Time-varying AR model for EEG

This example presents some illustrations to demonstrate the applicability of the proposed modelling framework for characterising epileptic seizure EEG signals by using time-varying AR models. Again, channel C3 of the EEG signal of a subject for an epileptic seizure activity lasting 20 seconds, was considered. A total of 10000 samples, recorded with a sampling rate of 500Hz, are shown in Figure 4.7.

Similar to the previous example, the third, fourth and fifth order B-splines were adopted to establish TVAR models for the EEG recordings. Numerical experiments have suggested that a TVAR model of order p4 is appropriate for representing

the EEG recordings here. The topographical diagram of the time-dependent spectrum estimated from the TVAR (4) model is shown in Figure 4.8, and the 2-D image diagram and the contour plot of the time-dependent spectrum produced from the 3-D topographical diagram are given in Figure 4.9.

From Figures 4.8 and 4.9, the power spectrum of the EEG signal with an epileptic seizure activity of 20 seconds is mainly distributed in the frequency range from zero to 10 Hz. Three frequency bands, i.e. Delta band (0-4 Hz), Theta band (4-7 Hz) and Alpha band (8-12 Hz), can clearly be observed as follows: 1) the low frequency band (around 3 Hz); 2) the frequency band that is centralized around 6 Hz; 3) around 10 Hz representing the high frequency band component. The image and contour plot of the time-dependent spectrum shown in Figure 4.9 clearly reflects the distribution of these frequency components along with the time course. It is clear that the variations of the time course signals can be observed from the contour diagram of the transient spectrum. For example, the power spectrum is mainly dominated by a 3-Hz frequency component during the period of the first 2 seconds, 7 to 8 seconds, and 11 to 12 seconds, respectively, while the high frequency (around 10 Hz) activity is dominated by the time course from 16 to 18 seconds.

Any time-invariant parametric modelling framework such as the commonly applied conventional AR models cannot reveal these properties.

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Figure 4.3 Estimates of the time-varying coefficients a ti

 

for i1,2,3,4 and b t for n

 

Figure 4.2 The EEG recordings (F3 Channel: Input signal, F4 Channel: Output signal), for a seizure activity of a patient, recorded over 7 seconds, with a sampling rate of 500 Hz.

0 1 2 3 4 5 6 7

-400 -200 0 200 400

Input Signal

EEG(uV)

0 1 2 3 4 5 6 7

-500 0 500

Output Signal

EEG(uV)

Sample Time(sec)

Figure 4.5 The 3-D topographical map of the time-dependent spectrum estimated from the TVARX model for the EEG signal.

Figure 4.4 A comparison between the recovered signal from the identified TVARX (4, 3) model and the original observations for the EEG signal. Solid (blue) line represents the observations and the dashed (red) line represents the signal recovered from the TVARX (4, 3) model with One-Step-Ahead prediction. For a clear visualization only the data points of the period from 0 to 2 seconds are displayed.

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(a)

(b)

Figure 4.6 The 2-D image and the contour diagram of the time-dependent spectrum produced by the 3-D topographical map for the EEG signal given in Figure 4.2.

(a) the 2-D image;

(b) the contour diagram.

Figure 4.8 The 3-D topographical map of the time- dependent spectrum estimated from the TVAR (4) model for the EEG signal given in Figure 4.7.

Figure 4.7 The EEG recordings of C3 Channel, for a seizure activity of an epileptic patient, recorded over 20 seconds, with a sampling rate of 500 Hz.

0 2 4 6 8 10 12 14 16 18 20

-400 -300 -200 -100 0 100 200 300 400

Output Signal

EEG(uV)

Sample Time(sec)

79

(a)

(b)

Figure 4.9 The 2-D image and the contour diagram of the time-dependent spectrum produced by the 3-D topographical map for the EEG signal given in Figure 4.7.

(a) The 2-D image;

(b) The contour diagram.

Time (Sec)

Freq (Hz)

0 5 10 15 20

0 5 10 15 20 25 30 35 40 45 50

10 20 30 40 50 60 70

4.6 Conclusions

In this Chapter, a novel TV parametric modelling approach has been presented based on time-dependent coefficients approximated multi-wavelet basis functions to account for the transient spectrum information, whereas the basis functions involved are locally defined. Using the method given in Chapter 3, a time varying multi-wavelet basis function expansion model can be constructed. The orthogonal least squares algorithm is then applied to select significant model terms (wavelet basis functions) from the initially pre-specified model set. Finally, the time-varying model parameters can be recovered by using the selected significant wavelet basis functions.

The time-dependent spectrum based on TVARX and TVAR model, with multi-wavelet basis functions, can reflect the global frequency behaviour of the signal and to reveal the local variations of the signal along the time course. One advantage of the proposed model, compared with traditional time-invariant models, is that it can capture much more transient information of the inherent nonstationary dynamics of the associated processes.

It is believed that the proposed time-varying modelling framework, coupled with results obtained from some other useful nonparametric methods for example higher-order statistics approach (Zhou et al. 2008), can promise significant new results which can reveal new and important features buried in EEG signals, which can in turn enable further studies and analysis, for example for disease detection and diagnosis. In addition, it is well known that EEG signals are nonlinear processes, and thus linear models may not be sufficient to represent such nonlinear processes; nonlinear representations should be more suitable for EEG data modelling and analysis. Time varying nonlinear modelling and identification of EEG signals will be discussed in the next Chapter.

81 where the symbol .,. denotes the inner product of two vectors, that is

   



 ^{ }