^Department of Clinical Neurophysiology, Charing Cross Hospital, Charing Cross & Westminster Medical School, University of

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1 Relation between singular values and graph dimensions of deterministic epileptiform EEG signals V. Cabukovski,* N. de M. Rudolph,* N. Mahmood* ^Institute of Informatics, Faculty of Sciences, University of Skopje, ^Department of Clinical Neurophysiology, Charing Cross Hospital, Charing Cross & Westminster Medical School, University of Abstract Computerised detection and prediction of epileptic discharges from EEG data is a problem whose solution may lead to the prediction of epileptic seizures and planning of treatment. The recently confirmed fact that the EEG has a fractal nature enables a new approach to analysis of epilepsy. A conventional signal processing approach is not appropriate for highly complex signals, such as chaotic deterministic signals including epileptiform EEG. In our previous work we found that the graph dimension is the most appropriate measure for real-time fractal dimension estimation of EEG signals, and that it could be used for differentiation between parts of EEG signals with and without epileptic discharges. However, we also found that this measure is especially sensitive to signal artefacts and signal noise. In this paper we present the results of our search for a more robust measure for differentiation. The mean singular value is our new quantitative measure which in combination with the graph dimension can be used for differentiation of different EEG states (with and without epileptic discharges). Introduction Several recent studies have shown that EEG signals may exhibit chaotic behaviour with complex dynamics (Babloyantz [1], Layne et al. [2], Babloyantz and Destexhe [3], Pijn et al. [4], Pijn and Lopes da Silva [5], Jinghua and Xiang-bao [6]). The fact that the fractal dimension of the EEG signal is not very big (4 to 8 for different normal brain states and 2 to 4 for some pathological states) (Babloyantz [11, Babloyantz and Destexhe [3], Jinghua and Xiang-bao [6])

540 Computer Simulations in Biomedicine confirms that modelling of the EEC by a deterministic dynamical system with a finite number of variables may be possible.

2 540 Computer Simulations in Biomedicine confirms that modelling of the EEC by a deterministic dynamical system with a finite number of variables may be possible. We attained successful modelling of epileptiform EEG by a non-linear logistic expression. Phase portrait and graph dimension measurement methods when applied to data derived from this model agreed with those from actual EEG data of epileptic patients, confirming that the model indeed represents the EEG of such patients (Rudolf and Mahmood [7,8]). Analysing sleep recordings by plotting Y vs a (a is a parameter from the non-linear logistic expression and Y the amplitude of the signal), we obtained regular shapes for normal parts of the EEG data and marked differences in the shapes for parts of the signal with epileptic discharges (Rudolf, Cabukovski and Mahmood [9]). Measuring the fractal dimensions of consecutive 1.5 or 4 second epochs, we have observed a high-dimension period beforehand with a fall at or just before the onset of a continued epileptic discharge (Rudolf and Mahmood [10]). We concluded that differentiation between the normal part of the EEG signal and the part with epileptic discharges is possible. In our further experiments we applied several methods of fractal dimension measurement and concluded that the most reasonable of them is the little known Higuchi graph method, both in terms of computation time and sample size. This method (Higuchi [11]) gives a good approximation to the fractal dimension, using the length of the irregular curve, from a small number of sampled points. It is applicable to EEG data sets and is a better solution for fractal dimension estimation in real time EEG data analysis (Cabukovski, Rudolf and Mahmood [12]) than the capacity dimension, the correlation dimension, the information dimension and Lyapunov exponents. Unfortunately, there is an inherent limitation that dimensions above a value of two cannot be measured and the graph dimension is too sensitive to signal artefacts and inherent signal noise. These limitations on having a suitable quantitative measure led us to look for other forms of data representation such as phase portraits and singular values of the signal (Mahmood, Rudolf, Cabukovski [13]). Many of these results were obtained with a program for fractal analysis of EEG signals which we have developed for our research purposes (Cabukovski, Mahmood and Rudolf [14]). Methods and material We measured the graph dimension of consecutive overlapping small epochs (1 second) from the EEG signal using the Higuchi graph method which gives a good approximation to the fractal dimension from a small number of points (Higuchi [11]). It uses the length of the irregular plane curve, but we applied it for consecutive small trajectory parts of the n-dimensional EEG at tractor. The EEG signal attractor can be obtained by embedding the data from the original EEG signal into an n-dimensional space (Packard et al. [15]). Practically, our modification of the Higuchi graph method is a real-time application of this met-

