Thesis - Instituto de Telecomunicações

108 CHAPTER 5. FEATURE SELECTION AND CLASSIFICATIONFigure 5.8: Distribution of ˆp(x|w i ). The mean of p(x) is the solid line, and the dashed linesrepresent ˆp(x|w i )± its standard deviation.Our goal is to be able to compute the variance of the discriminant functions, ˆp(w i |x) thatfollow equation 5.16. This is an even more complex form of operations over distributionsthan the ones previously presented.We decided to estimate the discriminant functions variance via the bootstrap approach[58, 59, 111], given the observed difficulties. We estimate the distribution of ˆp(w i |x ∗ ) (wherex ∗ is the sample we are classifying). We can look at the distribution ĝ i (x ∗ ) determiningit based on bootstrap estimates. We create bootstrap sets of size n, represented byx b = x b 1 ,xb 2 , ···xb n, obtained by sampling with replacement from the training populationof m training samples x = [x 1 ,x 2 ···x m ]. With each bootstrap set, x b , we generate asample ĝi b(x∗ ) to extract statistical properties, providing insight into the random variableĝ i (x ∗ ). This approach can be done in either a parametric or non-parametric estimationmethodologies.Using the bootstrap approach we estimate the variance of ĝ i (x ∗ ) for all the classes. Fora sufficiently large n, the mean of ĝ i (x ∗ ) is equal to g i (x ∗ ).5.4.2 Uncertainty Based Reject Option ClassifierThe reject option classification is based on a classifier that either select one of the classes,or rejects a sample based on the estimated error probability. We propose a simple extension