SVM.zip_SVM对于数据的预测_svm预测资源-CSDN文库

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支持向量机（Support Vector Machine，简称SVM）是一种广泛应用于分类和回归分析的机器学习算法。SVM的核心思想是找到一个最优超平面，将不同类别的数据尽可能地分开，同时保持两类样本的距离最大化，以达到最好的泛化能力。在这个" SVM.zip_SVM对于数据的预测_svm预测 "的压缩包中，很可能包含了一个小型的SVM预测程序，用于帮助用户直观地理解和应用SVM。 1. **SVM的基本原理**： SVM通过构建一个超平面，使得训练数据集中距离超平面最近的样本（支持向量）到超平面的距离最大。这个最大距离被称为间隔（Margin）。SVM的决策边界是基于这些支持向量确定的，而不是所有数据点，这使得SVM在处理高维数据时仍能保持较好的性能。 2. **线性SVM与非线性SVM**：当数据可以被线性划分时，SVM直接构建线性超平面进行分类。若数据非线性可分，SVM采用核函数（如高斯核、多项式核、Sigmoid核等）将低维数据映射到高维空间，使原本在低维空间中难以划分的数据在高维空间变得容易区分。 3. **软间隔与硬间隔**：在实际应用中，我们可能无法找到一个完全将数据点分离的超平面。因此，SVM引入了松弛变量，允许一部分数据点违反间隔约束，即引入了软间隔。这有助于模型对异常值的容忍和提高泛化能力。 4. **优化问题**： SVM的求解过程是一个凸二次规划问题，目标是最小化间隔的同时，使误分类的数据点数量最小。这个问题可以通过拉格朗日乘数法和Karush-Kuhn-Tucker（KKT）条件来解决，得到的支持向量将参与构建决策边界。 5. **SVM的应用**： SVM不仅在分类问题中表现出色，还可用于回归任务（如支持向量回归-SVR）。在文本分类、图像识别、生物信息学等领域，SVM都展现出强大的性能。 6. **预测流程**： SVM预测时，新数据会被映射到同样的高维空间，然后根据决策边界判断其类别或回归值。预测过程快速且简单，因为仅依赖于支持向量。 7. **优缺点**： SVM的优点包括高效处理高维数据、优秀的泛化能力、对噪声和异常值的鲁棒性。但其缺点也很明显，如对大规模数据集的训练时间较长，选择合适的核函数和参数调优较为困难。 "SVM.zip"中的程序可能是为了演示如何利用SVM进行数据预测，并帮助用户了解和支持向量机的工作机制。通过运行和分析这个程序，你可以更深入地理解SVM的预测过程和其在实际问题中的应用。

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SVM.zip （10个子文件）

SVM

svmTrain.m 4KB

Main_SVR_Nu.m 3KB

Main_SVR_Epsilon.m 3KB

Main_SVC_Nu.m 3KB

svmSim.m 4KB

工具箱说明.txt 1KB

Main_SVC_C.m 3KB

kernel.m 1KB

Main_SVM_One_Class.m 3KB

LIBSVM a Library for Support Vector Machines.pdf 284KB

LIBSVM: a Library for Support Vector Machines

Chih-Chung Chang and Chih-Jen Lin

∗

Last updated: December 27, 2006

Abstract

LIBSVM is a library for support vector machines (SVM). Its goal is to help

users to easily use SVM as a tool. In this document, we present all its imple-

mentation details. For the use of LIBSVM, the README ﬁle included in the

package and the LIBSVM FAQ provide the information.

1 Introduction

LIBSVM is a library for support vector machines (SVM). Its goal is to let users can

easily use SVM as a tool. In this document, we present all its implementation details.

For using LIBSVM, the README ﬁle included in the package provides the information.

In Section 2, we show formulations used in LIBSVM: C-support vector classiﬁca-

tion (C-SVC), ν-support vector classiﬁcation (ν-SVC), distribution estimation (one-

class SVM), -support vector regression (-SVR), and ν-support vector regression

(ν-SVR). We discuss the implementation of solving quadratic problems in Section 3.

Section 4 describes two implementation techniques: shrinking and caching. We also

support diﬀerent penalty parameters for unbalanced data. Details are in Section 5.

Then Section 6 discusses the implementation of multi-class classiﬁcation. Parameter

selection is important for obtaining good SVM models. LIBSVM provides simple and

useful tools, which are discussed in Section 7. Section 8 presents the implementation

of probability outputs.

2 Formulations

2.1 C-Support Vector Classiﬁcation

Given training vectors x

∈ R

, i = 1, . . . , l, in two classes, and a vector y ∈ R

such that y

∈ {1, −1}, C-SVC (Cortes and Vapnik, 1995; Vapnik, 1998) solves the

∗

Department of Computer Science, National Taiwan University, Taipei 106, Taiwan (http://

www.csie.ntu.edu.tw/

∼

cjlin). E-mail: cjlin@csie.ntu.edu.tw

following primal problem:

min

w,b,ξ

w + C

i=1

(2.1)

subject to y

φ(x

) + b) ≥ 1 − ξ

≥ 0, i = 1, . . . , l.

