BREAST CANCER DIAGNOSIS AND PROGNOSIS USING

DIFFERENT KERNEL-BASED CLASSIFIERS

Tingting Mu and Asoke K. Nandi

Department of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill, Liverpool, UK, L69 3GJ

Keywords:

Breast cancer, diagnosis, prognosis, pattern classiﬁcation, kernel method.

Abstract:

The medical applications of several advanced, kernel-based classiﬁers to breast cancer diagnosis and progno-

sis are studied and compared in this paper, including kernel Fisher’s discriminative analysis, support vector

machines (SVMs), multisurface proximal SVMs, as well as the pairwise Rayleigh quotient classiﬁer and the

strict 2-surface proximal classiﬁer that we recently proposed. The radial basis function kernel is employed

to incorporate nonlinearity. Studies are conducted with the Wisconsin diagnosis and prognosis breast cancer

datasets generated from ﬁne-needle-aspiration samples by image processing. Comparative analysis is pro-

vided in terms of classiﬁcation accuracy, computing time, and sensitivity to the regularization parameters for

the above classiﬁers.

1 INTRODUCTION

Despite the increasing public awareness and scientiﬁc

research, breast cancer continues to be the most com-

mon form of cancer and the second most common

cause of cancer deaths in females; the disease affects

approximately 10% of all women at some stage of

their life in the western world (Marshall, 1993). The

long-term survival of a patient with breast cancer are

improved by the early detection of the disease, which

is enhanced by an accurate diagnosis. The choice

of appropriate treatments following surgery is inﬂu-

enced by the expected long-term behavior of the dis-

ease, so-called prognosis.

Deﬁnitive diagnosis of a breast mass can only

be established through ﬁne-needle aspiration (FNA)

biopsy, core needle biopsy, or excisional biopsy.

Among these methods, FNA is the easiest and fastest

method of obtaining a breast biopsy, and is effec-

tive for women who have ﬂuid-ﬁlled cysts. Research

works on the Wisconsin Diagnosis Breast Cancer

(WDBC) data grew out of the desire of Dr. Wolberg

to diagnose breast masses accurately based solely on

FNA (Wolberg et al., 1993; Street et al., 1993). Later,

a number of research projects have been developed

with the WDBC dataset, focusing on computer-aided

diagnosis (CAD) using machine learning techniques

(Wolberg et al., 1994; Wolberg et al., 1995; Man-

gasarian et al., 1995; Guo and Nandi, 2006; Mu and

Nandi, 2007). Breast cancer prognosis is a more dif-

ﬁcult problem, that is, the long-term outlook for the

disease for patients whose cancer has been surgically

removed. Till now, few works have been developed

on predicting the time to recur (TTR) for a patient

for whom cancer has not recurred and may never re-

cur (Wolberg et al., 1995; Mangasarian et al., 1995;

Street et al., 1995). The detection of malignant breast

tumors from a set of benign and malignant samples

for diagnosis, and the simple prediction of patients

as ’recurred’ or ’not recurred’ without predicting the

TTR for prognosis, both belong to the pattern classi-

ﬁcation problems.

The idea of using kernel functions as inner prod-

uct in a feature space was introduced into machine

learning in 1964 by the work of Aizerman, Braver-

man and Rozonoer (Aizerman et al., 1964). Ker-

nel methods to pattern analysis embeds the data in

a suitable feature space, and then uses algorithms

based on linear algebra, geometry, and statistics to

discover patterns in the embedded data. Different

kernel-based classiﬁers have been proposed. Boser,

Guyon, and Vapnik (Boser et al., 1992) ﬁrst combined

the kernel function with the large margin hyperplanes,

leading to support vector machines (SVMs) that are

highly successful in solving various nonlinear and

non-separable problems in machine learning. In addi-

tion to the original C-SVM learning method (Cortes

and Vapnik, 1995), the ν-SVM learning method was

proposed by Sch

olkopf et al. (Sch

olkopf et al., 2000),

which is closely related to the C-SVM but with a dif-

ferent optimization risk. The famous Fisher’s linear

discriminant analysis (FLDA), dating back to 1936

342

Mu T. and K. Nandi A. (2008).

BREAST CANCER DIAGNOSIS AND PROGNOSIS USING DIFFERENT KERNEL-BASED CLASSIFIERS.

