模式识别Chapter 2-2.pptVIP

* Discriminant functions Case 1 Euclidean Distance Face? * Discriminant functions Case 2 Another simple case arises when the covariance matrices for all of the classes are identical but otherwise arbitrary. * Discriminant functions Case 2 Mahalanobis Distance * Discriminant functions Case 2 Expansion of the quadratic form, the resulting discriminant functions are again linear. * Discriminant functions Case 2 * Discriminant functions Case 3 In general multi-variate normal case, the covariance matrices are different for each category. * Discriminant functions Case 3 * Discriminant functions Case 3 The discrininant functions are inherently quadratic: * Discriminant functions Case 3 * Example Determine the Decision boundary for two-dimensional Gaussian data. * Discriminant functions Example: Decision region for two-dimensional Gaussian data. * Discriminant functions Example: Decision region for two-dimensional Gaussian data. * * Bayes decision theory The basic ideas underlying Bayes decision theory are very simple. To minimize the overall risk, one should always choose the action that minimize the conditional risk. In particular, to minimize the probability of error in a classification problem, one should choose the state of nature that maximizes the posterior probability. Bayes formula allows us to calculate such probabilities from the prior probabilities and the conditional probabilities. * 作业 三 1. 已知甲类：P(ω1) = 0.7和类条件概率密度函数p(x|ω1) ,乙类：P(ω2) = 0.3和类条件概率密度函数p(x|ω2) 今有待分类样本特征观察值x = 10，且由函数曲线查得p(10|ω1) = 0.2， p(10|ω2) = 0.5 ⑴试用最小错误率Bayes决策对样本x = 10进行分类 ⑵试用最小风险Bayes决策对该样本进行分类，设λ11=λ22=0，λ12=2，λ21=1 试给出决策面方程 * 作业三 2. 在 的情况下，说明： (1) 若 ，则分类超平面靠近先验概率较小的类。 (2) 在什么情况下，先验概率对超平面的位置影响不大。 * Pattern Recognition 北京交通大学 电子信息工程学院 Chapter 2 Bayesian Decision Theory Introduction Bayesian decision theory -- error/risk minimum Classifiers, discriminant function The normal density Discriminant function for normal density