A New Wavelet-Based Neural Network for Classification of Epileptic-Related States using EEG

Abstract

In this paper, we present a novel neural network able to classify epileptic seizures using electroencephalogram (EEG) signals, called “Multidimensional Radial Wavelons Feed-Forward Wavelet Neural Network” (MRW-FFWNN). The network is part of a classification system, which distinguishes among three brain states related to epilepsy namely ictal, interictal and healthy. Efficient methods for pre-processing EEG’s, extracting features and getting the final class decisions were selected using a statistical three-fold cross-validation method, which assures the robustness of the system and its generalization ability. The following methods were systematically analyzed to find the most appropriate for this problem: 1) Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) filters for noise reduction; 2) discrete Wavelet Transform (DWT) and Maximal Overlap Discrete Wavelet Transform (MODWT) for frequency decomposition of the EEG signals; 3) average correlation and maximum voting correlation for selecting a suitable mother wavelet for frequency decomposition; 4) Binary-tree and one-vs-one (OVO) decomposition strategies for primary binary classification; 5) voting and weighted-voting strategy aggregation strategies for the final classification. The integrated system was assessed using a three-fold cross validation, applied to a benchmark provided by the University of Bonn, getting an average accuracy of 93.33% when tested using sets Z, S and F and 95.0% when sets Z, S, F and O were used. The proposed network got competitive accuracy, compared with other state-of-the art classifiers, training in almost a half of the time than the ones with similar accuracy.

Appendix: Description of partial derivatives for the proposed model MRW-FFWNN

• 1. Weights between the input layer and output layer

  The partial derivative of $\frac {\partial y(n)}{\partial a_{k}}$ is computed by using the output of the network as follows:

  $$ \frac{\partial y(n)}{\partial a_{k}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})+\sum \limits_{k=1}^{N_{i}}a_{k} x_{k} }}{\partial a_{k}} $$

  (39)

  The weights between the output of MRW nodes and the output of the MRW-FFWNN do not depend on a_k. Therefore, the previous equation is converted into

  $$ \frac{\partial y(n)}{\partial a_{k}} = \frac{\partial {\sum \limits_{k=1}^{N_{i}}a_{k} x_{k} }}{\partial a_{k}} = x_{k} $$

  (40)

  Equation 40 is the partial derivative of the output with respect to a_k in the MRW-FFWNN model.

• 2. The translation parameters

  $\frac {\partial y(n)}{\partial t_{jk}}$, is defined as:

  $$ \frac{\partial y(n)}{\partial t_{jk}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})+\sum \limits_{k=1}^{N_{i}}a_{k} x_{k} }}{\partial t_{jk}} $$

  From this equation, the second term in the addition does not depend on the parameters of translation of the units of the wavelet layer, so that it is written as

  $$ \frac{\partial y(n)}{\partial t_{jk}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})}}{\partial t_{jk}} = w_{j} \frac{\partial \psi (c_{j})}{\partial t_{jk}} $$

  By the chain rule, we obtain

  $$ \frac{\partial y(n)}{\partial t_{jk}} = w_{j} \frac{\partial \psi (c_{j})}{\partial c_{j}} \frac{\partial c_{j}}{\partial t_{jk}} $$

  Developing the third term of the product we have,

  $$ \frac{\partial y(n)}{\partial t_{jk}} = w_{j} \frac{\partial \psi (c_{j})}{\partial c_{j}} \frac{\partial \left[ {d_{j}^{2}}(x_{1} - t_{j1})^{2} + ... + {d_{j}^{2}} (x_{k} - t_{jk})^{2} \right]^{\frac{1}{2}}}{\partial t_{jk}} $$

  which gives

  $$ \frac{\partial y(n)}{\partial t_{jk}} = w_{j} \frac{\partial \psi (c_{j})}{\partial c_{j}} \left[ \frac{1}{2 c_{j}} (-2{d_{j}^{2}} x_{k} + 2 {d_{j}^{2}} t_{jk}) \right] $$

  (41)

  with c_j = R(x_k,d_j,t_j).

  Equation 41 is the partial derivative of the output with respect to t_jk in the MRW-FFWNN model.

• 3. The dilation parameters

  Likewise, it is defined:

  $$ \frac{\partial y(n)}{\partial d_{j}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})+\sum \limits_{k=1}^{N_{i}}a_{k} x_{k} }}{\partial d_{j}} $$

  From this equation, the second term in the addition does not depend on the parameters of dilation of the units of the wavelet layer, so that it is converted into

  $$ \frac{\partial y(n)}{\partial d_{j}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})}}{\partial d_{j}} = w_{j} \frac{\partial \psi (c_{j})}{\partial d_{j}} $$

  Applying the chain rule, we obtain:

  $$ \frac{\partial y(n)}{\partial d_{j}} = w_{j} \frac{\partial \psi (c_{j})}{\partial c_{j}} \frac{\partial c_{j}}{\partial d_{j}} $$

  Developing the third term of the product we have,

  $$ \frac{\partial y(n)}{\partial d_{j}} = w_{j} \frac{\partial \psi (c_{j})}{\partial c_{j}} \frac{\partial \left[ {d_{j}^{2}}(x_{1} - t_{j1})^{2} + ... + {d_{j}^{2}} (x_{k} - t_{jk})^{2} \right]^{\frac{1}{2}}}{\partial d_{j}} $$

  Therefore, using the Eq. 3,

  $$ \frac{\partial y(n)}{\partial d_{j}} = w_{j} \frac{\partial \psi (c_{j})}{\partial c_{j}} \left[\frac{d_{j}}{c_{j}} \left( (x_{1} - t_{j1})^{2} + ... + (x_{k} - t_{jk})^{2}) \right) \right] $$

  (42)

  with c_j = R(x_k,d_j,t_j).

  Equation 42 is the partial derivative of the output with respect to d_j in the MRW-FFWNN model.

• 4. Weights of the connections between the MRW nodes and the output layer

  Let:

  $$ \frac{\partial y(n)}{\partial w_{j}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})+\sum \limits_{k=1}^{N_{i}}a_{k} x_{k} }}{\partial w_{j}} $$

  Finally, we calculate the partial derivative as follows:

  $$ \frac{\partial y(n)}{\partial w_{j}} = \frac{\partial {\sum \limits_{j=1}^{N_{w}}w_{j}\psi_{j}(c_{j})}} {\partial w_{j}} $$
  $$ \frac{\partial y(n)}{\partial w_{j}} = \psi_{j} (c_{j}) $$

  (43)

  Equation 43 is the partial derivative of the output with respect to w_j in the MRW-FWWNN model.