Instrumentation & Measurement Magazine 24-2 - 26

processed by the nonlinear activation function. In our study,
the sigmoid function is chosen as the activation function and
it is given by:
	

Sigmoid  x  

1
	(3)
1  e x

Finally, the output neuron is expressed as follows:
	






y Sigmoid  h  X  	(4)

Recall that the information fed to each neuron in each hidden layer is processed by a sigmoid activation function. The
resilient backpropagation algorithm is chosen as the training
function since it is fast and does not require large memory during computation. Finally, the number of epochs is set to 10, and
the learning rate is set to 0.01.
Recall that there is no rule of thumb on how to configure
the entire architecture of the DFFNN along with its parameters. In general, they are fixed, according to the experience of
the user. In our study, the number of hidden layers is set to four
for deeper analysis of the cepstum coefficients compared to
two and three hidden layers and to obtain acceptable computation processing time. Also, the number of epochs is set to 10
for fast processing of the input signal. Finally, a learning rate of
0.01 is a good compromise between convergence and required
processing time. Indeed, our goal is to design an effective and
fast DFFNN system for analysis and classification of ceptrum
coefficients.

Baseline Classifiers: SVM, NB, and PNN
The SVM classifier [22] is a supervised learning algorithm
based on statistical learning theory used to determine a hyper
plane. The latter optimally splits two classes by learning a train
n
data set. For instance, let xi , y i i1 where x is the input vector,
and y is the class label. The classification decision function is
expressed as follows:
 N

f  x  sign   i yi K xi , x j  b 	(5)
	 
 i1






where αi is the Lagrange multipliers, K(xi,xj) is a linear kernel
function, and b is a constant parameter.
The Naïve Bayes classifier [23] is based on estimation of
probabilities to assign the membership of an input vector x to
a particular class y. The Bayes' theorem can be used to express
the conditional probability of class label y given input vector
x as follows:
	

P  y|x 

P  x| y  P  y
P  x

	(6)

where P(y|x) is the posterior probability of the compound
class, P(x|y) is the conditional probability that a compound
has certain features given its class y, P(y) is the prior probability estimated from the training set, and P(x) is the marginal
probability of observing the given features in the dataset.
The probabilistic neural network (PNN) [24] is composed
of three layers: the input layer, pattern layer, summation layer,
26	

and output layer. The input layer has 1000 neurons, corresponding to the number of cepstrum coefficients. The second
layer consists of 1000 neurons where each one is represented
by a Gaussian transfer function. The pattern layer has two neurons where each one is used to represent a specific class. The
pattern layer is used to perform an average operation of the
outputs from the pattern layer for each class. Finally, the output layer has one neuron used to compute the maximum sum
as follows:





y  argmaxi Pi  y  	(7)

	
where,

Pi  y  

	
and,
	

Qij  y  

1
Ni

Ni

Q  y 	(8)
j 1



ij



T

y  xij
  y  xij
exp

0.5
2
2
 2   

1

 

	(9)



where Pi(y) represents the probability that a test sample x belongs to class y, Qij represents the standard probability density
function (PDF), δ is a smoothing parameter set to unity, and N
is the number of neurons in the input layer.

Experimental Protocol and Performance
Measures
The performance of each classifier is measured by computing
the accuracy, sensitivity, and specificity. The accuracy is the ratio of the correct predictions to the total number of predictions,
sensitivity measures the proportion of positive predictions
that are correctly identified over all positive cases, and specificity measures the proportion of negative predictions that are
correctly identified. Hence, accuracy, sensitivity, and specificity are expressed as follows:
TP  TN
Accuracy 
TP  FN  TN  FP 	(10)
	

	

	

Sensitivity 

TP
TP  FN 	(11)

Specificity 

TN
TN  FP 	(12)

where TP, TN, FN and FP indicate the number of true positives,
true negatives, false negatives and false positives, respectively.
To evaluate the performance of each classifier while avoiding overfitting, 10-fold cross-validation protocol is adopted
in our study. For instance, under 10-fold cross-validation protocol, the data set is divided into 10 subsets. Each time, one
different subset is used as the test set, and the remaining nine
subsets are put together to form a training set. Then the average performance across all 10 trials is calculated. In the current
work, the average and standard deviation of accuracy, sensitivity, and specificity are calculated across the 10 folds. Also,

IEEE Instrumentation & Measurement Magazine	

April 2021



Instrumentation & Measurement Magazine 24-2

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