模式识别Chapter 2-1课件

1、1Pattern 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 Discriminant function The normal density Discriminant function for normal density Pattern Recognition System4An example character Recognitionab?5An example character Recogn

2、itionabTwo classes , denoted by C1, C2C1C27举例 字符识别Corresponding to a;Corresponding to b;8举例 字符识别10Features 输入矢量的维数很高 设法降低维数特征字符高与宽的比率 v11Features字符高与宽的比率 v如何选择合适的阈值？12Features 特征T直方图字符高与宽的比率14The overall system can be viewed as a mapping from a set of input variables , to an output variables .The pr

3、oblem is how to get the mapping function.Classify （分类）15ClassifyClassifyx1x217Classify18Classify19模式识别系统的主要构成怎样才能得最好的分类效果呢？20模式识别系统的主要构成统计模式识别神经网络支持向量机 21Bayes theorem Supposing that we wish to classify a new character but as yet we have made no measurements on the image of that character. The goal

4、is to classify the character in such a way as to minimize the probability of misclassification. 22Bayes theoremIf we have collected a large number of examples of the character,we could find the fractions which belong in each of the two classes. 24Bayes theoremPriori probabilities （先验概率）If the letter

5、 a occurs three times as often as the letter b 25Bayes theoremIf we were forces to classify a new character without being allowed to see the corresponding imageThen the best we can do is to assign it to the class having higher prior probability. 先验概率27Bayes theoremNow suppose that we have measured t

6、he value of the feature variables It is clear that this give us further information on which to base our classification decision基于特征分类28Bayes theorem直方图29Bayes theorem Priori probabilities （先验概率）30Bayes theorem Joint probability - the probability that the image has the feature value and belongs to c

7、lass 联合概率 - 属于类 而拥有特征值 的概率31Bayes theoremConditional probability - specifies the probability that the observation falls in column of the array given that it belongs to the class条件概率 - 类 ，同时特征值为 32Bayes theorem33Bayes theorem34Bayes theorem35Bayes theorem36Bayes theoremWhat isThe probability that the

8、 class is given that the measured value falls in the cell Posterior probability/ 后验概率37Bayes theorem in general 38Bayes theoremThe posterior probability gives the probability of the pattern belonging to class once we have observed the feature vector . The probability of misclassification is minimize

9、d by selecting the class having the largest posterior probability39Bayes theorem in general 40Decision boundaries A feature vector x is assigned to class if for all or41Decision boundaries 42R1R243R1R2分类错误最小！44Bayes DecisionThe probability of misclassification is minimized by selecting the class hav

10、ing the largest posterior probability45Example:假设在某个局部地区细胞识别中正常 和异常 两类的先验概率分别为 正常： 异常： 现有一待识别的细胞，其观察值为 ，从类条件概率密度曲线上查得 试对该细胞分类。 46Example:利用贝叶斯公式，分别计算 及 的后验概率。 归类于正常状态47字符识别问题48Bayes DecisionThe probability of misclassification is minimized by selecting the class having the largest posterior probabil

11、ity49Decision functions A feature vector x is assigned to class if for all分类错误最小50Decision functions A feature vector x is assigned to class if for all51Decision functions A feature vector x is assigned to class if for all52Decision functionsA feature vector x is assigned to class if53Decision funct

12、ions54字符识别问题基于最小风险的贝叶斯决策55基于最小错误率的决策基于最小风险的贝叶斯决策56基于最小风险的贝叶斯决策57基于最小风险的贝叶斯决策58基于最小风险的贝叶斯决策59基于最小风险的贝叶斯决策60Take action : decide , if Bayesian decision rule is stated as:基于最小风险的贝叶斯决策61Take action : decide , if 基于最小风险的贝叶斯决策62Bayes decision rule : Take action : decide , If equal to: is called : Likeliho

13、od ratio. 基于最小风险的贝叶斯决策63 Likelihood ratio : Decision rule: If the likelihood ratio of class and exceeds a threshold value (that is independent of the input pattern ), the optimal action is to decide . 64Decision functions A feature vector x is assigned to class if 分类风险最小Minimum-error rate classifica

14、tion65正确类别的后验概率=正确率True state : , correct rate=错误率=1 - 正确率Minimum-error rate classification66Minimum-error rate classification67error rateMinimum-error rate classification6869Bayes Decision: error minimumThe probability of misclassification is minimized by selecting the class having the largest post

15、erior probability70Bayes Decision: risk minimum If the likelihood ratio of class and exceeds a threshold value (that is independent of the input pattern ), the optimal action is to decide . 71小结 分类问题： 数学问题： 如何解决这个数学问题？数学问题从输入到输出的映射概率、 Bayes 决策最小错误率：后验概率最小风险：似然比72作业 二1. 两类样本C1， C2的特征分布直方图如下：试求每类样本的先验

16、概率、(类)条件概率及后验概率。73作业 二2. 已知甲类：P(1) = 0.7和类条件概率密度函数p(x|1) ,乙类：P(2) = 0.3和类条件概率密度函数p(x|2)今有待分类样本特征观察值x = 10，且由函数曲线查得p(10|1) = 0.2， p(10|2) = 0.5试用最小错误率Bayes决策对样本x = 10进行分类试用最小风险Bayes决策对该样本进行分类，设11=22=0，12=2，21=174Bayesian Decision TheoryBayesian decision theory is a fundamental statistical approach to the problem