Combining Pattern Classifiers

8 FUNDAMENTALS OF PATTERN RECOGNITION
maximum membership rule (1.3), x will be labeled to the same class by any of the
equivalent sets of discriminant functions.
If the classes in Z can be separated completely from each other by a hyperplane
(a point in R, a line in R 2 , a plane in R 3 ), they are called linearly separable. The two
classes in Figure 1.5 are not linearly separable because of the dot at (5,6.6) which is
on the wrong side of the discriminant function.
1.3 CLASSIFICATION ERROR AND CLASSIFICATION ACCURACY
It is important to know how well our classifier performs. The performance of a
classifier is a compound characteristic, whose most important component is the
classification accuracy. If we were able to try the classifier on all possible input
objects, we would know exactly how accurate it is. Unfortunately, this is hardly a
possible scenario, so an estimate of the accuracy has to be used instead.
1.3.1 Calculation of the Error
Assume that a labeled data set Z ts of size N ts n is available for testing the accuracy
of our classifier, D. The most natural way to calculate an estimate of the error is to
run D on all the objects in Z ts and find the proportion of misclassified objects
Error(D) ¼ N error
N ts
(1:7)
where N error is the number of misclassifications committed by D. This is called the
counting estimator of the error rate because it is based on the count of misclassifications.
Let s j [ V be the class label assigned by D to object z j . The counting
estimator can be rewritten as
Error(D) ¼ 1 X N ts
1 Iðl(z j ), s j Þ , zj [ Z ts (1:8)
N ts
j¼1
where I(a, b) is an indicator function taking value 1 if a ¼ b and 0 if a = b.
Error(D) is also called the apparent error rate. Dual to this characteristic is the
apparent classification accuracy which is calculated by 1 Error(D).
To look at the error from a probabilistic point of view, we can adopt the following
model. The classifier commits an error with probability P D on any object x [ R n
(a wrong but useful assumption). Then the number of errors has a binomial distribution
with parameters (P D , N ts ). An estimate of P D is
^P D ¼ N error
N ts
(1:9)