In Proceedings of the First International Conference on Bio-inspired Systems and Signal Processing, pages 342-348

DOI: 10.5220/0001059303420348

 SciTePress

(Fisher, 1936), seeks separating hyperplanes which

best separate two or more classes of samples based

on the Fisher criterion with the between- and within-

class scatters built on individual samples. Mika et al.

(Mika et al., 1999) combined kernels functions with

FLDA leading to kernel Fisher’s discriminant analy-

sis (KFDA). Mu et al. (Mu et al., 2007a) proposed to

seek the optimal separating hyperplane based on the

pairwise Rayleigh quotient (PRQ) criterion with the

between- and within- class scatters built on the pair-

wise information; they also proposed to combine ker-

nels functions with the linear PRQ classiﬁer leading

to the nonlinear PRQ classiﬁer. Multiplane learning

is a comparatively new machine learning method de-

veloped in recent years. Mangasarian and Wild (Man-

gasarian and Wild, 2006) proposed the kernel-based

multisurface proximal SVM (MPSVM) that seeks two

cross proximal planes by optimizing a regularized

optimization objective with Tikhonov regularization

term employed. More recently, Mu et al. (Mu et al.,

2007b) proposed the strict 2-surface proximal (S2SP)

classiﬁer that seeks two cross proximal planes by em-

ploying a “square of sum” optimization factor with-

out any regularization term, which is mathematically

stricter than the optimization objective of MPSVM;

and kernel functions were employed to incorporate

nonlinearity.

In this paper, studies are conduced on the WDBC

and WPBC datasets to investigate the beneﬁts of ap-

plying different kernel-based classiﬁers to breast can-

cer diagnosis and prognosis, including SVM, KFDA,

PRQ classiﬁer, MPSVM, regularized δ-MPSVM

(Mangasarian and Wild, 2006), and S2SP classiﬁer.

The detecting accuracies, computing times, and sensi-

tivities to regularization parameters are compared for

the above kernel-based classiﬁers.

2 CLASSIFICATION METHODS

Given a set of l labeled training samples z =

{(x

)}

i=1

∈ (R

× Y ), where R

is the n-

dimensional real feature space with a binary label

space Y = {1,−1}, and y

∈ Y is the label assigned

to the sample x

∈ R

, the purpose of classiﬁcation is

to seek the best prediction of the label for an input

sample x. All the kernel-based classiﬁers are devel-

oped in the kernel-transformed feature space κ, with

a nonlinear mapping φ : R

→ κ.

2.1 Discriminant Classiﬁcation

The basic idea of the discriminant classiﬁcation is

to seek one optimal hyperplane that best separates

the two classes of samples in a corresponding feature

space. In the kernel-transformed feature space κ, by

expanding the direction vector of the hyperplane into

a linear summation of all training samples, the sepa-

rating hyperplane can be given as

f (x) =

∑

i=1

K(x

,x) + b, (1)

where {α

}

i=1

denote the summating weights, b de-

notes the bias of the separating hyperplane, and K(·,·)

is a kernel function used to compute the inner product

matrix, the so-called kernel matrix, on pairs of sam-

ples in the kernel-transformed feature space κ. Dif-

ferent classiﬁcation methods lead to different ways to

determine the optimal separating hyperplane f

∗

(x).

The label of a given test sample x can be predicted by

p(x) = sgn( f

∗

(x)), (2)

where sgn(x) is equal to 1 when x ≥ 0, and −1 other-

wise.

2.1.1 Support Vector Machines

The basic idea of SVMs is to construct a separat-

ing hyperplane as the decision surface in such a way

that the margin of separation between the positive and

negative samples is maximized in an appropriate fea-

ture space. To determine f

∗

(x) based on the maxi-

mal margin rule, the following constrained quadratic

programming problem is solved (Cortes and Vapnik,

1995), as

O(β) =

∑

i=1

−

∑

i=1

∑

j=1

βiβ

K(x

), (3)

subject to

∑

i=1

= 0,

0 ≤ β

≤ C, i = 1,2,...,l,

where {β

}

i=1

are Lagrange multipliers, and C is

the regularization parameter set by the user. Letting

{β

∗

}

i=1

denote the optimal solution of O(β), the op-

timal value of the summating weights {α

}

i=1

and the

bias b are obtained by

∗

= y

∗

, i = 1, 2, . . . , l, (4)

∗

= −

∑

x∈S

−

∑

i=1

∗

K(x,x

), (5)

where S

and S

−

are two sets of support vectors with

the same size of S but different labels of +1 and −1.

