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Pattern Classification, Chapter 2 (Part 2) 0 Pattern Classification All materials in these slides were taken from Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons, 2000 with the permission of the authors and the publisher

Chapter 2 (Part 2): Bayesian Decision Theory (Sections ) Minimum-Error-Rate Classification Classifiers, Discriminant Functions and Decision Surfaces The Normal Density

Pattern Classification, Chapter 2 (Part 2) 2 Minimum-Error-Rate Classification Actions are decisions on classes If action  i is taken and the true state of nature is  j then: the decision is correct if i = j and in error if i  j Seek a decision rule that minimizes the probability of error which is the error rate

Pattern Classification, Chapter 2 (Part 2) 3 Introduction of the zero-one loss function: Therefore, the conditional risk is: “The risk corresponding to this loss function is the average probability error” 

Pattern Classification, Chapter 2 (Part 2) 4 Minimize the risk requires maximize P(  i | x) (since R(  i | x) = 1 – P(  i | x)) For Minimum error rate Decide  i if P (  i | x) > P(  j | x)  j  i

Pattern Classification, Chapter 2 (Part 2) 5 Regions of decision and zero-one loss function, therefore: If is the zero-one loss function wich means:

Pattern Classification, Chapter 2 (Part 2) 6

7 Classifiers, Discriminant Functions and Decision Surfaces The multi-category case Set of discriminant functions g i (x), i = 1,…, c The classifier assigns a feature vector x to class  i if: g i (x) > g j (x)  j  i

Pattern Classification, Chapter 2 (Part 2) 8

9 Let g i (x) = - R(  i | x) (max. discriminant corresponds to min. risk!) For the minimum error rate, we take g i (x) = P(  i | x) (max. discrimination corresponds to max. posterior!) g i (x)  P(x |  i ) P(  i ) g i (x) = ln P(x |  i ) + ln P(  i ) (ln: natural logarithm!)

Pattern Classification, Chapter 2 (Part 2) 10 Feature space divided into c decision regions if g i (x) > g j (x)  j  i then x is in R i ( R i means assign x to  i ) The two-category case A classifier is a “dichotomizer” that has two discriminant functions g 1 and g 2 Let g(x)  g 1 (x) – g 2 (x) Decide  1 if g(x) > 0 ; Otherwise decide  2

Pattern Classification, Chapter 2 (Part 2) 11 The computation of g(x)

Pattern Classification, Chapter 2 (Part 2) 12

Pattern Classification, Chapter 2 (Part 2) 13 The Normal Density Univariate density Density which is analytically tractable Continuous density A lot of processes are asymptotically Gaussian Handwritten characters, speech sounds are ideal or prototype corrupted by random process (central limit theorem) Where:  = mean (or expected value) of x  2 = expected squared deviation or variance

Pattern Classification, Chapter 2 (Part 2) 14

Pattern Classification, Chapter 2 (Part 2) 15 Multivariate density Multivariate normal density in d dimensions is: where: x = (x 1, x 2, …, x d ) t (t stands for the transpose vector form)  = (  1,  2, …,  d ) t mean vector  = d*d covariance matrix |  | and  -1 are determinant and inverse respectively