支持向量机与最优核函数参数选择在破产预测中的应用

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"本文探讨了使用支持向量机（Support Vector Machines, SVM）以及最优核函数参数选择在破产预测中的应用。通过5折交叉验证的网格搜索技术确定了SVM的最优参数，以此提高模型的解释能力和稳定性。同时，该研究对比了SVM与其他传统方法，如多变量判别分析（Multiple Discriminant Analysis, MDA）、逻辑回归分析（Logistic Regression, Logit）和三层全连接反向传播神经网络（Back-Propagation Neural Networks, BPNs）的预测准确性。实验结果显示，SVM在破产预测中表现出更高的性能。" 在这篇论文中，作者专注于如何利用支持向量机这一机器学习技术来改进破产预测模型。支持向量机是一种强大的分类工具，尤其在处理小样本和非线性问题时表现出色。其核心思想是通过构造最大边距超平面将数据分隔开，以实现最优分类。在实际应用中，选择合适的核函数及其参数对模型的性能至关重要。为了找到最佳的核函数参数，作者采用了网格搜索和5折交叉验证策略。网格搜索是一种遍历所有可能参数组合的方法，通过比较不同参数组合下的模型性能，找出最优的参数设置。5折交叉验证则是一种评估模型性能的有效方式，它将数据集分为五部分，每次用四部分训练模型，剩下的部分用于测试，这样循环五次，最后取平均结果，以减少过拟合风险。论文中还对比了SVM与其他几种常见的预测模型。多变量判别分析是一种统计方法，通过寻找类间距离最大、类内距离最小的投影方向进行分类。逻辑回归分析则是基于概率的分类方法，适用于处理二分类问题。而三层全连接的反向传播神经网络是深度学习的一种基本模型，能处理复杂的非线性关系。通过比较这些方法与SVM的预测准确率，可以证明SVM在破产预测问题上的优势。这篇论文展示了支持向量机在破产预测中的高效性和准确性，以及通过优化核函数参数提升模型性能的重要性。这种方法对于金融风险管理和企业决策具有实际价值，有助于提前识别潜在的财务危机，降低经济损失。

(Cristianini & Shawe-Taylor, 2000; Gunn, 1998; Hearst

et al., 1998; Vapnik, 1998).

SVM is simple enough to be analyzed mathematically

since it can be shown to correspond to a linear method in

a high dimensional feature space nonlinearly related to

input space. In this sense, SVM may serve as a promising

alternative combining the strengths of conventional

statistical methods that are more theory-driven and easy

to analyze, and more data-driven, distribution-free and

robust machine learning methods. Recently, the SVM

approach has been introduced to several ﬁnancial

applications such as credit rating, time series prediction,

and insurance claim fraud detection (Fan & Palaniswami,

2000; Gestel et al., 2001; Huang, Chen, Hsu, Chen, &

Wu, 2004; Kim, 2003; Tay & Cao, 2001; Viaene, Derrig,

Baesens, & Dedene, 2002). These studies reported that

SVM was comparable to and even outperformed other

classiﬁers including ANN, CBR, MDA, and Logit in

terms of generalization performance. Motivated by these

previous researches, we apply SVM to the domain of

bankruptcy prediction, and compare its prediction per-

formance with those of MDA, Logit, and BPNs.

A simple description of the SVM algorithm is provided

as follows. Given a training set DZ fx

; y

iZ1

with input

vectors x

Z ðx

ð1Þ

; .; x

ðnÞ

and target labels

2fK1;C1g, the support vector machine (SVM) classiﬁer,

according to Vapnik’s original formulation, satisﬁes the

following conditions

fðx

Þ C bRC1; if y

Z C1

fðx

Þ C b%K1; if y

Z K1

(

(1)

which is equivalent to

½w

fðx

Þ C bR 1; i Z 1; .; N (2)

where w represents the weight vector and b the bias.

Nonlinear function fð,Þ : R

/ R

maps input or

measurement space to a high-dimensional, and possibly

inﬁnite-dimensional, feature space. Eq. (2) then comes

down to the construction of two parallel bounding

hyperplanes at opposite sides of a separating hyperplane

f(x)CbZ0 in the feature space with the margin width

between both hyperplanes equal to 2/(jjwjj

). In primal

weight space, the classiﬁer then takes the decision

function form (3)

sgnðw

fðxÞ C bÞ (3)

Most of classiﬁcation problems are, however, linearly

non-separable. Therefore, it is general to ﬁnd the

weight vector using slack variable (x

)topermit

misclassiﬁcation. One deﬁnes the primal optimization

problem as

Min

w;b;x

w C C

iZ1

(4)

subject to

ðw

fðx

Þ C bÞR 1 K x

; i Z 1; .; N

R 0; i Z 1; .; N

(

(5)

where x

’s are slack variables needed to allow misclassi-

ﬁcations in the set of inequalities, and C 2R

is a

tuning hyperparameter, weighting the importance of

classiﬁcation errors vis-a

-vis the margin width. The

solution of the primal problem is obtained after

constructing the Lagrangian. From the conditions of

optimality, one obtains a quadratic programming (QP)

problem with Lagrange multipliers a

’s. A multiplier a

exists for each training data instance. Data instances

corresponding to non-zero a

’s are called support vectors.

On the other hand, the above primal problem can be

converted into the following dual problem with objective

function (6) and constraints (7). Since the decision variables

are support vector of Lagrange multipliers, it is easier to

interpret the results of this dual problem than those of the

primal one.

Max

Qa K e

a (6)

subject to

0% a

% C; i Z 1; .; N

a Z 0

(

(7)

In the dual problem above, e is the vector of all ones, Q is

a N!N positive semi-deﬁnite matrix, Q

K(x

), and

K(x

)hf(x

)

f(x

) is the kernel. Here, training vectors x

’s

are mapped into a higher (maybe inﬁnite) dimensional space

by function f. As is typical for SVMs, we never calculate w

or f(x). This is made possible due to Mercer’s condition,

which relates mapping function f(x) to kernel function

K($,$) as follows.

Kðx

; x

Þ Z fðx

fðx

Þ (8)

For kernel function K($,$), one typically has

several design choices such as the linear kernel of

Kðx

; x

ÞZ x

, the polynomial kernel of degree d of

Kðx

; x

ÞZ ðgx

C rÞ

; gO 0, the radial basis function

(RBF) kernel of K(x

)Zexp{Kgjjx

}, gO0, and

the sigmoid kernel of Kðx

; x

ÞZ tanhfgx

C rg, where d;

r 2N and g 2R

are constants. Then one constructs the

ﬁnal SVM classiﬁer as

sgn

Kðx; x

Þ C b

(9)

The details of the optimization are discussed in (Chang &

Lin, 2001; Cristianini & Shawe-Taylor, 2000; Gunn, 1998;

Vapnik, 1998).

J.H. Min, Y.-C. Lee / Expert Systems with Applications 28 (2005) 603–614 605

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