BREAST CANCER DIAGNOSIS AND PROGNOSIS USING DIFFERENT KERNEL-BASED CLASSIFIERS

343

2.1.2 Kernel Fisher’s Discriminant Analysis

KFDA determines f

∗

(x) by maximizing the follow-

ing Fisher criterion (Shawe-Taylor and Cristianini,

2004), as

O( f ) =

(µ

− µ

−

)

(σ

)

+ (σ

−

)

, (6)

where

∑

i=1

f (x

)

−

∑

i=1

f (x

)

(σ

)

∑

i=1

( f (x

) − µ

)

(σ

−

)

−

∑

i=1

( f (x

) − µ

−

)

where µ

and µ

−

denote the mean projections of the

positive and negative samples, respectively; σ

and

−

are the corresponding standard deviations; and l

and l

−

denote the number of samples from the pos-

itive and negative classes, respectively. By incorpo-

rating Eq. (1) into Eq. (6), the optimal values of

{α

}

i=1

and b can be calculated by solving a general-

ized eigenvalue problem (Shawe-Taylor and Cristian-

ini, 2004).

2.1.3 Pairwise Rayleigh Quotient Classiﬁer

The PRQ classiﬁer helps in classiﬁcation with insuf-

ﬁcient training samples by employing pairwise con-

straints instead of individual samples. To determine

the optimal separating hyperplane f

∗

(x), the follow-

ing PRQ criterion is maximized (Mu et al., 2007a),

O( f ) =

−

, (7)

where

d =

∑

i=1

(1 − z

)( f

− f

)

− 1)

∑

i=1

(1 + z

)(1 + y

)( f

− f

)

−

− 1)

∑

i=1

(1 + z

)(1 − y

)( f

− f

)

where

d denotes the differences of projections be-

tween samples from different classes;

denotes the

differences of projections between samples from the

positive class;

−

denotes the differences of projec-

tions between samples from the negative class; y

de-

notes the label of the sample x

; z

∈ {1,−1} is the

pairwise constraint assigned to the two samples in the

pair (x

), and z

= 1 if the two samples (x

)

belong to the same class, whereas z

= −1 if the two

samples (x

) belong to different classes; f

and

are used to denote f (x

) and f (x

); and m is the

total number of available pairwise constraints. By in-

corporating Eq. (1) into Eq. (7), the optimal values

of {α

}

i=1

and b can be simply calculated by matrix

computation (Mu et al., 2007a). Compared with the

Fisher criterion built on individual samples from a to-

tal number of l available samples, the PRQ criterion

offers more possibilities by employing pairwise con-

straints from a total number of l ∗ (l − 1) available

constraints.

2.2 Proximal Classiﬁcation

The basic idea of proximal classiﬁcation is to seek

two proximal planes in a corresponding feature space,

so that the ﬁrst plane is as close to the points of the

positive class while being as far as possible from the

points of the negative class, whereas the second plane

is as close to the points of the negative class while

being as far as possible from the points of the positive

class. In the kernel-transformed feature space κ, by

expanding the direction vector of the hyperplane into

a linear summation of all training samples, the two

proximal hyperplanes are given as

(x) =

∑

i=1

K(x

,x) + b

, (8)

(x) =

∑

i=1

K(x

,x) + b

, (9)

where the subscripts 1 and 2 denote the ﬁrst and sec-

ond proximal plane, respectively. Let d

and d

de-

note the Euclidean distance between the sample and

the two proximal planes, respectively, in the feature

space κ. The label of a given test sample x can be

predicted by considering values of d

, d

, and

to-

gether using linear discriminant analysis.

