SICE-ICASE International Joint Conference 2006

Oct.18-21,2006 in Bexco,Busan,Korea

Fuzzy Decision-making SVMwith An Offset for Real-world Lopsided Data

Classiﬁcation

Boyang LI,Jinglu HU and Kotaro HIRASAWA

Graduate School of Information,Production and Systems,Waseda University,Hibikino 2-7,Wakamatsu-ku,

Kitakyushu-shi,Fukuoka-ken,Japan

(Tel/fax:(+81)93-692-5271;E-mail:liboyang@akane.waseda.jp,jinglu@waseda.jp,hirasawa@waseda.jp)

Abstract:An improved support vector machine (SVM) classiﬁer model for classifying the real-world lopsided data is

proposed.The most obvious differences between the model proposed and conventional SVMclassiﬁers are the designs

of decision-making functions and the introduction of an offset parameter.With considering about the vagueness of the

real-world data sets,a fuzzy decision-making function is designed to take the place of the traditional sign function in the

prediction part of SVM classiﬁer.Because of the existence of the interaction and noises inﬂuence around the boundary

between different clusters,this ﬂexible design of decision-making model which is more similar to the real-world situations

can present better performances.In addition,in this paper we mainly discuss an offset parameter introduced to modify

the boundary excursion caused by the imbalance of the real-world datasets.Because noises in the real-world can also

inﬂuence the separation boundary,a weighted harmonic mean (WHM) method is used to modify the offset parameter.

Due to these improvements,more robust performances are presented in our simulations.

Keywords:SVM,Fuzzy decision-making function,WHMoffset,Real-world lopsided dataset,Classiﬁcation

1.INTRODUCTION

SVMis an algorithmbased on the structure of statisti-

cal theory,so it has general well performances for unseen

data.Heretofore,SVM has been applied to many actual

problems,especially in various classiﬁcation problems,

but the classiﬁcation and SVMitself still have some prob-

lems to be resolved [2][3].

For most real-world classiﬁcation problems,databases

are usually affected by interaction and noises between

different classes,so the real-world classiﬁcation prob-

lems are non-separable mostly.For dealing with these

non-separable cases,SVMalgorithmuse a regularization

parameter C in the training part,which can weigh the tol-

erance of SVM.Because this parameter is the unique ad-

justable parameter in SVMto control the chosen of sup-

port vectors,the changing of variable C always affects

the performance of SVM evidently.In order to reduce

the inﬂuences caused by improper choice of C and deal

with the misclassiﬁed problems caused by the interaction

and noises,a fuzzy decision-making model is proposed

to replace the traditional one in the prediction part of

SVM classiﬁers.By this way,the hard-shell boundary

between neighboring classes is transformed into a ﬂexi-

ble one.Then the misclassiﬁed cases caused by the inter-

action and noises can be reduced [1].

In addition,the number of samples in one of the

classes in the real-world datasets is usually much larger

than the others.This imbalance characteristic is the rea-

son of the excursion of the boundary,which is another

frequent problem in the actual classiﬁcation.An offset

parameter was introduced for modifying this excursion in

our model [1].In SVM,the support vectors are the near-

est samples to the separation boundary,so their prediction

values can be used to calculate this offset.However,it is

found fromexperiment results that the noises in the real-

world cases not only make the separation boundary to be

a gray zone but also can increase the difﬁculties to com-

pute a proper offset parameter.In other words,because

SVM admits the violations in non-separable cases,sup-

port vectors can be conﬁrmed by the bounding planes be-

longs to the different subsets.If these support vectors are

disturbed by the interaction or noises consumingly,it will

be difﬁcult to get a correct separation boundary.Based

on this consideration,in this paper we introduce a series

of weights β

1

,...,β

n

are introduced to built a Weighted

Harmonic Mean (WHM) offset,which can equipoise the

inﬂuences from all support vectors and make some de-

viant decision values of support vectors to be invalid.Us-

ing this WHMoffset we can get a new separation bound-

ary and then the performances of the this model is studied

in simulations with different kinds of real-world datasets.

