Thesis - Instituto de Telecomunicações

112 CHAPTER 5. FEATURE SELECTION AND CLASSIFICATIONAlgorithm 3: Uncertainty Based Classifier Fusion.Input: observations (x, y); claim: (x, y) ∈ w i ; training data for bootstrap.1 Let g i (x) =p c1 (w i |x) and ĝ i (x), std(ĝ i (x)) denote the mean and standard deviationof ĝ i (x) at point x estimated by bootstraping.2 Let g i (y) =p c1 (w i |y) and ĝ i (y), std(ĝ i (y)) denote the mean and standard deviationof ĝ i (y) at point y estimated by bootstraping.3 Define gi u(z) =k[ĝ i(z)+w std(ĝ i (z))], z = x, y, the modified discriminant function,4 with5 w: a weighting parameterk: a normalizing constant, such that ∑ 6 n ci=1 gu i (z) = 17 Fusion decision rule:{ true if guAccept(x, y ∈ w i )=i (x)gi u(y) 1p(w i ) >λ8false otherwise9 where λ is tuned for a particular value of FRR and FAR.This reject option was introduced by Chow [40], and is extended in [53] by assuming thatmore classes can exist than the ones in the training phase or that some of the classes didn’thave enough information on the training phase. Another extension to the work createdthe concepts of ambiguity rejection (the same defined by Chow), and distance rejection[167], creating a metric to detect if a sample is too far from the classes sample space to beclassified. In [8] the reject option was studied for the K-Nearest-Neighbor (K-NN) classifier.An interesting approach was followed in [95], where instead of rejecting or accepting asample, the concept of group classification is created. For a particular λ there is a set ofclasses to which the sample can be classified with error probability lower than 1 − λ.The uncertainty term was used in classification tasks mainly by using the Dempster-Shafer theory of evidence [212]. In [5] The Dempspter-Shafer theory is used to combine evidencefrom several classifiers and use a belief metric computed for each classifier. The termconfidence measures [185] is also related to uncertainty computation as used in BayesianNetworks. None of these approaches focus on the study of the estimated error probabilityuncertainty for a particular classifier, as proposed here.5.5 ConclusionsWe have presented the set of classification techniques that support our approach for biometricclassification. They have been studied, proposed and developed for the complex problem