2.2.1 Multisurface Proximal SVMs

MPSVMs obtain the ﬁrst proximal hyperplane by

maximizing the following objective function, as

(Mangasarian and Wild, 2006)

(α

) =

−

+ eb

; (10)

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

344

and obtain the second proximal hyperplane by maxi-

mizing (Mangasarian and Wild, 2006)

(α

) =

+ eb

−

+ eb

, (11)

where α

and α

are two column vectors each with

elements equal to {α

}

i=1

and {α

}

i=1

, respectively;

the l

× l matrix K

represents the kernel matrix be-

tween the samples from the positive class and all the

training samples; the l

−

× l matrix K

−

represents the

kernel matrix between the samples from the negative

class and all the training samples; and e is a column

vector with all elements equal to one. The optimal

values of α

, b

, α

, and b

can be calculated by

solving two generalized eigenvalue problems (Man-

gasarian and Wild, 2006), respectively.

Letting ˜α





, and ˜α





to improve the classiﬁcation performance of the

MPSVMs, Mangasarian and Wild (Mangasarian and

Wild, 2006) proposed to employ a Tikhonov regular-

ization term, the two optimization objective shown in

Eq. (10) and Eq. (11) become

(α

) =

−

+ eb

+ δk ˜α

+ eb

, (12)

(α

) =

−

+ eb

+ δk ˜α

+ eb

, (13)

where δ is a nonnegative regularization parameter set

by the user. However, similar to the regularization pa-

rameter of the SVM, such as C for the C-SVM (Cortes

and Vapnik, 1995) and ν for the ν-SVM (Sch

olkopf

et al., 2000), performance of the above regularized δ-

MPSVM is sensitive to the setting of the regulariza-

tion parameter δ.

2.2.2 Strict 2-Surface Proximal Classiﬁer

With consideration of the sign effect under the situa-

tion of misclassiﬁcation with large projections onto

the separating plane, the S2SP classiﬁer eliminates

the regularization term by employing the “square of

sum” numerator. To obtain the ﬁrst proximal hyper-

plane, the following objective function is to be maxi-

mized (Mu et al., 2007b), as

(α

) =

−

+ eb

, (14)

and obtain the second proximal hyperplane by maxi-

mizing (Mu et al., 2007b)

(α

) =

+ eb

−

+ eb

, (15)

where dvectore is used to denote the sum of the ele-

ments of the vector; and dmatrixe is used to denote

a column vector with the sum of each row. The op-

timal values of α

, b

, α

, and b

can be calculated

by matrix computation (Mu et al., 2007b). There is

no regularization parameter to be tuned for the S2SP

classiﬁer, which makes this method more convenient

for the users, as compared with MPSVMs.

3 FEATURE PREPARATION

The WDBC and WPBC datasets were obtained from

the University of Wisconsin Hospitals, Madison, of

which the features were computed from digitized

FNA samples. A portion of well-differentiated cells

was scanned using a digital camera. The image anal-

ysis software system Xcyt was used to isolate indi-

vidual nuclei (Wolberg et al., 1994; Wolberg et al.,

1995; Mangasarian et al., 1995). In order to evalu-

ate the size, shape, and texture of each cell nuclei, ten

characteristics were derived and described as follows.

• Radius is computed by averaging the length of

radial line segments from the center of mass of

the boundary to each of the boundary points.

• Perimeter is measured as the sum of the distances

between consecutive boundary points.

• Area is measured by counting the number of pix-

els on the interior of the boundary and adding one-

half of the pixels on the perimeter, to correct for

the error caused by digitization.

• Compactness combines the perimeter and area to

give a measure of the compactness of the cell, cal-

culated as

perimeter

area

• Smoothness is quantiﬁed by measuring the differ-

ence between the length of each radial line and the

mean length of the two radial lines surrounding it,

calculated by

∑

points

− (r

+ r

i+1

)/2|

perimeter

where r

is the length of the line from the center

of mass of the boundary to each boundary point.

• Concavity is captured by measuring the size of

any indentations in the boundary of the cell nu-

cleus.

• Concave points is similar to concavity, but counts

only the number of boundary points lying on the

concave regions of the boundary, rather than the

magnitude of such concavities.

• Symmetry is measured by ﬁnding the relative dif-

ference in length between pairs of line segments

BREAST CANCER DIAGNOSIS AND PROGNOSIS USING DIFFERENT KERNEL-BASED CLASSIFIERS

345

perpendicular to the major axis of the contour of

the cell nucleus, calculated by

symmetry =

∑

|left

− right

∑

(left

+ right

)

where left

and right

denote the lengths of perpen-

dicular segments on the left and right of the major

axis, respectively.