This paper is organized as follows:the next section

provides an overview of SVMand its applications in the

nonlinear nonseparable classiﬁcation problems.Section

3 discusses the decision-making part of SVMand a fuzzy

decision making method is proposed for ﬁtting the real-

world datasets.And In Section 4,an offset parameter

is introduced to modify the excursion of the boundary,

which can be calculated as a weighted harmonic mean

of the decision values of support vectors.And then,the

results with comparison to different kernels and different

classiﬁcation decision-making functions with details on

accuracy of classiﬁcation are presented in the Section 5.

Finally,concluding remarks are given in Section 6.

2.SVMFOR CLASSIFICATION

In recent years,SVMreveals its prominent capability

in many practical applications,especially in classiﬁcation

problems.In the elementary design of SVM classiﬁer,

bounding planes of each subsets are considered and the

distance between these bounding planes is deﬁned as the

Fig.1 Support Vectors in Non-separable Classiﬁcation

margin.The process of maximizing the margin equals

to the process of ﬁnding the optimal separation bound-

ary.In real-world the classiﬁcation problems are usually

non-separable and non-linear.So the violations can be

accepted in non-separable cases.But for the nonlinear

cases,the input vectors should be ﬁrstly mapped into a

high-dimensional feature space in which a separating hy-

perplane is found by solve a quadratic programming (QP)

problemin its dual form.

2.1 Basic Problemof SVMClassiﬁer

Any complex classiﬁcation problems can be divided

into several binary ones,so we just discuss the binary

cases in our paper.

If we have a training data set denoted as {x

i

,y

i

},

where x

i

∈ R

n

,i = 1,2,...,N.x

i

means the i-th in-

put vector and y

i

is its class label(+1 or -1).The training

data set can be divided into two different sets A and B

which have labels +1 and -1 respectively.As we dis-

cussed above,the distance between two sets bounding

planes is called the margin.It is obvious that maximiz-

ing this margin could improve the ability of the classiﬁer

model generally [4].

2.2 SVMfor Non-separable Classiﬁcation

But in the case where the training data are non-

separable,one should attempt to minimize the separation

error and to maximizing the margin simultaneously.

The support vector machine classiﬁer is obtained by

solving an optimization problem with an objective func-

tion which balances a termforcing separation between A

and B and a term maximizing the margin of separation,

so the tolerance is accepted in these cases [5].

As shown in the Fig.1,support vectors from A are

those A

i

in the halfspace {x ∈ R

n

|w

T

x ≤ b + 1} (i.e.

those points of A ’on’ or ’below’ the bounding plane

w

T

x = b + 1),where w and b are the weight and bias.

Support vectors from B are those points B

i

in the halfs-

pace {x ∈ R

n

|w

T

x ≥ b − 1} (i.e.those points of ’on’

or ’above’ the plane w

T

x = b −1).These points are the

only data points that are relevant in determining the op-

timal separating plane.The number of support vectors is

usually small and is also proportional to a bound on the

generalization error of the classiﬁer.But there is a prob-

lem in many actual applications.If these support vec-

tors are inﬂuenced by the noises,some of themmay have

large absolute decision values,and because the noises are

usually uncertain,we can not get the correct separation

hyperplane.

2.3 SVMfor Nonlinear Classiﬁcation

Because the common cases in the real-world classiﬁ-

cation are nonlinear nonseparable,so in the primal space,

we transformthe lowdimension large input data sets into

a high dimensional feature space by using a mapping

function ϕ(x).For non-separable cases in the feature

space,the boundary function has a nonnegative variables

ξ

i

to make the margin accept the violations,and then we

have a separating plane function

y

i

[w

T

ϕ(x

i

) +b] ≥ 1 −ξ

i

,∀i (1)

The optimal hyperplane problem becomes to ﬁnd the so-

lution of the following optimization problem,

min

w,b,ξ

J(w,ξ) =

1

2

w

T

w +C

N

i=1

ξ

i

(2)

s.t.

y

i

[w

T

ϕ(x

i

) +b] ≥ 1 −ξ

i

,

ξ

i

≥ 0,i = 1,...,N.

where parameter C is used to control the degree of toler-

ance,which is the only changeable parameter in SVM.