$Computer Simulations in Biomedicine 541 hod for continual estimation of the fractal dimension from short EEC signal epochs.$

3 Computer Simulations in Biomedicine 541 hod for continual estimation of the fractal dimension from short EEC signal epochs. This provides fine temporal resolution of changes in the EEG fractal dimension and it can be presented as a graph dimension time curve. The Higuchi graph method has an inherent limitation: fractal dimensions above a value of two cannot be measured. We found that for sets with fractal dimension below two this method gives better results, both in terms of computation time and sample size, compared with the capacity dimension, the correlation dimension, the information dimension and Lyapunov exponents, the usually used methods for fractal dimension estimation (Farmer, Ott and Yorke [16]). We also found that the estimated graph dimension time curve could be used for differentiation of different EEG states (with and without epileptic discharges), but it is very sensitive to signal artefacts and inherent signal noise (Cabukovski, Mahmood and Rudolf [12]). Singular value decomposition (SVD), usual in signal processing, is an accurate way of signal data compression, which is achieved by a linear mapping of the signal's representation vectors from the measurement space into a low dimensional space (Cohen [17]).We used this technique to calculate an optimal basis for the projection of the attractor reconstructed from the EEG signal. We found that after applying the Karhunen-Loeve Expansion (KLE) (see Kittler [18]) to EEG data and projecting the data on the first eigenplane, the phase portraits drawn with these data produced a clearer and a unique representative image (Mahmood, Rudolf and Cabukovski [13]). This technique removes irrelevant information resulting from artefacts and noise, and extracts representative pattern vectors, effecting maximum data compression. The trajectory vector basis was obtained by employing the embedded dimension method, first used by Packard et al. [15], to constitute each row vector from the data matrix A, and thereafter each row was normalised to make the singular values invariant with respect to amplitudes of the signal and the signal sample sizes. The singular values Sj and orthonormal eigenvectors Xj and YJ of the matrices AA^ and A^A respectively were computed by applying a very fast iterative method often called the power method or Shlien's method (Cohen [16], Shlien [19]). The matrices XjY^ are called eigenplanes or singular planes. The projection of the data matrix on the first eigenplane is calculated by Projection on the first plane = S^X^Y^. The mean singular value is calculated by Mean singular value = Sum of Sj / Number of singular values. The mean singular value was estimated for all consecutive overlapping epochs. The mean singular values obtained from the EEG signal are presented as a mean singular value time curve plotted together with the graph dimension time curve on the same coordinate system. The linear regression model for any estimated pair of points from the mean singular value time curve and the graph dimension time curve was fitted by a least square method (Press et al. [20]). The linear correlation coefficient rasa

4 542 Computer Simulations in Biomedicine quantitative measure of the linear correlation between two sets was also calculated (Press et al. [20]). In order to explore the relation between the graph dimension and the mean singular value and consider the possibility of using the mean singular value as an additional quantitative measure in detection of epileptic discharges, from five epileptic EEG recordings, we constructed five single-channel EEC signals with normal beginning (signal part without epileptic discharges), epileptic continuation (signal part with epileptic discharges) and normal end. These experimental samples were obtained from epileptic patients of different ages, sex, type of epilepsy, type of epileptic discharges, asleep or awake, by taking the clear non-epileptic and epileptic EEG parts and pasting them in one signal, for each patient separately. The durations of the signals are between 46 and 67 seconds with durations of the epileptic discharges between a fraction of a second (very short epileptic discharges) and 38 seconds (long epileptic discharges). The first three signals present spike-and-wave epileptic activities, the fourth signal repetitive sharp wave epileptic activity and the fifth signal iregular spike discharges. All EEG signals had been recorded on an Oxford Medilog recorder, subsequently sampled at 39 or 62 Hz. Results The five EEG signals constructed in the way described above were treated with our program for fractal analysis of EEG signals (Cabukovski, Mahmood and Rudolf [14]) in terms of obtaining the graph dimension time curve, mean singular value time curve, the line which fits these two time series and the corresponding linear correlation coefficient r. In figures 1 to 5, are presented graphically the results from the signals. In each of the figures on the left side are presented the graph dimension time curve and the mean singular value time curve with the epileptic parts marked; on the right side are presented the linear relation between the graph dimensions and the mean singular values, the analytical linear expression and the linear correlation coefficient. It is obvious that the fractal dimensions arid the mean singular values are inversely related, which means that the fractal dimension decreases when the mean singular value increases. In all analysed EEG samples, the data have been fitted to a straight line regression model and the values obtained for the correlation coefficients show how strongly the data are linearly correlated. We noticed some differences in the calculated correlation coefficients which mean that some data are more correlated (data in figures 2 and 3), and some less (data in figures 1 and 4). In figure 5, the smallest value for the linear correlation coefficient is a result of a periodicity in both the graph dimension time curve and the mean singular value time curve. In order to identify the presence of artefacts and noise in the signals we projected the EEG signals separately on the first eigenplane and after that we plotted the new estimated graph dimension time curves as well as the linear fitted models, analytical relations and new linear correlation coefficients. Big

