Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
38,
No.
2,
May
2025,
pp.
1383
∼
1391
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v38.i2.pp1383-1391
❒
1383
Chebyshe
v
distance-embedded
twin
support
v
ector
machine
f
or
sk
ewed
classication
pr
oblems
Sai
Lakshmi
Balasubramanian,
Gajendran
Ganesan
Department
of
Mathematics,
Colle
ge
of
Engineering
and
T
echnology
,
SRM
Institute
of
Science
and
T
echnology
,
Cheng
alpattu,
India
Article
Inf
o
Article
history:
Recei
v
ed
Apr
26,
2024
Re
vised
No
v
7,
2024
Accepted
No
v
24,
2024
K
eyw
ords:
Benchmark
dataset
Chebyshe
v
distance
Rectangular
k
ernel
Support
v
ector
machine
T
win
support
v
ector
machine
ABSTRA
CT
Support
v
ector
machine
(SVM)
is
a
pi
v
otal
classication
algorithm,
and
its
e
v
olutionary
counterpart,
the
twi
n
SVM
(TWSVM),
has
g
ained
acclaim
for
its
adv
anced
generalization
capabilities
,
particularly
i
n
handling
imbalanced
data.
TWSVMs
a
chie
v
e
swift
training
by
e
xplicitly
e
xploring
a
pair
of
non-
parallel
h
yperplanes,
yet
selecting
numerical
v
alues
for
h
yperparameters
poses
a
challenge
due
to
the
uncertainty
introduced
by
random
preferences.
This
paper
presents
a
no
v
el
approach,
the
Chebyshe
v
distance-based
TWSVM,
specically
designed
for
h
yperparameter
tuning
in
imbalanced
binary
classication.
This
inno
v
ati
v
e
model
mitig
ates
the
uncertainty
of
h
yperparameter
selection
by
le
v
er
-
aging
Chebyshe
v
distance,
the
reby
enhancing
the
generalization
capabilities
of
the
TWSVM.
T
o
e
v
aluate
its
ef
cac
y
,
computational
tests
were
conducted
on
publicly
a
ccessible
real-w
orld
benchmark
datasets
across
v
arious
domains,
including
non-linear
cases
.
The
results
demonstrate
that
the
Chebyshe
v
distance-based
TWSVM
outperforms
se
v
eral
e
xisting
methods,
achie
ving
supe-
rior
performance
with
reduced
computational
time
and
setting
a
ne
w
benchmark
in
the
eld.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Gajendran
Ganesan
Department
of
Mathematics,
Colle
ge
of
Engineering
and
T
echnology
SRM
Institute
of
Science
and
T
echnology
Kattankulathur
,
Cheng
alpattu,
603203,
T
amil
Nadu,
India
Email:
g
ajendr
g@srmist.edu.in
1.
INTR
ODUCTION
Support
v
ector
machine
(SVM)
is
a
rob
ust
binary
class
ication
method
introduced
by
Cortes
and
V
apnik
in
1995.
SVM
is
based
on
V
apnik-Cherv
onenkis
dimensional
theory
and
structural
risk
minimization.
It
utilizes
decision
h
yperplanes,
kno
wn
as
support
h
yperplanes,
to
separate
data
into
tw
o
classes.
The
distance
between
t
hese
h
yperplanes
is
maximized
through
quadratic
programming,
making
SVM
ef
fecti
v
e
e
v
en
in
non-
linearly
separable
cases
with
the
k
ernel
trick.
The
SRM
principle
enhances
generalization
and
reduces
training
errors.
SVM
nds
widespread
applications
in
te
xt
classication,
f
ace
recognition,
nance,
bio-medicine,
and
more.
Rob
ust
SVM,
with
a
ne
w
loss
function,
addresses
noise
for
better
generalization.
The
basic
idea
in
v
olv
es
nding
supporting
h
yperplanes
with
the
lar
gest
distance
between
them,
achie
v
ed
by
solving
a
con
v
e
x
quadratic
minimization
problem.
Modications
lik
e
the
least
squares
SVM
simplify
h
yperplane
determination
through
linear
equations,
of
fering
adv
antages
in
specic
scenarios
[1].