By introducing the vector of Lagrange multipliers α =

(α

1

,...,α

N

),the problem(Eq.2) can be rebuilt as a QP

problemin dual space [6]:

max

α

Q(α) = −

1

2

N

i,j=1

y

i

y

j

K(x

i

,x

j

)α

i

α

j

+

N

j=1

α

j

(3)

s.t.

N

i=1

α

i

y

i

= 0

0 ≤ α

i

≤ C,∀i

where K(x

i

,x

j

) = ϕ(x

i

)

T

ϕ(x

j

) is the kernel func-

tion [7].In our experiments,Polynomial kernel and

Gaussian RBF kernel are taken into account.Polynomial

mapping is a common method for non-linear modeling

shown as following

K(x

i

,x

j

) = (x

i

,x

j

+1)

d

(4)

where d is the exponential quantity of the polynomial.

And RBF has outstanding performances in applica-

tions

K(x

i

,x

j

) = exp(−

x

i

−x

j

2

2σ

2

) (5)

where σ

2

is the common width.

And then the decision making function can be gained,

y(x) = sign[

N

i=1

α

i

y

i

K(x,x

i

) +b] (6)

y(x) is the output prediction label of input vector x.

Although the sign function can divide the test data set

into two classes by detecting the signs of decision values

N

i=1

α

i

y

i

K(x,x

i

) + b,it is too hard-shelled to make

some mistakes when the decision value is close to zero.

3.FUZZY DECISION-MAKINGSVM

MODEL

3.1 Fuzzy Decision-making SVMProcess

We propose a fuzzy decision-making SVM process,

which is a model based on fuzzy method,SVMalgorithm

and analysis of a mass of database.As an extension of

traditional methods,the model proposed is more suitable

for actual applications.

In the training part,we still use the same method as

traditional one to train the SVM classiﬁer [8].As we

discussed before,the support vectors,which belong to

a subset extracted from the training data and used for

describing the separation boundary,can be found.Us-

ing the trained SVMwe can calculate the decision-value

of the input data.But being different from the tradi-

tional method,the decision value fromconventional sign

decision-making function is used as the independent vari-

able in fuzzy decision-making function to measure the

belief degree of each input point.The general structure

of the whole processing (Fig 2) can be divided into three

main stages:SVM straining,decision value prediction,

and fuzzy decision-making.

Fig.2 fuzzy decision-making SVMprocess

In the decision-making part,a fuzzy model is con-

structed to replace the sign function in conventional

model.This is because interaction usually exists in many

real conditions,especially around the boundary between

classes.So misclassiﬁed cases occur in the neighbor of

threshold.Being different from conventional methods,

the fuzzy boundary in our model makes zero not be the

only important value as a threshold,but clusters will be

considered as fuzzy sets.

3.2 Fuzzy Decision-making Functions

Assume that these two fuzzy sets are called A (the

value in this set are deemed to be predicted as -1) and

B (the values in this set are deemed to be predicted as

+1).A gray-zone should be built to take the place of the

hard boundary.The values of the function can indicate

their reliability which handles the concept of belief prob-

lem(belief degree between 1 (completely believable) and

0 (completely false)).Through several experiments,arc-

tangent function is conﬁrmed to construct the curvilinear

fuzzy functions.The boundaries of set A and set B can

be written as:

f

A

(v) =

arctan(−v · s −d · s)

π

+0.5;(7)

f

B

(v) =

arctan(v · s −d · s)

π

+0.5.(8)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-10

-5

0

5

10

Believable value

Decision value

Bondary Function of Fuzzy Set B

Bondary Function of Fuzzy Set A

Positive Set B

Minus Set A

-d

d

-s

s

Fig.3 curvilinear fuzzy decision model

where d indicates the discerption degree,s means the

scale factor and the decision value is denoted as v.The

cross section ﬁgure of the fuzzy sets is shown in Fig 3.

4.WEIGHTED HARMONIC MEAN

OFFSET

4.1 Introduction of WHMoffset

Because most real cases are lopsided,the borderline

can not be described by formulae (7) and (8) accurately.

