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SVM.zip （10个子文件）

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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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