• Fractal dimension is approximated using the

“coastline approximation” described by Mandel-

brot (Mandelbrot, 1997). The perimeter of the

nucleus is measured using increasingly larger

“rulers”. As the ruler size increases, the precision

of the measurement decreases, and the observed

perimeter decreases. Plotting these values on a

log-log scale and measuring the downward slope

gives the negative of an approximation to the frac-

tal dimension.

• Texture is measured by ﬁnding the variance of the

gray-scale intensities in the component pixels.

The mean value, standard error, and the extreme

(largest or “worst” ) value of each characteristic were

computed for each image, which resulted in 30 fea-

tures of 569 images, yielding a database of 569×30

samples representing 357 benign and 212 malignant

cases, for the WDBC dataset; and 30 features of 198

images, yielding a database of 198×30 samples rep-

resenting 151 nonrecurring and 47 recurring cases, for

the WPBC dataset.

4 EXPERIMENTS

Experiments and comparative analysis were con-

ducted on the WDBC and WPBC datasets, using

SVM, KFDA, PRQ classiﬁer, MPSVM, regularized

δ-MPSVM, and S2SP classiﬁer. The features were

normalized to have zero mean and unit variance be-

fore being used as the input of a classiﬁer. Clas-

siﬁcation performance is shown in terms of classi-

ﬁcation accuracy in percentage. The radial basis

function (RBF) kernel was employed to calculate the

inner-product matrix between samples in the kernel-

transformed feature space, given as

K(x

) = exp



−

− x

2σ



where σ is the kernel width set by the user. The SVM

was trained by using the “SVM and kernel methods

MATLAB toolbox” (Canu et al., 2003).

The 10-fold-cross validation was used to evaluate

the classiﬁers, which was executed by randomly di-

viding all the available samples into ten subsets each

Table 1: Performance comparison in percentage accuracy

and computing time for different kernel-based classiﬁers.

WDBC WPBC

Accu. Time Accu. Time

Methods (%) (Sec.) (%) (Sec.)

SVM 98.8 0.09 76.3 0.08

KFDA 97.2 0.09 76.3 0.02

PRQ 97.7 8.02 76.3 1.07

MPSVM 85.3 0.90 75.3 0.21

δ-MPSVM 91.6 0.67 76.3 0.09

S2SP 99.2 0.10 77.3 0.02

Lam et al. 95.6 N/A 76.3 N/A

with nearly the same number of samples. The same

ten sets of training-test trials were employed for every

classiﬁcation method, each with one subset for test

and the remaining nine subsets for training. Param-

eters of each classiﬁer were selected by using the 5-

fold-cross validation within the training set of the ﬁrst

trial. The same ﬁve sets of training-test trials were

conducted to select parameters for each classiﬁcation

method. Finally, the mean value of the ten test clas-

siﬁcation accuracies with the selected parameters was

used to represent the generalized performance.

The classiﬁcation performance and the corre-

sponding computing time of each classiﬁer are

recorded in Table 1 using the WDBC and WPBC

datasets; the results were also compared with the

10-fold-cross-validation performance obtained by the

edited nearest-neighbor (ENN) with pure ﬁltering

(Lam et al., 2002) using the same datasets. The S2SP

classiﬁer provided the best classiﬁcation accuracy of

99.2% as compared with the other ﬁve kernel-based

classiﬁers. Nearly all of our obtained results (above

97%) were better than the published result of 95.6%

(Lam et al., 2002) (see Table 1). For the more dif-

ﬁcult WPBC dataset, the S2SP classiﬁer provided

the best classiﬁcation accuracy of 77.3%. KFDA,

SVM, δ-MPSVMs, and the PRQ classiﬁer provided

the same performance of 76.3% as that obtained by

ENN (Lam et al., 2002) (see Table 1). KFDA, SVM,

and the S2SP classiﬁer possess faster training speed

than MPSVMs, δ-MPSVMs, and the PRQ classiﬁer,

and performs better than MPSVMs and δ-MPSVMs.

The classiﬁcation performance of the PRQ classiﬁer

is comparable to those obtained by KFDA, SVM, and

the S2SP classiﬁer.