In
non-linear
models,
a
feature
map
ϕ
:
R
n
→
H
maps
data
into
a
higher
-dimensional
Hilbert
space
H
,
where
the
search
for
separating
h
yperplanes
occurs,
and
the
associated
k
ernel
function
is
denoted
as
K
:
R
n
×
R
n
→
R
[2].
The
decision
function
for
linear
models
is
of
the
form:
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1384
❒
ISSN:
2502-4752
f
(
x
)
=
sg
n
(
⟨
w
,
z
⟩
+
b
)
(1)
and
the
decision
function
for
non-linear
models
will
be,
f
(
x
)
=
sg
n
(
i
=
I
X
i
=1
α
i
y
i
K
(
x
i
,
x
)
+
b
)
(
x
∈
R
n
)
(2)
The
primary
aim
of
SVM
(2)
and
(3)
is
to
minimize
a
higher
bound
on
generalization
error
.
In
linearly
separable
cases,
SVM
nds
the
h
yperplane
that
is
f
arthest
from
the
closest
learning
samples.
F
or
non-separable
data,
SVM
maximizes
the
mar
gin
while
k
eeping
the
error
lo
w
.
Ho
we
v
er
,
traditional
SVM
f
aces
challenges
due
to
comple
x
quadratic
programming
problem
(QPP)
calculations
and
dif
culties
handling
non-
linearly
independent
data,
which
may
lead
to
lo
wer
accurac
y
[3].
Despite
its
superior
performance
compared
to
other
systems,
SVM
struggles
with
comple
x
data
due
to
the
high
computational
cost
of
solving
QPPs.
T
o
tackle
this,
Plat
t
[4]
introduced
sequential
mini
mal
optimiza-
tion
(SMO),
a
specialized
algorithm
for
solving
QPPs
during
SVM
training.
V
arious
enhancements,
including
adv
anced
k
ernel
methods
[5],
ha
v
e
im
p
r
o
v
ed
SVM
accurac
y
o
v
er
the
past
decade.
In
2001,
Fung
and
Man-
g
asarian
[6]
proposed
proximal
SVM
(PSVM),
using
systems
of
linear
equations
instead
of
quadratic
or
linear
programming
problems
,
and
classifying
data
by
assigning
points
to
the
closest
of
tw
o
parallel
planes.
In
2006,
Mang
asarian
and
W
ild
[7]
introduced
generalized
eigen
v
alue
proximal
support
v
ector
machine
(GEPSVM),
which
classies
non-parallel
h
yperplanes
using
a
generalized
eigen
v
alue
approach
and
a
v
oids
solving
QPPs.
Y
e
and
Y
e
[8]
proposed
an
impro
v
ed
approach
in
2009,
laying
the
groundw
ork
for
twin
SVM
(TWSVM).
Jayade
v
a
et
al.
[9]
introduced
TWS
VM,
a
binary
SVM
classier
,
to
enhance
GEPSVM’
s
general-
ization
ability
.
TWSVM
determines
tw
o
non-parallel
h
yperplanes
and
solv
es
tw
o
small
QPPs,
reducing
com-
putational
cost.
TWSVM
of
fers
f
aster
performance
and
better
generalization
than
SVM
and
GEPSVM,
using
k
ernels
in
non-linear
classication
while
requiring
matrix
in
v
ersions.
In
2017,
Ma
et
al.
[10]
proposed
an
it-
erati
v
e
algorithm
based
on
L
2
p
norm
distance
to
enhance
TWSVM
for
tw
o-class
data
classication.
Although
numerous
methods
e
xist
in
the
literature
to
enhance
the
performance
of
the
TWSVM
model,
there
is
still
no
appropriate
method
to
tune
the
h
yperparameters,
which
are
considered
k
e
y
parameters
in
the
TWSVM
model.
T
o
address
this
issue,
in
2023
Balasubramanian
and
Gajendra
n
[11]
introduced
the
P-distance
based
re
gularized
TWSVM.
This
method
aims
to
impro
v
e
the
generalization
capacity
of
TWSVM
and
reduce
the
randomness
in
h
yperparameter
tuning.
