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模式识别(Pattern Classification)英文版课后答案与解析

机器学习

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"模式识别Pattern Classification的英文版课后习题答案分享，这份资源来自于CSDN，并由用户tommyod在GitHub上更新于2018年12月11日。" "Pattern Recognition"是由Richard O. Duda, Peter E. Hart和David G. Stork合著的一本经典书籍，尽管出版于2001年，但其第二版经受住了时间的考验，成为模式识别或机器学习领域的广泛引用和深受欢迎的入门教材。据谷歌学术统计，该书已获得近40,000次引用。提供的文档包含了书中部分习题的解答，这些解答是用LaTeX排版的，旨在帮助作者自己的学习过程。作者强调，虽然包含章节摘要，但它们不能替代原书的内容，如果没有原书，这些摘要可能难以理解。对于那些希望深入学习机器学习的人来说，作者推荐购买原著。近年来，机器学习领域获得了巨大的关注，尽管网上有许多有价值的资源，但在全面理解和掌握知识点时，书籍仍然是首选。作者分享的解答应能对遇到困难的读者有所帮助，鼓励读者在寻求答案前先自己尝试解决问题。这份资源涵盖了图像处理、模式分类和机器学习的相关习题解答，对于正在学习这些主题的读者来说，是一份宝贵的参考资料。通过解答，读者可以加深对模式识别算法的理解，提升解决实际问题的能力，包括但不限于特征提取、分类模型的构建、训练与验证以及性能评估等方面的知识。

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Figure 7: A dendrogram representation of hierarchical clustering. The source is

Wikipedia.

• Competitive learning is a clustering algorithm implemented by a neural network.

The data x ∈ R

is augmented to x := (1, x) and normalized so that it lies on

a (d + 1)-dimensional sphere. The weight w

∗

= arg max

x is updated to be

more similar to the sample x. The algorithm can also create new clusters w

certain conditions are met. It’s suitable for on-line learning, but will not necessarily

converge with constant learning rate.

• PCA can be implemented as a 3-layer neural network autoencoder. The ﬁrst k

eigenvalues/eigenvectors of X solves the minimization of kX − X

k, where X

a rank k approximation of X. Non-linear PCA is an extension of the idea, using a

5-layer non-linear neural network autoencoder.

Figure 8: A neural network autoencoder with 5 layers. The source is Wikipedia.

• Self-organizing feature maps transform from a high dimensional space to low dimen-

sional one, while preserving local topological information. They are implemented as

neural networks, and a key idea is to update neighboring areas when learning the

map. An additional beneﬁt is that they automatically place more points in regions

of high probability.

2 Solutions to “Pattern Classiﬁcation”

This section contains solutions to problems in “Pattern Classiﬁcation” by Duda et al.

There are approximately 8 solved problems from each chapter.

2.2 Bayesian Decision Theory

Problem 2.6

a) We want the probability of choosing action α

to be smaller than, or equal to, E

given that the true state of nature is ω

. Let’s assume that µ

< µ

and that the

decision threshold is x

∗

, so we decide α

if x > x

∗

. We then have

P (α

| ω

) ≤ E

p(x > x

∗

| ω

) ≤ E



1 −

∗

p(x | ω

) dx



≤ E

We let Φ : R → [0, 1] denote the cumulative Gaussian distribution, and Φ

−1

[0, 1] → R it’s inverse function. Making use of Φ we write

1 − Φ



∗

− µ



≤ E

∗

≥ µ

+ σ

−1

(1 − E

) .

