**5.3.1 Block adaptive weight adjustment**

Initially the classification was attempted by feeding cumulant slices of short ST segments of the order of 10 to 30 samples at 500 Hz sampling rate. This attempt was 80% successful as the network missed low profile petal patterns with low levels of signal-to-noise ratios in their vicinity as a result of short data segmentation. A function was introduced to strengthen the relative magnitude of the discriminant cumulant slice features. The function is sampled across the input layer and its parameters (,) can be adaptively changed for each cumulant slice fed during the training phase which usually takes up to 10 modified cumulant slices every 1000 cardiac cycles. Obviously the shape of function can be changed to cater for other types of abnormalities.

Detection and Classification of Adult and

1988).

control parameters.

**5.4 Experimental results** 

invasively examined and confirmed.

Fetal ECG Using Recurrent Neural Networks, Embedded Volterra and Higher-Order Statistics 247

has become the most popular learning algorithm for multi-layer networks, its rate of convergence is often found to be too slow for practical applications. Well established methods have been adopted (Hush and Salas, 1988). In the modified back-propagation method, every weight wij in the network is given its own learning rate ij, and the training data set is divided into a number of epochs each containing K training patterns (training patterns are 1-d cumulants from overlapping segments of the ST region). The weight wij and learning rate ij are updated every time after an entire training epoch (10 cumulants) has been presented to the network. The weight and learning rate updating rules of the modified back-propagation algorithm can be summarized as follows (Hush and Salas,

> 1 ( 1) ( ) ( 1) ( ) ( ) ( 1) *K*

> >

*if S n D n*

, ( 1) ( ) 0

*otherwise*

*k k ij <sup>n</sup> D n*

*w n* 

( 1) ( ) ( )

( ) ( ), ( 1) ( ) 0

1 ( ) ( ) ( ) *K*

The index n refers to the nth epoch in the training data; the index k refers to the kth pattern in an epoch containing K patterns; kj is the modulated error signal of neuron j with the kth pattern in an epoch; yki refers to the actual computed output of neuron I with the kth pattern in an epoch; k is the index performance to be minimised by the weight update rule with the kth input pattern; finally, , , and (all of which have values between 0 and 1) are the

Three orthogonal leads ECG were recorded from several subjects confirmed of having VT with a prior myocardial infarction (MI). Two subjects suspected of having MI but time- and frequency-domains analysis had not shown any abnormality, and several normal subjects. A total of 3,000 cardiac cycles for this pilot study. Their feature extraction and enhancement were performed. The parameter and of the exponential weight function applied across the input layer were chosen to fall in the region of 1-2 and 0.25 – 0.5 for and , respectively. The initial learning rates ij(0) were all chosen to be 0.06. The momentum factor, , was fixed at 0.09. The control parameters , and were chosen to be 0.03, 0.1 and 0.5 respectively. The classifier described here achieved very high (90%) classification rate. The remaining 10% failure mainly arose because the MI suspected cases were not

*n n if S n D n*

*ij n nn* 

 

*ij ij* (5.2)

(5.4)

*Sn Dn Sn* ( ) (1 ) ( ) ( 1). (5.5)

(5.3)

(5.1)

*ij ij ij kj ki ij k w n w n n ny n w n* 

0, *ij ij*

 

Fig. 5.1. Typical third-order cumulants and their 1-d diagonal and wall slices shown in insets (left, right) of (a) a normal subject and (b) a subject having infarction in the ventricular muscle.

Fig. 5.2. Architecture of the four-layerd neural network. (a) Neuron or processing unit in the network. (b) The four-layer neural network. Only 1-d slice of the weight function modifies the corresponding cumulant slice.

#### **5.3.2 Modification to the back-propagation algorithm**

The back propagation method (Hush and Salas, 1988) used in the supervised learning of a multi-layer neural network is basically a gradient descent method. Although this method has become the most popular learning algorithm for multi-layer networks, its rate of convergence is often found to be too slow for practical applications. Well established methods have been adopted (Hush and Salas, 1988). In the modified back-propagation method, every weight wij in the network is given its own learning rate ij, and the training data set is divided into a number of epochs each containing K training patterns (training patterns are 1-d cumulants from overlapping segments of the ST region). The weight wij and learning rate ij are updated every time after an entire training epoch (10 cumulants) has been presented to the network. The weight and learning rate updating rules of the modified back-propagation algorithm can be summarized as follows (Hush and Salas, 1988).

$$w\_{ij}(n+1) = w\_{ij}(n) + \eta\_{ij}(n+1) \sum\_{k=1}^{K} \delta\_{kj}(n) y\_{ki}(n) + \text{\textquotedbl{}l\textquotedbl{}} w\_{ij}(n-1) \tag{5.1}$$

$$
\eta\_{ij}(n+1) = \eta\_{ij}(n) + \Delta \eta\_{ij}(n) \tag{5.2}
$$

$$\Delta \eta\_{ij}(n) = \begin{vmatrix} \Omega\_{\prime} & \text{if } & S(n-1)D(n) > 0 \\ -\phi \eta\_{ij}(n) & \text{if } & S(n-1)D(n) < 0 \\ 0 & & \text{otherwise} \end{vmatrix} \tag{5.3}$$

$$D(n) = \sum\_{k=1}^{K} \frac{\partial \varepsilon\_k(n)}{\partial w\_{ij}(n)} \tag{5.4}$$

$$S(n) = (1 - \Theta)D(n) + \Theta S(n-1). \tag{5.5}$$

The index n refers to the nth epoch in the training data; the index k refers to the kth pattern in an epoch containing K patterns; kj is the modulated error signal of neuron j with the kth pattern in an epoch; yki refers to the actual computed output of neuron I with the kth pattern in an epoch; k is the index performance to be minimised by the weight update rule with the kth input pattern; finally, , , and (all of which have values between 0 and 1) are the control parameters.