As a result of the imbalance of dataset,the midpoint of

the gray boundary zone always not equals to zero.So

an offset constant δ is needed to be introduced to denote

the distance between the real borderline and the theoretic

one.

Because support vectors are the nearest samples

around the boundary,so one way to calculate the offset δ

is to compute the mean of decision values of support vec-

tors which has been proposed in our former models [1].

The formula can be written as follows:

δ =

n

i=1

S

i

n

(9)

where S

1

,...,S

n

are the decision values of supports.

This mean value can modify the separation boundary to

a better position than the conventional one,but if some

support vectors are also inﬂuenced by the noises in the

real-world datasets,this offset will be inauthentic.

For ﬁnding a more proper offset,it is necessary to

ignore these false support vectors,so we introduce a

weighted harmonic mean of decision values of support

vectors(SVs).As the previous offset parameter proposed

by us,SVs are used as the test data to gain their decision

values S

1

,...,S

n

,where n is the number of support vec-

tors.Suppose the corresponding weights are β

1

,...,β

n

then the offset has a conﬁguration shown as follows

δ =

n

i=1

β

i

n

i=1

β

i

S

i

(10)

As we concerned in the subsection 2.2,the support

vectors from subset A are the points of A ’on’ or ’be-

low’ the bounding plane w

T

x = b +1,and similarly the

support vectors fromBare those points of ’on’ or ’above’

the plane w

T

x = b −1.So if some of the support vectors

is inﬂuenced by the interaction and noises strongly,then

these samples may apart from the separation boundary.

Therefore,we need to give these support vectors smaller

weights so that they will become invalid in the calculation

of the offset.Based on this consideration,we employ the

Blackman equation to calculate the weights of support

vectors,which is shown in the following,

β

i

= 0.42 −0.5cos(

π(S

i

+S

max

)

S

max

)

+0.08cos(

2π(S

i

+S

max

)

S

max

) (11)

where S

max

is the largest absolute value of all the sup-

port vectors’ decision values,S

i

means a certain support

vector’s decision value,and β

i

denotes its corresponding

weight.

4.2 Decision-making model with WHMoffset

Fig.4 fuzzy decision model with an offset

Introduce the proposed WHM offset to the fuzzy

boundary SVMclassiﬁer model concerned above,a mod-

iﬁed fuzzy boundary can be obtained,and then the formu-

las (7) and (8) can be rewritten as:

f

A

(v) =

arctan(−v · s −d · s +δ · s)

π

+0.5 (12)

f

B

(v) =

arctan(v · s −d · s −δ · s)

π

+0.5 (13)

where d,and s are chosen from a great deal of experi-

ments.We can use the values f

A

(v) and f

B

(v) to esti-

mate whether an input vector should be labeled as +1 or

-1.Based on the formulas (12) and (13),the bounding

curves of the model with WHM offset can be drawn in

Fig.4.

So the whole process of the fuzzy decision-making

SVMwith WHMoffset can be described as shown in Fig.

5.From the ﬁgure we can ﬁnd the whole procedure con-

sists of four steps as follows,

Step 1:SVMtraining process.We can ﬁnd the support

vectors and compute the parameters for building SVM

classiﬁer model.

Step 2:Prediction process for support vectors.We can

calculate the decision values of support vectors and the

weights corresponding to them.

Fig.5 model with weighted harmonic mean offset

Step 3:Prediction process for the test datasets.Deci-

sion values of test input vectors can be gained.

Step 4:Final decision-making process.Using the

decision values of the support vectors and the weights

from step 2 to calculate the offset.And using the fuzzy

decision-making method to predict the ﬁnal output labels

for the input vectors.

