**3.4 Estimation of the bispectral contour matching variance**

The variance of the BIC is defined as the expected value of the squared difference in frequency (in Hz) between the computed BIC of the 250 msec flag window of the transabdominal ECG signal and the computed BIC from the synchronised fetal scalp electrode ECG 250 msec window.

Detection and Classification of Adult and

**perceptron** 

in Section 2.

for training varied from 6 to 14.

Fetal ECG Using Recurrent Neural Networks, Embedded Volterra and Higher-Order Statistics 237

 Varb = E [(Bis(1,2) Transabdominal – Bis(1,2) fetal scalp ] 2 (3.3) The above variance ranges from 0.47 – 3.3, average = 1.716, when calculated for 120,000 FHBs. The variance indicates the deviation of the frequency of the BIC (in Hz) of the

Single-hidden-layer perceptron classifiers are trained in a supervised manner with the back-propagation algorithm which is based on the error-correction learning rule. The back-propagation algorithm provides a computationally efficient method for the training of the classifiers. The back-propagation algorithm is a first-order approximation of the steepest descent technique. It depends on the gradient of the instantaneous error surface in weight space. The algorithm is therefore stochastic in nature. It has a tendency to zigzag its way about the true direction to a minimum on the error surface. Consequently, it suffers from a slow convergence property. A momentum term is employed to speed up the performance of the algorithm. The classifier used here is exactly the same as that used

Fig. 3.2 shows the effect of changing the learning rate (), the momentum constant () and the middle layer size on the classification of the maternal QRS-complexes and the fetal heartbeats using the BIC template matching technique. The effect of changing the learning rate on the classification rate is shown in Fig. 3.2 (a). Small values of are not able to track the variations in the bispectral contours. For classification of the bispectral contours, reaches its optimum value at 0.2. For values larger than 0.2, the output values are too large so that the difference with respect to the reference signal (template) will increase. This leads to larger error that will be fed back to the network, which will lead to slower convergence. The network will take long time to converge, or it might not converge at all. The optimum value of the momentum constant is found to be 0.2, as depicted in Fig. 3.2 (b). Smaller values are not enough to push the adaptations to avoid local minima. While larger values tend to affect the routine detrimentally by bypassing the global minimum. The performance deteriorates significantly as the learning rate and the momentum constant diverge from their optimum values. The number and size of the middle layers were investigated by trial and error. There is a trade off between networks that should be small enough to allow faster implementation, and larger networks in size and number of hidden layers which are very slow and can not be implemented on-line using the current technology. Large networks could have complex relationships that represent nonlinearities that might not exist in the real signals at all. The optimum parameters indicated in Fig. 3.2 are calculated without considering the CPU time factor which might render those parameters undesirable for real-time applications. The CPU time for training is in the range of 17 to 60 sec. The average mean-squared error (MSE) is 0.04. The worst error is 0.1, which is the criterion for convergence. The implemented neural network has a single middle layer size of 6 x 6 as shown in Fig. 3.2 (c). The number of passes (epochs) required

transabdominal ECG signal from that of the fetal scalp electrode around 30 Hz.

**3.5 The back-propagation with momentum algorithm in single-hidden-layer** 

**3.6 Optimisation of the parameters of the back-propagation algorithm** 

Fig. 3.1. Dual-band-pass filtered bispectra, Kaiser shaped window, (l.h.s.) and their contour maps normalised to the maternal QRS spectral peak (r.h.s.) for the transabdominallymeasured ECG segments I, II, III, and IV of Fig. 2.1. Segment I: maternal QRS-complex, Segment II: the first fetal heartbeat with maternal contribution; Segment III: QRS-free ECG; and Segment IV: the second fetal heartbeat with maternal contribution. The dual band-pass filter consists of two fifth-order Butterworth filters with cut-off frequencies of 10 Hz to 20 Hz, and 25 Hz to 40 Hz, respectively, and a pass-band attenuation of 0.5 dB, a stop-band attenuation larger than 70 dB. The sampling rate is 500 Hz.

$$\text{Var}\_b = \mathbb{E}\left[ \left( \text{Bis}(\text{o}\_{\text{l}\prime} \text{ o}\_2)\_{\text{Transabdorminal}} - \text{Bis}(\text{o}\_{\text{l}\prime} \text{ o}\_2)\_{\text{fel scalar}} \right) \mathbb{1} \right] \tag{5.3}$$

The above variance ranges from 0.47 – 3.3, average = 1.716, when calculated for 120,000 FHBs. The variance indicates the deviation of the frequency of the BIC (in Hz) of the transabdominal ECG signal from that of the fetal scalp electrode around 30 Hz.

## **3.5 The back-propagation with momentum algorithm in single-hidden-layer perceptron**

236 Recurrent Neural Networks and Soft Computing

Fig. 3.1. Dual-band-pass filtered bispectra, Kaiser shaped window, (l.h.s.) and their contour maps normalised to the maternal QRS spectral peak (r.h.s.) for the transabdominallymeasured ECG segments I, II, III, and IV of Fig. 2.1. Segment I: maternal QRS-complex, Segment II: the first fetal heartbeat with maternal contribution; Segment III: QRS-free ECG; and Segment IV: the second fetal heartbeat with maternal contribution. The dual band-pass filter consists of two fifth-order Butterworth filters with cut-off frequencies of 10 Hz to 20 Hz, and 25 Hz to 40 Hz, respectively, and a pass-band attenuation of 0.5 dB, a stop-band

attenuation larger than 70 dB. The sampling rate is 500 Hz.

Single-hidden-layer perceptron classifiers are trained in a supervised manner with the back-propagation algorithm which is based on the error-correction learning rule. The back-propagation algorithm provides a computationally efficient method for the training of the classifiers. The back-propagation algorithm is a first-order approximation of the steepest descent technique. It depends on the gradient of the instantaneous error surface in weight space. The algorithm is therefore stochastic in nature. It has a tendency to zigzag its way about the true direction to a minimum on the error surface. Consequently, it suffers from a slow convergence property. A momentum term is employed to speed up the performance of the algorithm. The classifier used here is exactly the same as that used in Section 2.
