模式识别课后习题(英文)

P ( x | i ) ~ N ( i , i ) .
(1) (2)
Deduce h( x) and find the decision rule; Let

1 2
.Find the decision rule;
(3)Analyze the decision surface types in question(1) and question(2).
1 2
rule may be expressed as x
2.7 2.7
if
p ( x | 1 ) (12 22 ) p(2 ) £ p ( x | 2 ) (21 11 ) p(1 )
.
若 11 22 0 ， 12 21 ，证明此时最小最大决策面是来自两类的错误率相等。
2
x12 0, 0,1
2 x2 0,1, 0 2 x3 0,1,1 2 x4 1,1,1 T
x1 2 1, 0, 0
1 x3 1, 0,1
T
T
T
T
x1 4 1,1, 0
T
T
Respectively reduce the feature space dimension to d 2 and d 1 , then describe the positions of the samples in the feature space. 10.5 Let x1 , x2 , x3 and x4 , and consider the 5 4 1 0
习题 2.24 的情况下，若考虑损失函数 11 22 0 ， 12 21 ，画出似然比阈与错
误率之间的关系。 (1) 求出 pi (error ) 0.05 时完成 Neyman-Pearson 决策时总的错误率；
(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 11 22 0 ， 12 21 . (1) Consider the Neyman-Pearson criterion , what is the error rate for pi (error ) 0.05 ; (2) Calculate the threshold of the minimax decision and overall error rate. 3.1 Consider the sample set = x1 , x2 ,…,xN with the distribution is
付对数似然比为 h( x ) ln l ( x ) ， 当 P ( x | i ) 是均值向量为 i 和协方差矩阵为 的正态分布时： （1）试推导出 h( x ) ，并指出其决策规则；

i
（2）当

1 2
时，推导 h( x ) 及其决策规则；
（3）分析（1） ， （2）两种情况下的决策面类型。 2.22
ˆ of the parameter P . maximum likelihood estimate P
3.4
Suppose that the loss function is the quadratic function
ˆ, P P ˆP P



2
and the prior density of P
follows the uniform
ˆ distribution f P 1 , 0 P 1 . Calculate the Bayesian estimation P
under the condition of exercise 3.3. 3.14 Consider the sample set
2 1 2
T
， 2 1, 0 ，
T
I , p( ) p( ) 。Find the minus-log-likelihood ratio decision rule。
2.24 在 2.23 中，若
1 1 1 1 2 2 ，写出负对数似然比规 ， , 1 2 1 1 2 1 1 1 2 2
2.6 若 对两类问题，证明最小风险贝叶斯决策规则可表示为
p ( x | 1 ) (12 22 ) p(2 ) ¤ , 则x 1 p ( x | 2 ) (21 11 ) p(1 ) 2
。
2.6
In the two-category case, show that the minimum risk Bayes decision
2.23 二维正态分布， 1 1, 0 ， 2 1, 0 ，
T T
I , p( ) p( ) 。试写出负
1 2 1 2
对数似然比决策规则。 2.23
1
Let P ( x | i ) ~ N ( i , i ) ， 1 1, 0
4 1 0 5
following three partitions: ⑴ 1 = x1 , x2 , 2 = x3 , x4 ⑵ 1 = x1 , x4 , 2 = x2 , x3 ⑶ 1 = x1 , x2 , x3 , 2 = x4 Show that by the sum-of-square error J c criterion, the third partition is favored, whereas by the S criterion the first two partitions are favored.
3.3
Consider the sample set = x1 , x2 ,…,xN drawn from a binomial
distribution f x, P P xQ 1 x , x 0,1 , 0 P 1 , Q 1 P . Calculate the
Consider minimax criterion for 11 22 0 and 12 21 .
Prove that in this case p1 (error ) p2 (error ) .
2.22 似然比决策准则为 若 l ( x)
1 p ( x | 1 ) p (2 ) 则 x ¤ p ( x | 2 ) p (1 ) 2
= x1 , x2 , …,xN
drawn from a is known.
multivariate normal distribution p x ~ N ,
where
ˆ of . Calculate the maximum likelihood estimate
3.2
Consider the sample set = x1 , x2 ,…,xN drawn from a multivariate
N ， 2 . Respectively calculate the maximum
normal population
ˆ， ˆ 2 of ， 2 . likelihood estimate
4.4 Consider a two-dimensional linear discriminant（判别） function
g x x1 2 x2 2
⑴ Transform the discriminant function into the form of g x wT x 0 , and describe the geometric figure（几何图形） of g x 0 ; ⑵ Map the discriminant function to obtain the generalized( 广 义 ) homogeneous（齐次） linear discriminant function g x aT y . ⑶ Show that the X-space is actually a subspace of the Y-space, and the partition of the X-space by aT y 0 is the same as the partition of the X-space by wT x 0 0 in the original space. Describe it by a figure. 8.1 Given three partitions 1，2，3 as shown in the figure below.
density N ,1 , - , where the prior distribution of
p x ~ N 0,1 . Respectively calculate the maximum likelihood estimate
ˆ. and the Bayesian estimation
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
P (i | x ) P ( j | x )
2.5
对一切 j i成立时，x 1 。
则。
2.24
Let 1 2
1 1 1 1 2 2 . ， 1 , 2 1 1 1 1 2 2
Find the minus-log-likelihood ratio decision rule under the condition of exercise 3.3. 2.25
Likelihood ratio decision rules can be expressed as if l ( x)
p ( x | 1 ) p (2 ) . ¤ p ( x | 2 ) p (1 )
x 1 2
minus-log-likelihood ratio can be expressed as h( x) ln l ( x) ,where
1 x3 1, 0,1
T
T
T
T
x1 4 1,1, 0
T
T
Calculate the transform to obtain the biggest
1 by J 2 =tr S Sb .
J2
expressed
9.1 Given two sample sets
1
1 x1 0, 0, 0 T
(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 P (i | x) P ( j | x) where j i and x 1 .
Calculate S , Sb . 8.7 Given two sample sets
ห้องสมุดไป่ตู้
1
1 x1 0, 0, 0 T
2
x12 0, 0,1
2 x2 0,1, 0 2 x3 0,1,1 2 x4 1,1,1 T
x1 2 1, 0, 0