Thesis - Instituto de Telecomunicações

5.3. SEQUENTIAL CLASSIFICATION 105n f∏p(x j |w i )= p(x l,j |w i ). (5.10)l=1We can take into account that we are implementing an authentication system, accordingto which a decision must be made about the authenticity of the identity claim, the twopossible decisions are acceptance or rejection, based on its a posteriori probability. Sincep(w i |X j ) represents the probability of the classification being correct, we establish a limit,λ, to select one of the decisions, using the decision rule:⎧⎨ true if p(w i |X) >λAccept(X ∈ w i )=⎩ false otherwise. (5.11)The limit λ is adjusted to select the operating point of the classifier with a specific FRRand FAR. To present results about the classifier performance we adjust λ to operate at theequal error rate point.The user produces a sequence of samples, X =[x 1 , ··· , x ns ], and the resulting featurevectors can be used to improve the classification performance. Assuming independencebetween the sequential feature vectors:∏n sp(X|w i )= p(x j |w i ), (5.12)j=1and that the classes are equiprobable (p(w i ) = 1/n c , with i =1...n c where n c is thenumber of classes), we can deduce:p(w i |X) =∏ ns∑ nck=1j=1 p(x j|w i )∏ nsj=1 p(x j|w k ) . (5.13)We used the rule in equation 5.11 to decide about the subject’s acceptance or rejectionbased on the a posteriori probability function of equation 5.13.We will see that in some cases, even using several samples, the sequential classifier doesnot produce a good result. This effect can be overcome by integrating the classifier withother classifiers with higher discriminative data, in a multimodal biometrics framework.