Its dual is

min

Qα − e

subject to y

α = 0, (2.2)

0 ≤ α

≤ C, i = 1, . . . , l,

where e is the vector of all ones, C > 0 is the upper bound, Q is an l by l positive

semideﬁnite matrix, Q

≡ y

K(x

, x

), and K(x

, x

) ≡ φ(x

)

φ(x

) is the kernel.

Here training vectors x

are mapped into a higher (maybe inﬁnite) dimensional space

by the function φ.

The decision function is

sgn

i=1

K(x

, x) + b

2.2 ν-Support Vector Classiﬁcation

The ν-support vector classiﬁcation (Sch¨olkopf et al., 2000) uses a new parameter ν

which controls the number of support vectors and training errors. The parameter

ν ∈ (0, 1] is an upper bound on the fraction of training errors and a lower bound of

the fraction of support vectors.

Given training vectors x

∈ R

, i = 1, . . . , l, in two classes, and a vector y ∈ R

such that y

∈ {1, −1}, the primal form considered is:

min

w,b,ξ,ρ

w − νρ +

i=1

subject to y

φ(x

) + b) ≥ ρ − ξ

≥ 0, i = 1, . . . , l, ρ ≥ 0.

The dual is:

min

Qα

subject to 0 ≤ α

≤ 1/l, i = 1, . . . , l, (2.3)

α ≥ ν, y

α = 0.

where Q

≡ y

K(x

, x

The decision function is:

sgn

i=1

(K(x

, x) + b)

In (Crisp and Burges, 2000; Chang and Lin, 2001), it has been shown that e

α ≥ ν

can be replaced by e

α = ν. With this property, in LIBSVM, we solve a scaled version

of (2.3):

min

Qα

subject to 0 ≤ α

≤ 1, i = 1, . . . , l,

α = νl,

α = 0.

We output α/ρ so the computed decision function is:

sgn

i=1

(α

/ρ)(K(x

, x) + b)

and then two margins are

φ(x

) + b) = ±1

which are the same as those of C-SVC.

2.3 Distribution Estimation (One-class SVM)

One-class SVM was proposed by Sch¨olkopf et al. (2001) for es timating the support of

a high-dimensional distribution. Given training vectors x

∈ R

, i = 1, . . . , l without

any class information, the primal form in (Sch¨olkopf et al., 2001) is:

min

w,b,ξ,ρ

w − ρ +

νl

i=1

subject to w

φ(x

) ≥ ρ − ξ

≥ 0, i = 1, . . . , l.

The dual is:

min

Qα

subject to 0 ≤ α

≤ 1/(νl), i = 1, . . . , l, (2.4)

α = 1,

where Q

= K(x

, x

) ≡ φ(x

)

φ(x

In LIBSVM we solve a s caled version of (2.4):

min

Qα

subject to 0 ≤ α

≤ 1, i = 1, . . . , l,

α = νl.

The decision function is

sgn(

i=1

K(x

, x) − ρ).

2.4 -Support Vector Regression (-SVR)

Given a set of data points, {(x

, z

), . . . , (x

, z

)}, such that x

∈ R

is an input and

∈ R

is a target output, the standard form of support vector regression (Vapnik,

1998) is:

min

w,b,ξ,ξ

∗

w + C

i=1

+ C

i=1

∗

subject to w

φ(x

) + b − z

≤  + ξ

− w

φ(x

) − b ≤  + ξ

∗

, ξ

∗

≥ 0, i = 1, . . . , l.

The dual is:

min

α,α

∗

(α − α

∗

)

Q(α − α

∗

) + 

i=1

(α

+ α

∗

) +

i=1

(α

− α

∗

)

subject to

i=1

(α

− α

∗

) = 0, 0 ≤ α

, α

∗

≤ C, i = 1, . . . , l, (2.5)

where Q

= K(x

, x

) ≡ φ(x

)

φ(x

The approximate function is:

i=1

(−α

+ α

∗

)K(x

, x) + b.

2.5 ν-Support Vector Regression (ν-SVR)

Similar to ν-SVC, for regression, Sch¨olkopf et al. (2000) use a parameter ν to control

the number of support vectors. However, unlike ν-SVC, where ν replaces with C,

here ν replaces with the parameter  of -SVR. The primal form is

min

w,b,ξ,ξ

∗

,

w + C(ν +

i=1

(ξ

+ ξ

∗

)) (2.6)

subject to (w

φ(x

) + b) − z

≤  + ξ

− (w

φ(x

) + b) ≤  + ξ

∗

, ξ

∗

≥ 0, i = 1, . . . , l,  ≥ 0.

and the dual is

min

α,α

∗

(α − α

∗

)

Q(α − α

∗

) + z

(α − α

∗

)

subject to e

(α − α

∗

) = 0, e

(α + α

∗

) ≤ Cν,

0 ≤ α

, α

∗

≤ C/l, i = 1, . . . , l, (2.7)

Similarly, the inequality e

(α + α

∗

) ≤ Cν can be replaced by an equality. In

LIBSVM, we consider C ← C/l, so the dual problem solved is:

min

α,α

∗

(α − α

∗

)

Q(α − α

∗

) + z

(α − α

∗

)

subject to e

(α − α

∗

) = 0, e

(α + α

∗

) = Clν,

0 ≤ α

, α

∗

≤ C, i = 1, . . . , l. (2.8)

The decision function is

i=1

(−α

+ α

∗

)K(x

, x) + b,

which is the same as that of -SVR.

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