5 Computer Simulations in Biomedicine 543 differences between fractal dimensions before and after projection were not observed. We did find some differences in the new linear correlation coefficients compared with the old values. GD' y=l % r= Graph dimension 1 - Me an singular value Epileptic discharges Figure 1. Graph dimension and mean singular value time curves for patient with atypical petit mat, 10 years old, asleep; Linear correlation between the graph dimensions (GD) and the mean singular values (), r;=-6u9, and rf= after the projection of the signal on the first eigenplane. 3 - GD" Epileptic disch. Figure 2. Graph dimension and mean singular value time curves for patient with petit mal, 22 years old, awake; Linear correlation between the graph dimensions (GD) and the mean singular values (), ^=-0.64, and r2**=-0.60 after the projection of the signal on the first eigenplane. Graph dimension GD y= x r= ean singular value Epileptic discharges Figure 3. Graph dimension and mean singular value time curves for patient with secondary major epilepsy, 17 years old, awake; Linear correlation between the graph dimensions (GD) and the mean singular values (), r^-0.88, and rf=-0.78 after the projection of the signal on the first eigenplane.

6 544 Computer Simulations in Biomedicine 3 -i 2 - Graph Dimension 1 - Mean singular value Epileptic discharges Figure 4. Graph dimension and mean singular value time curves for patient with secondary major epilepsy, 17 years old, awake; Linear correlation between the graph dimensions (GD) and the mean singular values (), r^--oa5, and ^=-0.26 after the projection of the signal on the first eigenplane. 3 -, y-l x r= Graph dimension 1 - Epileptic discharge Mean singular value Figure 5. Graph dimension and mean singular value time curves for patient with temporal lobe epilepsy, 27 years old, asleep; Linear correlation between the graph dimensions (GD) and the mean singular values (), r$=-0.33, and r^=-0.24 after the projection of the signal on the first eigenplane.. Discussion The mean singular value time curves increase in amplitude when an epileptic discharge occurs and the changes in the mean values are evidently greater than the changes in the graph dimensions. This is especially of interest in the situations presented in figures 1 and 4, where the differentiation between the normal and epileptic part of the EEG signal by serial graph dimension estimations is not an easy task. In figure 5 one can observe small graph dimension values from normal as well as from epileptic parts of the signal. Looking at the changes of the linear correlation coefficients calculated before and after the projection of the EEG signals on the first eigenplane, one can observe a very big change for the data presented in figure 1. The linear correlation coefficient before the projection increases in value from to (the data, from being linearly con-elated, become linearly uncorrelated). The reason for this may be the presence of artefacts and/or noise in the analysed EEG signal which have great influence on the graph dimension values.