For a reasonable comparison of the classiﬁcation

capabilities, a score is calculated by averaging the

classiﬁcation performance over the two datasets and

timing by 100 for each classiﬁer, and recorded in

Table 2. It can be seen from Table 2 that the S2SP

classiﬁer provides the highest score and requires

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

346

−10 −5 0 5 10

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

log

10−Fold Accuracy

WDBC Dataset

Figure 1: Performance variations of the δ-MPSVM classi-

ﬁer versus different values of log

10..

δ, with the RBF kernel

width σ ﬁxed as the selected values, for the WDBC dataset.

−10 −5 0 5 10

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.8

log

10−Fold Accuracy

WPBC Dataset

Figure 2: Performance variations of the δ-MPSVM classi-

ﬁer versus different values of log

10..

δ, with the RBF kernel

width σ ﬁxed as the selected values, for the WPBC dataset.

the least parameters to be tuned. Both the SVM

and δ-MPSVM classiﬁers require to determine one

extra regularization parameter. For SVM, C controls

the tradeoff between the complexity of a SVM and

the number of non-separable points. The SVM

performance is not very sensitive to the setting of

C. The performance variations of the δ-MPSVM

are provided in Fig. 1 and Fig. 2, by varying the

value of log

δ from -10 to 10, for the WDBC and

WPBC datasets, respectively. It can be seen from

Fig. 1 and Fig. 2 that performance of the δ-MPSVM

classiﬁer is sensitive to the setting of δ. Without

using the regularization term, the average score of the

MPSVM classiﬁer falls down from 84.0 to 80.3 (see

Table 2). However, tuning of the values of the kernel

parameters is unavoidable for all these kernel-based

classiﬁers.

Table 2: Comparison of classiﬁcation capability in average

percentage accuracy for different classiﬁers.

Rank Classiﬁers Score Parameters

1 S2SP 88.3 1 (σ)

2 SVM 87.6 2 (σ, C)

3 PRQ 87.0 1 (σ)

4 KFDA 86.8 1(σ)

5 δ-MPSVM 84.0 2 (σ, δ)

6 MPSVM 80.3 1 (σ)

5 CONCLUSIONS

Five recently developed, kernel-based, nonlinear

classiﬁers, including SVM, KFDA, PRQ classiﬁer,

MPSVMs (unregularized MPSVM and regularized δ-

MPSVM), and S2SP classiﬁer, have been applied to

breast cancer diagnosis and prognosis. We have stud-

ied and compared the beneﬁts of the above classi-

ﬁers in terms of classiﬁcation accuracy, computing

time, and sensitivity to the regularization parameter.

Studies were conducted with the WDBC and WPBC

datasets. Experimental results demonstrate that the

classiﬁcation accuracies of SVM, KFDA, S2SP, and

PRQ classiﬁers are comparable. However, the PRQ

classiﬁer possesses the slowest computing speed, as

the PRQ criterion built on pairwise constrains leads to

an increase of the computing speed by l

as the size

(l) of the training samples increases. The classiﬁca-

tion performance of MPSVM is unsatisfactory, and

sensitive to the setting of the regularization parameter

δ. From an overall consideration, the S2SP classiﬁer

is more favorable to users with not only higher clas-

siﬁcation accuracy but also faster computing speed;

furthermore, there is no regularization parameter to

be tuned for the S2SP classiﬁer.

ACKNOWLEDGEMENTS

T. Mu would like to acknowledge ﬁnancial support

from the Overseas Research Students Awards Scheme

(ORSAS), the Hsiang Su Coppin Memorial Scholar-

ship Fund, and the University of Liverpool, UK. We

thank the Medical Research Council (the Interdisci-

plinary Bridging Awards), UK, for ﬁnancial support.

REFERENCES

Aizerman, M., Braverman, E., and Rozonoer, L.

(1964). Theoretical foundations of the potential

BREAST CANCER DIAGNOSIS AND PROGNOSIS USING DIFFERENT KERNEL-BASED CLASSIFIERS

347

function method in pattern recognition learning.

Automation and Remote Control, 25:821–837.

Boser, B. E., Guyon, I. M., and Vapnik, V. N. (1992).

A training algorithm for optimal margin classi-

ﬁers. In Proc. of the 5th Annual ACM Workshop

on Computational Learning Theory, pages 144–

152.

Canu, S., Grandvalet, Y., and Rakotomam, A. (2003).