T
o
be
precise,
the
proposed
method
specically
addresses
the
issue
of
randomly
tuning
h
yperparameters
by
using
Chebyshe
v
distance.
This
study
in
v
estig
ates
the
ef
fect
of
a
method
designed
to
impro
v
e
the
generalization
capacity
of
TWSVM
and
reduce
the
randomness
in
h
yperparameter
tuning.
2.
TWIN
SUPPOR
T
VECT
OR
MA
CHINE
Consider
all
data
samples
in
the
positi
v
e
class
of
a
matrix
A
∈
R
m
1
×
n
,
where
i
th
r
ow
A
i
∈
R
n
and
the
matrix
B
∈
R
m
2
×
n
represents
the
ne
g
ati
v
e
class.
From
(6),
the
linear
TWSVM
determines
tw
o
non-parallel
h
yperplanes,
where
w
1
∈
R
m
,
w
2
∈
R
m
,
b
1
∈
R
and
b
2
∈
R
.
The
h
yperplane
of
TWS
VM
is
positioned
close
to
the
data
of
one
class
[1]
or
[
−
1]
and
distanced
from
the
data
of
another
class
[1]
or
[
−
1]
as
much
as
possible.
The
formulation
of
TWSVM
can
be
e
xpressed
as
follo
ws:
min
(
w
1
,b
1
,q
1
)
∈
R
(2
n
+1)
∥
Aw
1
+
e
1
b
1
∥
2
+
c
1
e
T
2
q
1
,
such
that
−
(
B
w
1
+
e
2
b
1
)
+
q
1
≥
e
2
,
q
1
≥
0
(3)
min
(
w
2
,b
2
,q
2
)
∈
R
(2
n
+1)
∥
B
w
2
+
e
2
b
2
∥
2
+
c
2
e
T
1
q
2
,
such
that
(
Aw
2
+
e
1
b
2
)
+
q
2
≥
e
1
,
q
2
≥
0
(4)
In
TWSVM,
c
1
,
c
2
>
0
are
h
yperparameters,
e
1
,
e
2
are
v
ectors
of
ones
of
appropriate
dimensions
and
q
1
,
q
2
are
slack
v
ariables.
Apparently
,
the
goal
of
TWSVM
is
to
solv
e
the
QPP’
s
(7)
and
(8).
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
38,
No.
2,
May
2025:
1383–1391
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1385
The
dual
problems
can
be
obtained
as
follo
ws:
max
α
e
T
2
α
−
1
2
α
T
G
(
H
T
H
)
−
1
G
T
α
,
such
that
0
≤
α
≤
c
1
(5)
max
γ
e
T
1
γ
−
1
2
γ
T
H
(
G
T
G
)
−
1
H
T
γ
,
such
that
0
≤
γ
≤
c
2
(6)
where
α
and
γ
are
Lagrange
multipliers,
H
=
[
Ae
1
]
and
G
=
[
B
e
2
]
.
The
solutions
of
equations
utilized
to
nd
tw
o
h
yper
planes
by
,
[
w
1
b
1
]
T
=
−
(
H
T
H
)
−
1
G
T
α
and
[
w
2
b
2
]
T
=
−
(
G
T
G
)
−
1
H
T
γ
Although
the
matrices
G
T
G
or
H
T
H
are
positi
v
e
semi
denite
and
singular
.
The
y
introduced
a
re
g-
ularization
parameter
and
the
in
v
erse
matrices
(
G
T
G
)
−
1
and
(
H
T
H
)
−
1
are
replaced
by
(
G
T
G
+
δ
I
)
−
1
and
(
H
T
H
+
δ
I
)
−
1
respecti
v
ely
,
where
δ
and
I
are
positi
v
e
scalar
and
unit
matrix
respecti
v
ely
[11].