7 Computer Simulations in Biomedicine 545 The independence of the mean singular value from artefacts and noise is apparent. Small differences in the linear correlation coefficients before (-0.64) and after the projection of the signal (-0.60) in figure 2, and and in figure 3 are the result of the uncontaminated nature of the original EEC signal. In both cases the differentiation is possible with both measures: the graph dimension and mean singular value. The signal from figure 4 has a small quantity of artefacts and noise (the linear correlation coefficient increases from to -0.26) and it is more difficult to differentiate the normal and epileptic parts of the signal using the graph dimensions. In figure 5 one can observe a method for differentiation. Epileptic discharges occur where the mean singular values and graph dimensions approach each other and meet or overlap. The small reduction of the linear correlation coefficient before (-0.33) and after projection (-0.24) alerts one to the small quantity of artefacts and noise in the EEG signal. Conclusion The mean singular value may be a new quantitative measure which gives better differentiation of normal and epileptic parts of the EEG signal than the graph dimension in terms of robustness to noise and artefacts. It is easy to calculate and is applicable to short signal epochs. In combination with the graph dimension, differentiation may be possible in different types of epilepsy. References 1. Babloyantz, A. Evidence of chaotic dynamics of brain activity during the sleep cycle, Phys. Wf. 777 A, , Layne, S. P., Mayer-Kress, G., Holzfuss, J. Problems associated with dimensional analysis of electroencephalogram data, in Dimensions and Entropies in Chaotic Systems: Quantification of Complex Behaviour (ed. G. Mayer-Kress), pp , Springer Verlag, Babloyantz, A., Destexhe, A. Low - Dimensional chaos in an instance of epilepsy, Proc. AW. Acad. & /. C/&4, 83, , Pijn, J.P., Van Neeren, J., Noest, A., Lopes da Silva, F.H. Chaos or noise in EEG signals; dependence on state and brain site, Eleclroencephalography and Clinical Neurophysiologv, 79, , Pijn, J. P., Lopes da Silva, F. II. Chaos analysis, nonlinear associations and delay estimates for the dynamical analysis of the spread of an epileptic seizure, in Mathematical Approaches to Brain Functioning Diagnostics (ed I. Dvorak and A.V. Holden), pp , Manchester University Press, Jinghua, X., Xiang-bao, W. The non-linear dynamics of brain functions: the modelling, reconstruction and characterisation of human EEG data, in Mathematical Approaches to Brain Functioning Diagnostics (ed I. Dvorak and A.V. Holden), pp , Manchester University Press, 1991.

8 546 Computer Simulations in Biomedicine 7. Rudolf, N. de M., Mahmood, N. The role of fractals in modelling epilepsy, /. PhysioL, 438, 343P, Mahmood, N., Rudolf, N. de M. Ongoing dimensional and fractal characterisation of epileptiform EEG, Neurophysiologie Clinique, 22, Supp. 1, 125s, Rudolf, N. de M., Cabukovski, V., Mahmood, N. A fractal approach to detection of epileptic discharges, Electroencephalography and Clinical Neurophysiology, 87(2), S78, Rudolf, N. de M., Mahmood, N. Fractal representation of the epileptiform EEG, in Proceedings of the Second Annual Conference on Non-linear Dynamical Analysis of the EEG, Houston (ed B.H. Jansen and M.E. Brandt), pp , World Scientific, London, Higuchi, T. Approach to an irregular time series on the basis of the fractal theory, Physica JO, 31, , Cabukovski, V., Rudolf, N. de M., Mahmood, N. Measuring the fractal dimension of EEG signals: selection and adaptation of method for real-time analysis, in Computational Biomedicine (ed K.D. Held, C.A. Brebbia, R.D. Ciskowski, and H. Power), pp , Computational Mechanics Publications, Southampton, Boston, Mahmood, N., Rudolf, N. de M., Cabukovski, V. Epileptiform EEG analysis with phase portraits improved by eigen projection, in Proceedings of the Twelfth International Congress of the European Federation for Medical Informatics - MIE '94, Lisbon (ed P. Barahona, M. Veloso and J. Bryant), pp , Cabukovski, V., Mahmood, N., Rudolf, N. de M. A computer program for fractal analysis of EEG signals, in Proceedings of the 8th International Conference on Biomedical Engineering, Singapore (ed J. C. H. Goh and A. Nather), pp , National University of Singapore, Packard, N.H., Crutchfield, IP., Farmer, J.D. and Shaw, R.S. Geometry from a time series, P/^. #fv. Wf., 45, , Farmer, J.D., Ott, E., Yorke, J.A. The dimension of chaotic attractors, Physica, ID, , Cohen, A., Biomedical Signal Processing, Volume H, CRC Press Inc., Raton, Florida, Kittler, J. and Young, P.C. A new aproach to feature selection based on the Karhunen- Loeve expansion, Pattern Recognition, 5, , Shlien, S. A method for computing the partial SVD, IEEE Trans. Pattern Anal Mach. cf, 4, 671, Press, W. II., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T. Numerical Recipes in Pascal, the Art of Scientific Computing, Cambridge University Press, 1992.

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