5.SIMULATION RESULTS

5.1 Simulation 1:Heart disease detection problem

5.1.1 Description of problem

The ﬁst problem we used to test our models is heart

disease detection,which is from Statlog datasets.The

whole database consists of two classes:absence class and

presence class.The total number of examples is 200,

in which the number of samples in the absence class is

150 and the number of samples in the presence class is

50.Each sample in the datasets has 13 main attributes,

which are extracted from a larger set of 75.Denote each

pair of input vector and output label as {X(n),Y (n)},

where X(n) is the input vector with 13 attributes and

Y (n) is the output label with two poles:-1 (absence)

Table 1 Comparison of two offsets in simulation 1

C

1

10

100

model A

94.615%

95.38%

96.15%

model B

88.46%

91.54%

92.3%

model C

95.38%

96.15%

96.15%

model D

90%

92.3%

93.08%

or +1 (presence),(n = 1,2,...,200).Assume that the

vector X(n) = (x

1

(n),x

2

(n),...,x

13

(n)),the signiﬁ-

cations of these 13 elements are:age,sex,chest pain

type,resting blood pressure,serum cholestoral,fasting

blood sugar,resting electrocardiographic results,maxi-

mum heart rate,exercise induced angina,oldpeak,the

slope of the peak exercise ST segment,number of major

vessels and thal(3=normal;6=ﬁxed defect;7=reversable

defect).

In brief,our purpose is to predict the bipolar output

Y (n) with as few sign errors as possible,through the

given input vectors X(n).

5.1.2 Results of Classiﬁcation

Because the database used in our simulations comes

from the real-world,it is non-separable apparently.In

non-separable SVM classiﬁer,not only the kind of ker-

nel but the regularization parameter C can also effect the

accuracy which is a value used to evaluate classiﬁers.

C is used to denote the tolerance of SVM,along with

changing of which,both the results of training and the

prediction are changed.For conﬁrming the widely ap-

plicability of our method,we let C equal to 1,10,100

in turn as common situations,and use two kinds of ker-

nels presented in a previous section.They are RBF kernel

and Polynomial kernel.Using the fuzzy decision-making

function and WHMoffset in the prediction part of SVM

classiﬁers based on these two kinds of kernels,we can

form two classiﬁcation models:fuzzy decision-making

RBF-SVMmodel with WHMoffset and fuzzy decision-

making Polynomial-SVMmodel with WHMoffset.

Using the ﬁrst 70 samples as the training data,and con-

sidering the remainder as the test data,the accuracies of

these two proposed models are gained.Comparing with

our previous models with mean offsets,the experiment

results are shown in Table 1.Compare model A (fuzzy

decision-making RBF SVMwith mean offset) with model

C (fuzzy decision-making RBF SVM with WHM offset)

and compare model B (fuzzy decision-making Poly SVM

with mean offset) with model D (fuzzy decision-making

Poly SVM with WHM offset),then we can ﬁnd that the

newmodels with WHMoffsets have better performances.

And then compare with traditional RBF-SVM classi-

ﬁer,traditional Polynomial-SVM classiﬁer,the accura-

cies (y-axis) with three different values of parameter C

(x-coordinate axis) are shown in Fig.6.

In the aspect of parameter selection,we choose a stan-

dard RBF kernel with common width σ

2

= 4 and Poly-

nomial kernel with d = 3.Using different combinations

of kernel and C value to train the SVM classiﬁers,we

80%

85%

90%

95%

100%

1

10

100

Accuracy

C

Fuzzy Decision RBF SVM with WHM Offset

Fuzzy Decision Poly SVM with WHM Offset

RBF SVM

Poly SVM

Fig.6 Accuracy Curves for heart disease detection

get some different models.And then the boundary off-

set δ can be worked out by the decision values of support

vectors and their weights.We set the scale coefﬁcient

s = 256 for the classiﬁers using Polynomial kernel,and

set s = 512 for the classiﬁers using RBF kernel.

5.2 Simulation 2:Misﬁre detection problem

5.2.1 Description of problem

The second database we used is also from real-world,

which is a misﬁre detection problem in internal combus-

tion engines.The whole database is divided into two sub-

sets,one is used as the training data and the other is the

test data.These data contain the information of time se-

ries of 50000 samples which are produced by physical

system (10-cylinder internal combustion engine).Sim-

ilarly as the simulation 1,each sample k of time series

consists of four inputs elements and one output label,

each pair of input vector and output label also can be writ-

ten as {X(k),Y (k)},where X(k) is the input vector and

Y (k) is the output label,(k = 1,2,...,50000).