The
nonlinear
TWSVM
are
gi
v
en
by
,
min
(
w
1
,b
1
,q
1
)
∈
R
2
n
+1
∥
K
(
A,
C
T
)
w
1
+
e
1
b
1
∥
2
+
c
1
e
T
2
q
1
,
such
that
−
(
K
(
B
,
C
T
)
w
1
+
e
2
b
1
)
+
q
1
≥
e
2
,
q
1
≥
0
(7)
min
(
w
2
,b
2
,q
2
)
∈
R
2
n
+1
∥
K
B
,
C
T
w
2
+
e
2
b
2
∥
2
+
c
2
e
T
1
q
2
,
such
that
−
K
(
A,
C
T
)
w
2
+
e
1
b
2
+
q
2
≥
e
1
,
q
2
≥
0
(8)
The
performance
of
TWSVM
e
xceeds
that
of
classical
SVM
and
GEPSVM
o
n
specic
benchmark
datasets.
During
simulation,
a
k
e
y
challenge
is
selecting
appropriate
parameters
for
model
construction.
Despite
the
e
xistence
of
v
arious
parameter
selection
methods
for
TWSVM,
directly
choosing
h
yperparameters
remains
a
laborious
e
xhausti
v
e
search
process.
T
o
address
these
issues,
the
proposed
Chebyshe
v-based
(CB-TWSVM)
formulation
pro
v
es
benecial
in
tuning
h
yperparameters
without
relying
on
randomness.
3.
CHEBYSHEV
DIST
ANCE
B
ASED
TWSVM
Hyperparameter
sensiti
vity
signicantly
af
fects
the
performance,
con
v
er
gence,
and
generalization
of
machine
learning
models,
including
TWSVM,
which
are
particularly
sensiti
v
e
to
the
re
gularizat
ion
parameter
c
i
.
Smaller
v
alues
of
c
i
lead
to
wider
mar
gins
b
ut
allo
w
more
misclassications,
while
lar
ger
v
alues
result
in
tighter
mar
gins
with
fe
wer
errors.
T
raditional
TWSVM
models
often
lack
systematic
methods
for
tuning
c
i
,
prompting
the
de
v
elopment
of
CB-TWSVM,
a
ne
w
approach
that
emplo
ys
Chebyshe
v
distance
to
impro
v
e
h
yperparameter
selection
without
rel
ying
on
random
v
alues
where
numerous
literature
re
vie
ws
suggest
that
distance
measures
can
be
emplo
yed
in
v
arious
steps
of
TWSVM
algorithm
[12].
Additionally
,
to
enhance
performance,
we
use
a
x
ed
rectangular
k
ernel.
The
Chebyshe
v
distance,
also
kno
wn
as
the
maxim
u
m
or
L
∞
metric,
measures
the
lar
gest
dif
ference
between
an
y
coordinate
dimension
of
tw
o
points
[13].
The
mathematical
foundation
of
Chebyshe
v
distance
is
deri
v
ed
from
Mink
o
wski
distance.
F
or
v
ariables
x
and
x
′
,
Mink
o
wski
distance
is
dened
as,
d
p
(
x,
x
′
)
=
∥
x
−
x
′
∥
p
=
(
d
P
i
=1
|
x
i
−
x
′
i
|
p
)
1
p
where
p
≥
1
.
When
p
→
∞
,
Chebyshe
v
metric
can
be
deri
v
ed
as,
D
che
(
x,
x
′
)
=
∥
x
−
x
′
∥
∞
=
max
i
|
x
i
−
x
′
i
|
(9)
F
or
tw
o
points
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
in
a
tw
o-dimensional
space,
the
Chebyshe
v
distance
is
dened
as:
D
∞
=
max
(
|
x
2
−
x
1
|
,
|
y
2
−
y
1
|
)
Chebyshe
v
distance-embedded
twin
support
vector
mac
hine
for
...
(Sai
Lakshmi
Balasubr
amanian)
Evaluation Warning : The document was created with Spire.PDF for Python.
1386
❒
ISSN:
2502-4752
general,
for
n-dimensional
space,
the
Chebyshe
v
distance
betwe
en
tw
o
points
(
x
1
,
y
1
,
.
.
.
,
z
1
)
and
(
x
2
,
y
2
,
.
.
.
,
z
2
)
is
gi
v
en
by:
D
∞
=
max
(
|
x
2
−
x
1
|
,
|
y
2
−
y
1
|
,
.
.
