IEEE Computational Intelligence Magazine - August 2021 - 85

high generalization performance of the
ensemble, which will be described in
the later subsection.
E. Fitness Function
An effective fitness function is crucial
for guiding GP to construct high-level
features for fault diagnosis. When the
number of training samples is small, the
features constructed by GP can easily
achieve good classification performance
on the training set, but poor generalization
performance on the test set. To
address this issue, a new fitness function
based on the classification accuracy and
the distance measure of the training
samples is proposed to guide the search
of MFCGPE. To calculate the diagnosis
accuracy, KNN is used to perform classification
based on the constructed features.
The reason for using KNN is that
it is a simple classification algorithm,
easy to implement, and treats each
feature equally without any feature
weighting or selection [22], [52]. With
the use of KNN, MFCGPE can automatically
construct discriminative features
and avoid redundant or irrelevant
features. The distance measure is to
minimize the intra-class distance and
maximize the inter-class distance of the
training samples based on the constructed
features. Using such a distance
measure will group the samples in the
same class and enlarge the differences of
the samples in different classes, which
potentially improves the discriminability
of the constructed features. This also
helps to improve the effectiveness of
KNN as it is based on distance. Based
on the above analysis, the new fitness
function to be maximised is formulated
as follows.
Fit =
()
AccDist
2
+
(1)
where Acc represents the diagnosis accuracy
of KNN using the constructed features
and Dist represents the distance
measure. Only the training set is used in
the fitness evaluation step, and the constructed
features are transformed into
the range of [, ]01 through the min-max
normalization method. The Acc is calculated
using the 5-fold cross-validation
F5
M
scheme. That is, the new generated features
and labels of the training set are
divided into 5 folds, evenly. Each time,
one fold is used as the sub-test set, and
the other two folds are used as the subtraining
set. The average value of the
three sub-test sets is set as the diagnosis
accuracy.
The Dist is calculated according to
Equations (2)-(5), which evaluate the
distance of the training samples with the
constructed features. The calculation of
Dist is based on the Euclidean distance.
For the given samples {, ,
f ,}xkn
and X {, ,, }, the
l xx xl
=
ll
12 n
X xxk kk1
f
=
Euclidean distance between them can
be calculated as Equation (2), where xko
and xlo
Xk and X ,l
represent the features of samples
respectively, and n is the
number of features. The calculation of
the intra-class and inter-class distances is
based on Equation (3) and Equation
(4), where X represents the samples in
one single class, and N is the number of
samples in the X class. Equation (5) is
the calculation of Dist, where the sigmoid
function is used to transform the differ2
The
proposed fitness function optimizes
the classification accuracy and the
distances of the training samples, simultaneously.
When the fitness value
approaches one, it indicates that the features
constructed by GP have the best
classification performance and the training
samples have a small intra-class distance
and a large inter-class distance.
F. Ensemble for Fault Diagnosis
The MFCGPE system is able to find
the three best GP trees/programs that
TABLE III Terminal set corresponding to View2.
SYMBOL
F1
FORMULA
M
/ fj
j=1
M
F2
M
/ Af j
j=1
j
M
/ ()
j=1
F3
M
/ Af j
j=1
M
/ ()
j=1
F4
M
/
/
j=1
M
j=1
M
/ [( )]
j=1
==GG
//
(( ))
4
j
F6
M
/
j=1
F7
M
/ fj F 3
-
j=1
[( )]
MF
1
()
6
3
[( )]
M
fj F
1
-
-
-
-
1
2
Af jf j
Af j
2
j
M
()
j 11
==
j
F13
M
/ AF fj
j=1
j
-
2
9
[( )]
MF
F12
M
/ [(
j=1
MF
AF fj
j
-
()
9
2)( )]
4
4
Af j
Af j
[( )]
[( )]
4
j
2
j
[( )]
fj
2
j
F11
(( ))
fj
F9
()
ence value of the inter-class and intraclass
distances into the range of [, ]01 .
n
dX Xx xlo
1
(, )( ) 2
o
kl
==
/
ko
Dintra ()iX =
D (, )( ,)
NN kij l=1 =1
inter XX =
ij
Dist =
1 / /dX Xl
()i
Ni Nj
k
(4)
1
1 e DDminmax
+ -((
)( ))
interintra
(5)
(3)
()
j
NN kii l=1 =1
1 / /dX X(, )
() ()
Ni Ni
i
k
l
i
(2)
SYMBOL
F8
FORMULA
M
/ fj F 4
-
j=1
[( )]
MF
M
1
()
6
/ [(AF fj5)( )]
2
j
j=1
M
F10
F
F
2
9
-
2
M
/ [(
j=1
MF
AF fj
j
-
()
9
2)( )]
3
3
AUGUST 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 85
IEEE Computational Intelligence Magazine - August 2021

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