain side

These results show that the applied feature representation can be accepted, especially considering that these signals were resulted from direct measurements of different clinical experiments.

ANN model for signal classification

In order to develop a model for classifying measured clinical signals, we employed a supervised, backpropagation neural network with sigmoid activation functions. The network has  one input and two output nodes, and one hidden layer with two nodes. The teaching set consists of  the 75% of the total number of the signals (30 signals) and rest served for testing (10 signals). 
The network learned the teaching set, however one inconsistent signal was recovered, see Figure 6.

Figure 6. Supervised classification with backpropagation network having two hidden nodes

More than two hidden nodes resulted  overlearning, see Figure 7. This figure shows, that the neural network tried to learn the outlier signal, two.

Figure 7. Supervised classification with backpropagation network having two hidden nodes

Our ANN model for signal classification can be expressed in analytical form,  thanks to the symbolic algebra built in Mathematica kernel:

SignalClassifier[{w1,w2}]=

\({\(-272.3220422482035`\) + 
        272.42209692625585`\/\(1 + e\^\(\(-33.98109110103116`\) + 176.4155619005012`\ w1 - 97.6413643468084`\ w2\)\)
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  + 
        273.155288144691`\/\(1 + e\^\(\(\(34.0372652504046`\)\(\)\) - 176.73608430965223`\ w1 + 97.81064305502265`\ w2\)\), 
273.3220422482036` - 
        272.42209692625596`\/\(1 + e\^\(\(-33.98109110103116`\) + 176.4155619005012`\ w1 - 97.6413643468084`\ w2\)\)
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \  - 
        273.1552881446911`\/\(1 + e\^\(\(\(34.0372652504046`\)\(\)\) - 176.73608430965223`\ w1 + 97.81064305502265`\ w2\)\)}\)

where w1 and w2 are the wavelet coefficients of the approximation subband of DWT of the time signal. This function, after rounding , gives {1, 0} and  {0, 1}as output for the two classes, respectively.
Employing this model only  two signals were misclassified from the 10 elements of the testing set having not been involved in the teaching process.

Summary

For clinical time signal classification, an ANN model based on supervised, backpropagation network was developed. Feature extraction of the time signal having (2^16)  samples at 200 Hz sampling rate, was carried out with discrete wavelet transformation using Daubechies filter of order two. Because of the dominating  average energy content of the approximation subband coefficients, these two coefficients were considered as a feature vector for the signals. Unsupervised classification showed that this feature vector is an acceptable representation of the time signals. It also turned out that the difference between normal signal and a signal indicating drug injection effect is significant, and  much more dominant than the difference between signals of the right and left brain sides. 
Avoiding overlearning, a simple ANN model with two hidden nodes was suggested for classification of  the signals, and using symbolic algebra, the analytical form of this model could be given.
 In the future, having more measurements, further investigations would be necessary to examine other feature extraction methods as well as ANN models for the classifcation. However, these clinical experiments are very expensive, therefore physiologycal modelling and simulations would be also needed.
 For the computations Mathematica 5. and its Neural Networks Application were used.

Reference

Goswami, J.C and Chan, A.K. (1999) Fundamentals of Wavelets, John Wiley, New York.