Computational Intelligence - February 2015 - 27

At the end of the preprocessing, we obtain a set of conditions
C = {c 1, c 2, ..., c k}, where each condition refers to a unique
interval for one of the explanatory variables.
Representation and Initial Population: As illustrated in
Fig. 5b, candidate solutions are Boolean expressions built with
the set of operators {and, or, not} . These expressions are coded
as GP trees where the leaves correspond to the conditions
C = {c 1, c 2, ..., c k} retrieved in the preprocessing step. The population is initialized with a ramped half-and-half strategy.
Evaluation: The implemented multi-objective approach is
based on Non-Dominated Sorting Genetic Algorithm-II
(NSGA-II) [13] and targets both performance and complexity.
Classifier performance is calculated by evaluating (2). The complexity measure employed is the Subtree Complexity introduced in [14].
3) GP Function Classifier: The third binary classifier, first presented in [15], also implements a multi-objective Genetic Programming strategy based on NSGA-II. Given a set of explanatory variables X , we search for a nonlinear function y = f (Xr )
such the distributions p ( f (Xr )| H 0 ) and p ( f (Xr )| H 1 ) are best
separated. After learning the function, a threshold m is used to
build the decision rule
Lt i = )

1, if y i $ m
0, if y i <

m

3

Area under the ROC Curve: To compute the area
under the ROC curve, we evaluate the model for each
exemplar i of the dataset and vary the threshold m in the
decision rule
Lt i = )

1, if y i $ m
3.
0, if y i < m

We then evaluate the two errors for each threshold and compute the model's area under the ROC curve. We proceed as
follows:
1) Evaluate the GP tree on the training exemplars resulting
in y 1fn .
2) Retrieve the y max and y min values for the model outputs
y 1fn .
3) Normalize the outputs of the model with the obtained
boundaries.
4) Vary the threshold m ! [0; 1] and apply the decision rule
as above.
5) Obtain the False Positive and True Positive rates for each
value of m
6) Compute the area under the ROC curve with the
obtained rates.
Complexity: As in the Rule Tree classifier, we employ the
Subtree Complexity measure.
Threshold selection: The last step consists in identifying the threshold m for the best model f (Xr ) such that the
decision rule

that determines whether a given output represents a class 0 or
class 1 prediction.This approach is
graphically depicted in Fig. 6a.
Threshold
Representation and Initial
Class 1
Class 2
Class 2
Class 1
Class 1
Class 2
Population: As in the RuleSearch f(X)
Tree classifier, individuals are
Such That
coded with GP trees and the
population is initialized with a
X1
Xn
f(X)
ramped half-and-half strategy.
(a)
However, in this case the leaves
Moving Threshold
ROC Curve
of the trees are variables of the
Class 1 Class 2
problem and individuals are
numerical expressions built with
TP
the following set of operators:
Rate
ops = '

+, -, /, ) , sin, cos, log,
1
exp, square, cube, sqrt
(4)

Evaluation: Since the goal is
to find the model with the highest discriminatory power between
the two classes, the area under the
ROC curve is used as fitness
function to guide the search.
Simultaneously, we minimize the
complexity of the models to prevent bloating issues. Here in we
present the steps involved in evaluating the fitness function.

FP Rate

m0 m1 m2 m3 m4 m5 m6
0
f(X)
1
(b)
Best Model

Moving Threshold
Class 1 Class 2

+
log

sin

X4

Final Threshold
Class 1 Class 2
Cost Function

Eval Function

+
X1

exp
0

X5

f(X)

1

0

f(X)

1

(c)

r

Figure 6 GP Function classifier: (a) search for the nonlinear function f (X) such the distributions
p (f (X) | H 0) and p (f (X) | H 1) are best separated. (b) GP Function evaluation: a moving threshold is used
to compute the area under the ROC curve. (c) Threshold selection for the final model.

r

r

february 2015 | Ieee ComputatIonal IntellIgenCe magazIne

27



Table of Contents for the Digital Edition of Computational Intelligence - February 2015

Computational Intelligence - February 2015 - Cover1
Computational Intelligence - February 2015 - Cover2
Computational Intelligence - February 2015 - 1
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