模式识别课后习题(英文)

Pattern Recognition Theory and Its Application PROBLEMS 2 5 1 对 C 类情况推广最小错误率贝叶斯决策规则 2 指出此时使最小错误率最小等价于后验概率最大 即 对一切 ij PxPx 1 ji 成立时 x 2 5 1 Generalize the minimum error Bayes decision rule in case of class C 2 Show that the minimum error rate is equivalent to the maximum posterior probability namely where and ij PxPx ji 1 x 2 6 对两类问题 证明最小风险贝叶斯决策规则可表示为 若 1 12222 2211 1 121 p x p x p x p 则 2 6 In the two category case show that the minimum risk Bayes decision rule may be expressed as if 1 2 x 12222 21121 1 1 p p p x p x 2 7 若 证明此时最小最大决策面是来自两类的错误率相等 1122 0 1221 2 7 Consider minimax criterion for and 1122 0 1221 Prove that in this case 12 p errorp error 2 22 似然比决策准则为 若 则 1 2 2 1 px p xp p l x 1 2 x 付对数似然比为 当是均值向量为 和协方差矩阵为 ln h xl x i P x i 的正态分布时 i 1 试推导出 并指出其决策规则 h x 2 2 当时 推导及其决策规则 12 h x 3 分析 1 2 两种情况下的决策面类型 2 22 Likelihood ratio decision rules can be expressed as if 1 2 x 1 2 2 1 px p xp p l x minus log likelihood ratio can be expressed as where ln h xl x ii i P xN 1 Deduce and find the decision rule h x 2 Let Find the decision rule 12 3 Analyze the decision surface types in question 1 and question 2 2 23 二维正态分布 试写出 1 1 0 T 2 1 0 T 12 I 12 pp 负对数似然比决策规则 2 23 Let ii i P xN 1 1 0 T 2 1 0 T 12 I Find the minus log likelihood ratio decision rule 12 pp 2 24 在 2 23 中 若 写出负对数似然比 12 12 11 11 22 11 11 22 规则 2 24 Let 12 12 11 11 22 11 11 22 Find the minus log likelihood ratio decision rule under the condition of exercise 3 3 2 25 习题 2 24 的情况下 若考虑损失函数 画出似然比阈与错 1122 0 1221 误率之间的关系 1 求出时完成 Neyman Pearson 决策时总的错误率 0 05 i p error 3 2 求出最小最大决策的阈值和总的错误率 2 25 under the condition of exercise 3 3 let 1122 0 1221 1 Consider the Neyman Pearson criterion what is the error rate for 0 05 i p error 2 Calculate the threshold of the minimax decision and overall error rate 3 1 Consider the sample set with the distribution 12 N x xx density where the prior distribution of is 1N Respectively calculate the maximum likelihood estimate 0 1p xN and the Bayesian estimation 3 2 Consider the sample set drawn from a multivariate 12 N x xx normal population Respectively calculate the maximum 2 N likelihood estimate of 2 2 3 3 Consider the sample set drawn from a binomial 12 N x xx distribution Calculate the 1 xx f x PP Q 0 1x 01P 1QP maximum likelihood estimate of the parameter PP 3 4 Suppose that the loss function is the quadratic function and the prior density of follows the uniform 2 P PPP P distribution Calculate the Bayesian estimation under 1f P 01P P the condition of exercise 3 3 3 14 Consider the sample set drawn from a multivariate 12 N x xx normal distribution where is known Calculate the p xN maximum likelihood estimate of 4 4 Consider a two dimensional linear discriminant 判别 function 4 12 22g xxx Transform the discriminant function into the form of 0 T g xw x and describe the geometric figure 几何图形 of 0g x Map the discriminant function to obtain the generalized 广义 homogeneous 齐次 linear discriminant function T g xa y Show that the X space is actually a subspace of the Y space and the partition of the X space by is the same as the partition of the X 0 T a y space by in the original space Describe it by a figure 0 0 T w x 8 1 Given three partitions as shown in the figure below 123 Calculate b SS 8 7 Given two sample sets 1 2 5 1 1 0 0 0 T x 2 1 0 0 1 T x 1 2 1 0 0 T x 2 2 0 1 0 T x 1 3 1 0 1 T x 2 3 0 1 1 T x 1 4 1 1 0 T x 2 4 1 1 1 T x Calculate the transform to obtain the biggest expressed by 2 J 1 2 trb JS S 9 1 Given two sample sets 1 2 1 1 0 0 0 T x 2 1 0 0 1 T x 1 2 1 0 0 T x 2 2 0 1 0 T x 1 3 1 0 1 T x 2 3 0 1 1 T x 1 4 1 1 0 T x 2 4 1 1 1 T x Respectively reduce the feature space dimension to and then 2d 1d describe the positions of the samples in the feature space 10 5 Let and and consider the 1 4 5 x 2 1 4 x 3 0 1 x 4 5 0 x following three par