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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

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Chapter 2 (part 3) Bayesian Decision Theory (Sections 2-6,2-9) Discriminant Functions for the Normal Density Bayes Decision Theory – Discrete Features

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Pattern Classification, Chapter 2 (Part 3) 2 Discriminant Functions for the Normal Density We saw that the minimum error-rate classification can be achieved by the discriminant function g i (x) = ln P(x | i ) + ln P( i ) Case of multivariate normal

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Pattern Classification, Chapter 2 (Part 3) 3 Case i = 2 I ( I stands for the identity matrix) What does “ i = 2 I ” say about the dimensions? What about the variance of each dimension?

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Pattern Classification, Chapter 2 (Part 3) 4 We can further simplify by recognizing that the quadratic term x t x implicit in the Euclidean norm is the same for all i.

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Pattern Classification, Chapter 2 (Part 3) 5 A classifier that uses linear discriminant functions is called “a linear machine” The decision surfaces for a linear machine are pieces of hyperplanes defined by: g i (x) = g j (x) The equation can be written as: w t (x-x 0 )=0

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Pattern Classification, Chapter 2 (Part 3) 6 The hyperplane separating R i and R j always orthogonal to the line linking the means!

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Pattern Classification, Chapter 2 (Part 3) 7

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10 Case i = (covariance of all classes are identical but arbitrary!) Hyperplane separating R i and R j (the hyperplane separating R i and R j is generally not orthogonal to the line between the means!)

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Pattern Classification, Chapter 2 (Part 3) 11

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Pattern Classification, Chapter 2 (Part 3) 12

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Pattern Classification, Chapter 2 (Part 3) 13 Case i = arbitrary The covariance matrices are different for each category The decision surfaces are hyperquadratics (Hyperquadrics are: hyperplanes, pairs of hyperplanes, hyperspheres, hyperellipsoids, hyperparaboloids, hyperhyperboloids)

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Pattern Classification, Chapter 2 (Part 3) 14

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Pattern Classification, Chapter 2 (Part 3) 15

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Pattern Classification, Chapter 2 (Part 3) 16 Bayes Decision Theory – Discrete Features Components of x are binary or integer valued, x can take only one of m discrete values v 1, v 2, …, v m concerned with probabilities rather than probability densities in Bayes Formula:

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Pattern Classification, Chapter 2 (Part 3) 17 Bayes Decision Theory – Discrete Features Conditional risk is defined as before: R( |x) Approach is still to minimize risk:

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Pattern Classification, Chapter 2 (Part 3) 18 Bayes Decision Theory – Discrete Features Case of independent binary features in 2 category problem Let x = [x 1, x 2, …, x d ] t where each x i is either 0 or 1, with probabilities: p i = P(x i = 1 | 1 ) q i = P(x i = 1 | 2 )

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Pattern Classification, Chapter 2 (Part 3) 19 Bayes Decision Theory – Discrete Features Assuming conditional independence, P(x| i ) can be written as a product of component probabilities:

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Pattern Classification, Chapter 2 (Part 3) 20 Bayes Decision Theory – Discrete Features Taking our likelihood ratio

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Pattern Classification, Chapter 2 (Part 3) 21 The discriminant function in this case is:

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