IEEE Computational Intelligence Magazine - February 2021 - 89

TABLE 1 Function sets and terminal sets in ICS-GP.
TERMINAL SETS

LEFT SUB-TREE
(FOR CONSTRUCTING CLASSIFIERS)

RIGHT SUB-TREE
(FOR LEARNING
COST INTERVALS)

FUNCTION SETS

NAME

TYPE

NAME

TYPE (INPUT)

TYPE (OUTPUT)

* FEATURES OF A DATASET

Iput
(FLOAT)

*+

[Iput, Iput]

Iput

*-

[Iput, Iput]

Iput

*×

[Iput, Iput]

Iput

*%

[Iput, Iput]

Iput

* If

[Iput, Iput, Iput]

Iput

* Classifier

[Iput]

Predi (FLOAT)

* AddCost

[C_inter, C_inter]

C_inter

* SubCost

[C_inter, C_inter]

C_inter

* MulCost

[C_inter, C_inter]

C_inter

* DivCost

[C_inter, C_inter]

C_inter

* T_Cost

[C_inter]

Ocost (LIST)

* A RANDOM CONSTANT

C_inter
(LIST)

* THE INITIAL COST INTERVALS (C l, C u)
(CL AND CU ARE UNIFORMLY DISTRIBUTED
RANDOM NUMBERS IN THE RANGE OF [1, 2].)

1: Iput and Predi are different data types in an evolved tree program, even though they are essentially the same type (float).

1) The Left Sub-Tree
In a GP tree, the left sub-tree is essentially
used as a classifier. A terminal set includes
all the features and a random constant.
There are six functions in the function
set, including four basic arithmetic functions (i.e., +, -, # and protected division %), If and Classifier. Note that the
protected division % returns zero when
dividing by zero. If function takes three
arguments (if the first argument is negative, the second argument is returned;
otherwise the third argument is
returned). The function Classifier takes
one argument argu (the value of argu is
the output of an evolved classifier, and its
data type is Iput), and it directly returns
the value of argu (its data type is Predi).
Figure 3 shows an example of evolved
trees. For the tree in Figure 3, its left subtree is used as a classifier, which can be
translated into an arithmetic expression
Classifier ^ f 528 + f 59 # f 102h . In this
arithmetic expression, f 528, f 59 and f 102
are features from the terminal set, and +
and # are taken from the function set of
the left sub-tree.The output values of arithmetic expression ^ f 528 + f 59 # f 102 h
are used for predictions.
2) The Right Sub-Tree
The right sub-tree plays a role in learning
a cost interval, i.e., ^C l, C uh, which will
be later used to calculate classification
thresholds. The terminal set includes the
initial cost intervals, where its lower and

upper bounds are taken from the uniformly distributed random numbers in
the range of [1, 2] (this does not mean
that the evolved cost intervals are limited to 2, because GP individuals could
choose suitable functions from a function
set to increase the initial bounds). The
function set has four arithmetic functions
(i.e., AddCost, SubCost, MulCost and DivCost), and a function T_Cost. AddCost
and MulCost are defined as follows:

	

	

AddCost ^CI 1, CI 2h
lower : C1 l + C 2 l 
='
upper : C1 u + C 2 u
MulCost ^CI 1, CI 2h
lower : C1 l # C 2 l 
='
upper : C1 u # C 2 u

(4)

(5)

where CI1 and CI2 indicate two cost
intervals, and C1 l (or C2 l) and C1 u (or
C2 u) are the lower and upper bound
values of CI1 (or CI2), respectively.
SubCost and DivCost need to be
designed very carefully because they
might produce a cost interval whose
lower and upper bound values are less
than 1 (i.e., the misclassification cost of
the minority class is less than that of
the majority class) or even less than 0
(i.e., the misclassification cost of the
minority class is less than that of correct predictions). To avoid this risk, SubCost and DivCost are defined as follows:
SubCost ^CI 1, CI 2 h
lower : max ^1, abs ^C1 l - C 2 l hh 
(6)
=)
upper : max ^1, abs ^C1 u - C 2 uhh

Cost-Sensitive Classifier
(Cl, Cu)
Classifier

T_Cost

+

AddCost
×

f528
f59

MulCost
f102

(1.23, 1.51)

(1.76, 1.63)

(1.11, 1.53)

FIGURE 3 An example of the evolved trees.

FEBRUARY 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

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IEEE Computational Intelligence Magazine - February 2021

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