频谱特征选择：数据挖掘的最新指南

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《频谱特征选择在数据挖掘中的应用》是一本由郑艾伦、赵恒禄编著的专业书籍，收录于CRC出版社的数据挖掘与知识发现系列中。该书深入探讨了如何利用频谱特征选择技术来优化数据挖掘过程，特别是在处理复杂数据集时，频谱分析作为一种矩阵分解方法，能显著提升数据理解与处理效率。在当今的数据挖掘领域，特征选择是一个关键步骤，它涉及到从原始数据中挑选出最具信息价值的属性，以减少冗余和噪声，提高模型的性能。《SpectralFeatureSelection for Data Mining》详细介绍了如何通过频谱分析这一工具来实现这一目标，这种方法特别适用于处理大规模、高维度的数据集，如社交媒体数据、图像或文本数据，其中包含丰富的频域信息。作者们强调了矩阵分解技术在特征选择中的作用，如奇异值分解（SVD）等，这些技术能够揭示数据中的潜在结构，从而揭示隐藏的模式和规律。通过频谱特征选择，可以减少数据处理的时间复杂度，并有助于提高诸如分类、聚类、关联规则挖掘等数据挖掘任务的准确性。书中还提及其他相关领域的研究，例如《Computational Methods of Feature Selection》探讨了更广泛的特征选择方法论，而《Constrained Clustering: Advances in Algorithms, Theory, and Applications》则展示了约束聚类在反恐和执法等领域中的应用。此外，《Next Generation of Data Mining》和《Text Mining: Classification, Clustering, and Applications》则分别关注了数据挖掘的前沿技术和文本挖掘的实践。对于商业应用，《Data Mining for Design and Marketing》和《Datamining for Design and Marketing》展示了如何将频谱特征选择融入产品设计和市场策略中，以提升决策效率。《The Top Ten Algorithms in Data Mining》则提供了实践性很强的算法指南，便于读者理解和应用。《SpectralFeatureSelection for Data Mining》是一本极具实用价值的参考资料，不仅适合数据挖掘领域的研究人员，也对数据工程师、分析师和业务人员提供了深入理解并有效运用频谱特征选择进行数据分析的工具和策略。无论是在理论研究还是实际项目中，都能找到宝贵的知识和灵感。

Symbol Description

n Number of instances

m Number of features

C Number of classes

l Number of selected features

F A set of features

The i-th feature

X Data matrix

The i-th feature vector,

X = [f

, . . . , f

]

The i-th instance,

X = [x

, . . . , x

]

y Target vector

Y Target matrix

W Weight matrix

The i-th row of the weight

matrix W

R Residual matrix

A Active set

G A graph

S Similarity matrix

A Adjacency matrix

L Laplacian matrix

D Degree matrix

L Normalized Laplacian ma-

trix, L = D

−1/2

The i-th eigenvector

The i-th eigenvalue

K Kernel matrix

C Covariance matrix

I Identity matrix

1 1 = [1, . . . , 1]

λ A regularization parameter

F EA

Knowledge sources related

to features

SAM

Knowledge sources related

to instances

int

Internal knowledge

ext

External knowledge

exp (·) Exponential function

log (·) Logarithm function

k · k A norm

kak

norm of vector a

kak

norm of vector a

kak

norm of vector a

kAk

norm of matrix A

kAk

2,1

norm of matrix A

kAk

Frobenius norm of matrix A

M(·) Model function

Trace(·) Trace of a matrix

Card (·) Cardinality of a set

ϕ (·) Feature ranking function

Q (·) Q function

R Real numbers

Real n-vectors (n × 1 matri-

ces)

