SEKE 2012 Proceedings - Knowledge Systems Institute

uilding FNNs that satisfy different requirements regarding
classification rates.
C. FNN and MOO: Optimization Process
A multi-objective optimization process is used here in
such a way that a classification rate for a single class is a
single optimization objective. Therefore, a number of
objectives is equal to the num ber of classes. As the result of
such an optimization process, a set of solutions – data
models – is obtained. T hese models constitute a Pareto
surface.
Representation of Network Structures: An important aspect
of application of evolutionary-based optimization to
construct FNNs is to m ap a structure of an FNN to a
chromosome string. Two asp ects are important here: a
selection of attributes th at constitute inputs to AND/OR
nodes, and an adjustment of c onnection weights. Therefore,
a chromosome is built out of two segments:
- the segment that is “responsible” for selection of
attributes that become inputs to the neurons, let’s call it a
variable indexes part;
- the segment that determ ines connection w eights – let’s
call it a connection weights part.
Each position of the variable indexes part is an integer that
identifies the attributes’ index. T he values of these integers
are in the range from 0 to n-1 (for n-dimensional data). The
length of this segm ent is determined by the number of
classes c (equal to the number of OR/ AND nodes) i n the
dataset, and the maximum number of inputs to a single
OR/AND node – i. The overall length of the segment
variable indexes part is c*i. The connection weights part is
a string of floating point numbers. Each number represents a
single connection weight and can be in the range from 0 to
1. The number of connection weights for FNN is
(2*i+2*c)*c.
As it can be seen, the structure of the FN N is partially
fixed – i t contains so many OR/ AND nodes as di fferent
classes. It means each output of the FNN is associated with
a single class.
Optimization Objective Functions: The objective of the
optimization process is to construct FNNs that provide the
best classification rates for a single class. This m eans that
selection of the objective function that governs the
optimization process is important.
For any two-category classifi cation process, a confusion
or error matrix can be built, Table I. This matrix summarizes
classification capabilities of a m odel: values a (true
positives) and d (true negatives) represent proper
classifications, while b (false positive) and c (false negative)
misclassifications.
Based on this matrix a number of different measures can be
calculated. The measures that are used in our fitness
functions are:
a + d
accuracy =
a + b + c + d
that defines the ratio of all corr ectly classified data points to
all data points,
sensitivity (recall) =
a
a + c
that represents the percent of actual positive data points
classified as positive data points,
specificity =
d
b + d
that represents the percent of actual negative data points
classified as negative data points, and
precision =
a
a + b
that is the ratio of actual positive data points classified as
positive to all data point classified as positive . It can be said
that sensitivity and specificity are somehow reciprocal to
each other. The sensitivity is for positive data points, w hile
specificity is its equivalent for negative data points. Another
observation is related to sensitivity (called also recall) and
precision. Higher sensitivity means that alm ost all of the
positive data points w ill be included in the classification
results. However, at the sam e time, some negative data
points can be predicted as positive ones, w hat leads to low
values of precision.
Table I. Confusion matrix
actual POSITIVE NEGATIVE
predicted
POSITIVE a b
NEGATIVE c d
Of course, accuracy is a very important measure – it
means that the classifier is able to properly classify positive
and negative data points. However, in the case of large
imbalance in a number of data points that belong to each
class, accuracy measure is not able to assure a high
classification rate for each class. If 90 per cent of data points
belong to the class negative and only 10 per cent to the class
positive, then high accuracy can be achieved by correct
classification of the class negative only. At the same time,
this can lead to large misclassification (large b and c) for the
class positive. Therefore, there is a need to use som e other
measures that “take care” of large b and c.
As the result of these investigations, fitness functions
used here are constructed based on accuracy, sensitivity,
specificity and precision. Two fitness functions are defined:
- one that uses a product of sensitivity, specificity and
accuracy (FF SSA )
F SSA
= Sensitivity * Specificity * Accuracy
- one that uses a product of accuracy, recall and precision
– (FF APR )
F APR
= Accuracy * Recall * Precision
Each of the fitness functions represents different way of
suppressing b (false positives): F SSA does it in the reference
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