The four elements of input vectors x

1

(k),x

2

(k),

x

3

(k) and x

4

(k) represent cylinder identiﬁer (ﬁrst),en-

gine crankshaft speed in Revolutions Per Minute (RPM)

(second),load (third) and crankshaft acceleration (fourth)

respectively.Y (k) may have two values:-1 (normal ﬁr-

ing) or +1 (misﬁre).The amounts of the normal cases in

the training data and the test data are 45093 and 45395

and the numbers of samples in the misﬁre classes in the

training and test datasets are 4907 and 4605.We can

ﬁnd that this database is also an imbalance one.In this

problem,Our purpose is also to detect the value of output

Y (k) for a certain given input vectors X(k).

5.2.2 Results of Classiﬁcation

As described in the simulation 1,we also let C equal to

1,10,100,and use RBF kernel and Polynomial kernel in

our experiments respectively.The comparison of models

with different offsets is shown in Table 2.Model A,B,C

and D have same deﬁnitions as Table 1.In this problem

we can also obtain a better performance from the new

model with a WHMoffset.

The accuracies of traditional classiﬁers and proposed

models with WHMoffset are shown in Fig.7.

In the aspect of parameter selection,we also set σ

2

=

Table 2 Comparison of two offsets in simulation 2

C

1

10

100

model A

95.34%

95.654%

95.42%

model B

92.656%

93.858%

95.008%

model C

95.74%

96.02%

95.82%

model D

92.96%

94.27%

95.01%

90%

92%

94%

96%

98%

100%

1

10

100

Accuracy

C

Fuzzy Decision RBF SVM with WHM Offset

Fuzzy Decision Poly SVM with WHM Offset

RBF SVM

Poly SVM

Fig.7 Accuracy Curves for misﬁre detection problem

4 for RBF kernel and d = 3 for Polynomial kernel.As

same as the simulation 1,we set s = 256 for the clas-

siﬁers using Polynomial kernel,and set s = 512 for the

classiﬁers using RBF kernel.

5.3 Comparison among Different Classiﬁers

As shown in Fig.6 and Fig.7,with different values of

parameter C the accuracies of the models using RBF ker-

nel and the ones using Polynomial kernel are also differ-

ent.For different problems,one classiﬁer may presents

different performances,so how to choose a proper kernel

is still a problemin SVMmethod.In other words,for the

dataset in simulation one RBF kernel is better than the

Polynomial kernel but for the second problem,Polyno-

mial kernel is more proper.

Especially,from the ﬁgures we can also ﬁnd that the

method proposed in this paper can present a better ca-

pability than traditional SVM classiﬁer and our former

models for different regularization parameter C,and the

performance of this model is more robust too.

Even as what has been discussed,a more proper

decision-making function,a more valid offset,a more ap-

propriate width of gray zone and a more suitable kernel

would make the classiﬁer be more effective.

6.CONCLUSIONS

We have presented an fuzzy decision-making algo-

rithm for building SVM classiﬁers and introduced a

WHMoffset to modify the excursion of the boundary in

real-world datasets.

The model is proposed in this paper to improve the

performances of SVMin the classiﬁcation problems.By

using a fuzzy decision-making function,the prediction

boundary is transformed from a straitlaced conﬁguration

to a more ﬂexible structure so that the inﬂuence between

data sets can be reduced and many misclassiﬁed points

in the prediction part of traditional SVMclassiﬁer have a

chance to be relabeled.

In addition,we also construct a WHMoffset δ.The in-

troduction of this offset can modify the separation bound-

ary between each two sets to a more appropriate posi-

tion and then the error caused by the imbalance of data

sets could be overcome.Especially along with the em-

ployment of the weight harmonic mean method,an anti-

jamming offset can be calculated without considering the

support vectors inﬂuenced by the noises,since the distri-

bution of the weight values is used to control the effec-

tiveness of SVs to the offset parameter.

Although the model proposed presents some well per-

formances in simulations,there still remains some future

problems to be solved.How to choose a proper kernel

for a certain database,how to build a more robust kernel

function and how to conﬁrm the parameters in the fuzzy

decision-making part automatically will be main direc-

tions in our future research.

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