n×m

Real n × m matrices

Chapter 1

Data of High Dimensionality and

Challenges

Data mining is a multidisciplinary methodology for extracting nuggets of

knowledge from data. It is an iterative process that generates predictive and

descriptive models for uncovering previously unknown trends and patterns via

analyzing vast amounts of data from various sources. As a powerful tool, the

data mining technology has been used in a wide range of proﬁling practices,

such as marketing, decision-making support, fraud detection, and scientiﬁc

discovery, etc. In the past 20 years, the dimensionality of the data sets in-

volved in data mining applications has increased dramatically. Figure 1.1 plots

the dimensionality of the data sets posted in the UC Irvine Machine Learning

Repository [53] from 1987 to 2010. We can observe that in the 1980s, the max-

imal dimensionality of the data is only about 100; in the 1990s, this number

increases to more than 1500; and in the 2000s, it further increases to about

3 millon. The trend line in the ﬁgure is obtained by ﬁtting an exponential

function on the data. Since the y-axis is in logarithm, it shows the increasing

trend of the dimensionality of the data sets is exponential.

Data sets with very high (>10,000) dimensionality are quite common nowa-

days in data mining applications. Figure 1.2 shows three types of data that

are usually of very high dimensionality. With a large text corpus, using the

bag-of-words representation [49], the extracted text data may contain tens of

thousands of terms. In genetic analysis, a cDNA-microarray data [88] may

contain the expression of over 30,000 DNA oligonucleotide probes. And in

medical image processing, a 3D magnetic resonance imaging (MRI) [23] data

may contain the gray level of several million pixels. In certain data mining ap-

plications, involved data sets are usually of high dimensionality, for instance,

text analysis, image analysis, signal processing, genomics and proteomics anal-

ysis, and sensor data processing, to name a few.

The proliferation of high-dimensional data within many domains poses

unprecedented challenges to data mining [71]. First, with over thousands of

features, the hypothesis space becomes huge, which allows learning algorithms

to create complex models and overﬁt the data [72]. In this situation, the

performance of learning algorithms likely degenerates. Second, with a large

number of features in the learning model, it will be very diﬃcult for us to

understand the model and extract useful knowledge from it. In this case, the

interpretability of a learning model decreases. Third, with a huge number of

Data of High Dimensionality and Challenges 3

features, the speed of a learning algorithm slows down and their computational

eﬃciency declines. Below is an example that shows the impact of the data

dimensionality on learning performance.

Example 1 Impact of data dimensionality on learning performance

When data dimensionality is high, many of the features can be irrelevant

or redundant. These features can have negative eﬀect on learning models,

and decrease the performance of learning models signiﬁcantly.

To show this eﬀect, we generate a two-dimensional data set with three

classes, whose distribution is shown in Figure 1.3. We also generate diﬀerent

numbers of irrelevant features and add these features to the data set. We

then apply a k nearest neighbor classiﬁer (k-nn, k=3) with 10-fold cross-

validation on the original data set as well as the data sets with irrelevant

features. The obtained accuracy rates are reported in Figure 1.4(a). We can

observe that on the original data set, the k-nn classiﬁer is able to achieve

an accuracy rate of 0.99. When more irrelevant feature are added to the

original data set, its accuracy decreases. When 500 irrelevant features are

added, the accuracy of k-nn declines to 0.52. Figure 1.4(b) shows the com-

putation time used by k-nn when diﬀerent numbers of irrelevant features

are added to the original data. We can see when more features present in

the data, both the accuracy and the eﬃciency of the k-nn decrease. This

phenomenon is also known as the curse of dimensionality, which refers

to the fact that many learning problems become less tractable as feature

number increases [72].

1.1 Dimensionality Reduction Techniques

In data mining applications with high-dimensional data, dimensionality

reduction techniques [107] can be applied to reduce the dimensionality of the

original data and improve learning performance. By removing the irrelevant

and redundant features in the data, or by eﬀectively combining original fea-

tures to generate a smaller set of features with more discriminant power, di-

mensionality reduction techniques bring the immediate eﬀects of speeding up

data mining algorithms, improving performance, and enhancing model com-

prehensibility. Diﬀerent types of dimensionality reduction techniques generally

fall into two categories: feature selection and feature extraction.

Figure 1.5 shows the general idea of how feature selection and feature ex-

traction work. Given a large number of features, many of these features may

be irrelevant or redundant. Feature selection achieves dimensionality reduc-

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频谱特征选择：数据挖掘的最新指南

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