### **5.4 Experimental results**

246 Recurrent Neural Networks and Soft Computing

(a) (b)

Fig. 5.1. Typical third-order cumulants and their 1-d diagonal and wall slices shown in insets (left, right) of (a) a normal subject and (b) a subject having infarction in the ventricular

Fig. 5.2. Architecture of the four-layerd neural network. (a) Neuron or processing unit in the network. (b) The four-layer neural network. Only 1-d slice of the weight function modifies

The back propagation method (Hush and Salas, 1988) used in the supervised learning of a multi-layer neural network is basically a gradient descent method. Although this method

muscle.

the corresponding cumulant slice.

**5.3.2 Modification to the back-propagation algorithm** 

Three orthogonal leads ECG were recorded from several subjects confirmed of having VT with a prior myocardial infarction (MI). Two subjects suspected of having MI but time- and frequency-domains analysis had not shown any abnormality, and several normal subjects. A total of 3,000 cardiac cycles for this pilot study. Their feature extraction and enhancement were performed. The parameter and of the exponential weight function applied across the input layer were chosen to fall in the region of 1-2 and 0.25 – 0.5 for and , respectively. The initial learning rates ij(0) were all chosen to be 0.06. The momentum factor, , was fixed at 0.09. The control parameters , and were chosen to be 0.03, 0.1 and 0.5 respectively. The classifier described here achieved very high (90%) classification rate. The remaining 10% failure mainly arose because the MI suspected cases were not invasively examined and confirmed.

Detection and Classification of Adult and

third-order cumulant method.

**7. Conclusion** 

template.

Fetal ECG Using Recurrent Neural Networks, Embedded Volterra and Higher-Order Statistics 249

methods have AUC values of 0.731, 0.794, and 0.843, respectively. This suggests that the third-order cumulant is a better detection method than second-order statistics-based, and

Youden's index, defined as sensitivty + specificity – 1, has also been used for the detection methods. The second-order statistics-based methods have indices in the range of 0.42 to 0.55. The third-order cumulant method has an index of 0.72. The bispectral contour method has an index of 0.80. This suggests that the third-order cumulant is a better detection method than second-order statistics-based, and that the bispectral contour method outperforms the

Also, the Partial Area Under Curve (PAUC) measure has been used for a False-Positive Rate (FPR) of 10% and sensitivty larger than 75%. The second-order statistics-based method gives a PAUC of 0.043. The third-order cumulant method has a corresponding value of 0.125 whilst that of the bispectral contour method is 0.137. This suggests that the third-order cumulant is a better detection method than second-order statistics-based, and that the

The sensitivity, specificity and classification rate for the third-order slice cumulant matching hybrid system have been calculated . The technique has been evaluated for diagonal, wall, or arbitrary TOC slices, employing both the LMF-based quadratic and cubic Volterra filters. The results indicate that a linear combination of diagonal and wall slices of the TOC can improve the detection rate by up to 1% over and above the 77.8% obtainable using only either slice. Using two more arbitrary slices off-diagonal and off-wall would result in a further improvement of up to 1%. Using two slices instead of only one results in an two-fold increase in the CPU time of 1 msec using Unix WS. Further improvement of 6% to 8% is attainable with maternal transabdominal ECG signal linearisation employing second- and third-order Volterra synthesisers, respectively. Based on the first hybrid system using TOC slices for signal processing and subsequent single-hidden-layer classification, 100% and 86.16% classification rates have been achieved for maternal QRS-complex and fetal heartbeats, respectively. Note that the classification rates for coincident and non-coincident maternal and fetal QRS-complexes are 0% and 95.55%, respectively. The remaining undetected 13.84% fetal heartbeats include 9.8% overlap with the maternal QRS-complexes and 4% occur during depolarisation of the maternal T-waves. Those events unavoidably lead to significant distortion of the fetal TOCs. This means that the cumulant signatures will not be close to the TOC template signature stored in the database. Examples of false negatives and false positives have been found in the following cases, respectively, (i) a fetal heartbeat with maternal contribution TOC diagonal slice was wrongly matched to a QRSfree ECG TOC diagonal slice template, and (ii) a QRS-free ECG TOC diagonal slice was wrongly matched to a fetal heartbeat with maternal contribution TOC diagonal slice

Results obtained for the bispectral contour matching hybrid system from 30 cases using the non-invasive transabdominally-measured ECG signal, with the simultaneous fetal scalp electrode ECG signal as a reference, show that the method has a classification rate of 100% for normal, healthy maternal QRS-complexes and 90.12% for fetal heartbeats. It has been

that the bispectral contour method outperforms the third-order cumulant method.

bispectral contour method outperforms the third-order cumulant method.
