Thesis - Instituto de Telecomunicações

102 CHAPTER 5. FEATURE SELECTION AND CLASSIFICATIONwhere p(w i |x) is obtained from the conditional probabilities and the a priori probabilitiesp(x|waccording to the Bayes rule: p(w i |x) = P i )p(w i )ncj=1 (p(x|w j)p(w j. Assuming identical a priori))probability for all classes, p(w i )= 1 n c, we get p(w i |x) =kp(x|w i ) ) where k is a normalizationconstant.We call g i (x) the i-th discriminant function defined as g i (x) =p(w i |x), knowing that∑ nci=1 g i(x) = 15.3.1 Sequential ClassifierThe idea to use the sequence of data arriving from a source is that we can improve theclassification if we have more information. This type of classification, where we have asequence of data arriving from the same source, is called sequential classification [39, 229].We will present a simple example to provide insight into the improvements obtainedby using a sequence of samples from the same population to enhance the classification.We present a two class problem of two univariate normal distributed populations that, forsimplicity, have equal standard deviation, σ, and distinct means, µ 1 = −1 and µ 2 = 1 ata distance d = |µ 1 − µ 2 |, and equal a priori probability. We depict the PDF of the twodistributions in figure 5.5. The point b defines the decision boundary of a two class classifier.Figure 5.5: Probability density functions of two random variables with means µ 1 = −1and µ 2 = 1 both with standard deviation σ = 1. The intersection point forms the decisionboundary, b, between classes w 1 and w 2 . The two gray areas correspond to the classificationerror probability.If we have a vector x =[x 1 ,x 2 ··· ,x ns ] of samples belonging to the same class (the