1、Pattern Classification Chapter 2(Part 3),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,Pattern Classification Chapter 2(Par
2、t 3),1,Chapter 2 (part 3) Bayesian Decision Theory (Sections 2-6,2-9),Discriminant Functions for the Normal Density Bayes Decision Theory Discrete Features,Pattern Classification Chapter 2(Part 3),2,2.6 Discriminant Functions for the Normal Density,We saw that the minimum error-rate classification c
3、an be achieved by the discriminant functiongi(x) = ln p(x | i) + ln P(i) Case of multivariate normal,Pattern Classification Chapter 2(Part 3),3,Case i = 2.I (I stands for the identity matrix),Pattern Classification Chapter 2(Part 3),4,A classifier that uses linear discriminant functions is called “a
4、 linear machine” The decision surfaces for a linear machine are pieces of hyperplanes defined by: gi(x) = gj(x),Pattern Classification Chapter 2(Part 3),5,Pattern Classification Chapter 2(Part 3),6,The hyperplane separating Ri and Rj always orthogonal to the line linking the means!,Pattern Classific
5、ation Chapter 2(Part 3),7,Pattern Classification Chapter 2(Part 3),8,Pattern Classification Chapter 2(Part 3),9,Case i = (covariance of all classes are identical but arbitrary!) Hyperplane separating Ri and Rj(the hyperplane separating Ri and Rj is generally not orthogonal to the line between the me
6、ans!),Pattern Classification Chapter 2(Part 3),10,Pattern Classification Chapter 2(Part 3),11,Pattern Classification Chapter 2(Part 3),12,Case i = arbitrary The covariance matrices are different for each category(Hyperquadrics which are: hyperplanes, pairs of hyperplanes, hyperspheres, hyperellipsoi
7、ds, hyperparaboloids, hyperboloids),Pattern Classification Chapter 2(Part 3),13,Pattern Classification Chapter 2(Part 3),14,Pattern Classification Chapter 2(Part 3),15,Example R1(3,8),(3,4),(2,6),(4,6); R2(3,0),(3,-4),(1,-2),(5,-2),Pattern Classification Chapter 2(Part 3),16,Pattern Classification C
8、hapter 2(Part 3),17,2.9 Bayes Decision Theory Discrete Features,Components of x are binary or integer valued, x can take only one of m discrete values v1, v2, , vm,Pattern Classification Chapter 2(Part 3),18,Case of independent binary features in 2 category problemLet x = x1, x2, , xd t where each xi is either 0 or 1, with probabilities: pi = P(xi = 1 | 1)qi = P(xi = 1 | 2),Pattern Classification Chapter 2(Part 3),19,The discriminant function in this case is:,Pattern Classification Chapter 2(Part 3),20,Assignment :2.6.25, 2.9.43,