If the desired error is close to zero, then x

∗

goes to positive inﬁnity. If the desired

error is close to one, then x

∗

goes to negative inﬁnity.

b) The error rate for classifying ω

as ω

P (α

| ω

) = p(x ≤ x

∗

| ω

) =

∗

p(x | ω

) dx = Φ



∗

− µ



Making use of x

∗

from the previous problem, we obtain



+ σ

−1

(1 − E

) − µ



= Φ



− µ

−1

(1 − E

)



c) The overall error rate becomes

P (error) = P (α

, ω

) + P (α

, ω

)

= P (α

| ω

)P (ω

) + P (α

| ω

)P (ω

)

[P (α

| ω

) + P (α

| ω

)]



+ Φ



− µ

−1

(1 − E

)



In the last equality we used the results from the previous subproblems.

d) We substitute the given values into the equations, and obtain x

∗

≈ 0.6449. The

total error rate is P (error) ≈ 0.2056.

e) The Bayes error rate, as a function of x

∗

, is given by

P (error) = P (α

| ω

)P (ω

) + P (α

| ω

)P (ω

)

[p(x > x

∗

| ω

) + p(x < x

∗

| ω

)]



1 − Φ



∗

− µ



+ Φ



∗

− µ



The Bayes error rate for this problem is depicted in Figure 9.

4 3 2 1 0 1 2 3 4

0.0

0.1

0.2

0.3

0.4

0.5

(

)

(

)

(

error

)

Figure 9: Graf accompanying problem 2.6.

Problem 2.12

a) A useful observation is that the maximal value P (ω

max

| x) is greater than, or equal

to, the average. Therefore we obtain

P (ω

max

| x) ≥

i=1

P (ω

| x) =

where the last equality is due to probabilities summing to unity.

b) The minimum error rate is achieved by choosing ω

max

, the most likely state of nature.

The average probability of error over the data space is therefore the probability that

max

is not the true state of nature for a given x, i.e.

P (error) = E

[1 − P (ω

max

| x)] = 1 −

P (ω

max

| x)p(x) dx.

c) We see that

P (error) = 1 −

P (ω

max

| x)p(x) dx ≤ 1 −

p(x) dx = 1 −

c − 1

where we have used the fact that

p(x) dx = 1.

d) A situation where P (error) = (c − 1)/c arises when P (ω

) = 1/c for every i. Then

the maximum value is equal to the average value, and the inequality in problem a)

becomes an equality.

Problem 2.19

a) The entropy is given by H [p(x)] = −

p(x) ln p(x) dx. The optimization problem

gives the synthetic function (or Lagrange function)

= −

p(x) ln p(x) dx +

k=1



(x)p(x) dx − a



and since a probability density function has

p(x) dx = 1 we add an additional

constraint for k = 0 with b

(x) = 1 and a

= 1. Collecting terms we obtain

= −

p(x) ln p(x) dx +

k=0

(x)p(x) dx −

k=0

= −

p(x)

ln p(x) −

k=0

(x)

dx −

k=0

which is what we were asked to show.

b) Diﬀerentiating the equation above with respect to p(x) and equating it to zero we

obtain

−

ln p(x) −

k=0

(x)

+ p(x)



p(x)



dx = 0.

This integral is zero if the integrand is zero for every x, so we require that

ln p(x) −

k=0

(x) + 1 = 0,

and solving this equation for p(x) gives the desired answer.

Problem 2.21

We are asked to compute the entropy of the (1) Gaussian distribution, (2) triangle distri-

bution and (3) uniform distribution. Every probability density function (pdf) has µ = 0

and standard deviation σ, and we must write every pdf parameterized using σ.

Gaussian We use the deﬁnition H [p(x)] = −

p(x) ln p(x) dx to compute

H [p(x)] = −

√

2πσ

exp



−





√

2πσ



−



dx.

Let us denote the constant by K =

√

2πσ

to simplify notation. We obtain

−

K exp



−



ln K −



dx =

−K ln K

exp



−



dx + K

exp



−



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好的，您的问题是关于手写字母识别的KNN模型和classification_report表的使用。我可以为您提供一些相关的信息和指导。首先，KNN是一种常用的机器学习算法，可用于分类和回归问题。在手写字母识别中，我们可以将每...

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