SVM and Kernel Methods Matlab Toolbox. Per-

ception Systems et Information, INSA de Rouen,

Rouen, France.

Cortes, C. and Vapnik, V. (1995). Support-vector net-

works. Machine Learning, 20(3):273–297.

Fisher, R. A. (1936). The use of multiple measure-

ments in taxonomic problems. Annals of Eugen-

ics, 7(2):179–188.

Guo, H. and Nandi, A. K. (2006). Breast cancer diag-

nosis using genetic programming generated fea-

ture. Pattern Recognition, 39:980–987.

Lam, W., Keung, C., and Ling, C. X. (2002). Learning

good prototypes for classiﬁcation using ﬁltering

and abstraction of instances. Pattern Recogni-

tion, 35(7):1491–1506.

Mandelbrot, B. B. (1997). The Fractal Geometry of

Nature, Chapter 5. W. H. Freeman and Com-

pany, New York.

Mangasarian, O. L., Street, W. N., and Wolberg, W. H.

(1995). Breast cancer diagnosis and prognosis

via linear programming. Operations Research,

43(4):570–577.

Mangasarian, O. L. and Wild, E. W. (2006). Multisur-

face proximal support vector machine classiﬁca-

tion via generalized eigenvalues. IEEE Trans-

actions on Pattern Analysis and Machine Intelli-

gence, 28:69–74.

Marshall, E. (1993). Search for a killer: Focus shifts

from fat to hormones in special report on breast

cancer. Science, 259:618–621.

Mika, S., R

atsch, G., Weston, J., Sch

olkopf, B., and

Muller, K. (1999). Fisher discriminant analysis

with kernels. In Proc. of IEEE Neural Networks

for Signal Processing Workshop, pages 41–48.

Mu, T. and Nandi, A. K. (2007). Breast cancer de-

tection from FNA using SVM with different pa-

rameter tuning systems and SOM–RBF classi-

ﬁer. Journal of the Franklin Institute, 344(3-

4):285–311.

Mu, T., Nandi, A. K., and Rangayyan, R. M. (2007a).

Pairwise Rayleigh quotient classiﬁer with appli-

cation to the analysis of breast tumors. In Proc.

of the 4th IASTED Int’l Conf. on Signal Process-

ing, Pattern Recognition, and Applications, SP-

PRA, pages 356–361, Innsbruck, Austria.

Mu, T., Nandi, A. K., and Rangayyan, R. M. (2007b).

Strict 2-surface proximal classiﬁer with applica-

tion to breast cancer detection in mammograms.

In Proc. of the 32nd Int’l Conf. on Acoustics,

Speech, and Signal Processing, ICASSP, vol-

ume 2, pages 477–480, Honolulu, HI.

Sch

olkopf, B., Smola, A. J., Williamson, R., and

Bartlett, P. (2000). New support vector algo-

rithms. Neural Computation, 12:1207–1245.

Shawe-Taylor, J. and Cristianini, N. (2004). Kernel

Methods for Pattern Analysis. Cambridge Uni-

versity Press, Cambridge, UK.

Street, W. N., Mangasarian, O. L., and Wolberg, W. H.

(1995). An inductive learning approach to prog-

nostic prediction. In Proc. of the 12th Int’l Conf.

on Machine Learning, ICML, pages 522–530,

Morgan Kaufmann.

Street, W. N., Wolberg, W. H., and Mangasarian, O. L.

(1993). Nuclear feature extraction for breast tu-

mor diagnosis. In Proc. of IST/SPIE Symposium

on Electronic Imaging: Science and Technology,

volume 1905, pages 861–870, San Jose, CA.

Wolberg, W. H., Street, W. N., and Mangasarian, O. L.

(1993). Breast cytology diagnosis via digital im-

age analysis. Analytical and Quantitative Cytol-

ogy and Histology, 15(6):396–404.

Wolberg, W. H., Street, W. N., and Mangasarian, O. L.

(1994). Machine learning techniques to diagnose

breast cancer from ﬁne-needle aspirates. Cancer

Letter, 77:163–171.

Wolberg, W. H., Street, W. N., and Mangasarian, O. L.

(1995). Image analysis and machine learning ap-

plied to breast cancer diagnosis and prognosis.

Analytical and Quantitative Cytology and His-

tology, 17(2):77–87.

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

348