.
,
|
z
2
−
z
1
|
)
Chebyshe
v
distance
essentially
computes
the
maximum
absolute
dif
ference
along
an
y
dimension.
Consequently
,
Chebyshe
v
distance
serv
es
as
a
measure
of
the
maximum
separation
along
an
y
axis
between
tw
o
points
in
the
learning
set
[14].
Applying
(7)
to
(9)
in
place
of
c
1
and
c
2
for
the
positi
v
e
and
ne
g
ati
v
e
training
sample
sets,
respecti
v
ely
,
we
obtain
the
follo
wing:
for
the
positi
v
e
training
sample
set,
we
use
(9)
in
place
of
c
1
in
(7),
and
for
the
ne
g
ati
v
e
training
sample
set,
we
use
(9)
in
place
of
c
2
in
(8).
This
substitution
allo
ws
us
to
uti
lize
the
relationship
dened
in
(9)
to
e
xpress
c
1
and
c
2
for
the
respecti
v
e
positi
v
e
and
ne
g
ati
v
e
sample
sets
[15].
Proposed
CB-TWSVM
sho
wn
in
Algoritm
1.
Algorithm
1.
Proposed
CB-TWSVM
1.
X
=
{
¯
x
i
|
¯
x
i
=
(
x
i
1
,
x
i
2
,
.
.
.
)
}
∀
x
i
1
∈
[0
,
1]
where,
X
1
⊆
X
and
X
2
⊆
X
are
tw
o
classes
of
X
are
labeled
as
class
I
and
class
II.
2.
Set
h
yperparameters
c
I
,
c
I
←
c
1
,
c
2
for
each
c
I
,
From
(3.),
D
che
(
x,
x
′
)
=
∥
x
−
x
′
∥
∞
=
max
i
|
x
i
−
y
′
i
|
3.
Compute
c
I
c
I
=
σ
[
D
che
(
x
i
,
x
j
)
/i
̸
=
j
]
∀
I
=
1
,
2
for
x
i
,
x
j
∈
X
I
4.
Apply
c
I
←
c
1
,
c
2
,
v
alidating
the
model
5.
Update
and
xing
k
ernel
parameter
as
rectangular
k
ernel
k
(
A,
C
T
)
=
k
(
B
,
C
T
)
=
1
2
,
6.
Generate
h
yperplane,
(a)
[
w
1
b
1
]
T
=
−
(
H
T
H
)
−
1
G
T
α
(b)
[
w
2
b
2
]
T
=
−
(
G
T
G
)
−
1
H
T
γ
end
4.
SIMULA
TION
RESUL
TS
AND
DISCUSSIONS
After
testing
v
arious
pairwise
distance
measures,
Chebyshe
v
distance
pro
v
ed
ef
fecti
v
e
for
tuning
c
i
v
alues.
The
performance
of
the
CB-TWSVM
w
as
tested
on
se
v
en
benchmark
datasets
[16],
which
were
nor
-
malized
to
[0,1]
scale
using
Z-score
normalization.
MA
TLAB
2023
w
as
used
f
o
r
performance
e
v
aluations,
with
g
aussian
k
ernels.
The
ef
fecti
v
eness
of
CB-TWSVM
w
as
also
e
xamined
under
conditions
of
data
imbalance,
a
common
issue
in
classication
tasks
characterized
by
unequal
class
distrib
utions,
quantied
by
the
imbalance
ratio
(IR).
4.1.
Dataset
description
Breast
cancer
699:
breast
cancer
,
a
leading
cause
of
female
cancer
deaths,
has
dri
v
en
research
into
pre
v
ention.
Da
ta
mining
t
echniques
are
uti
lized
to
e
xtrac
t
insights
for
prediction
and
classication.
The
WBCD
dataset,
containing
699
samples
(with
16
missing
v
alues
as
of
July
15,
1992),
includes
10
attrib
utes.
Among
these,
241
cases
(65.5%)
are
malignant
and
458
(34.5%)
are
benign.
The
class
attrib
ute
uses
inte
gers
2
and
4
to
denote
benign
and
malignant
cases,
respecti
v
ely
[17].
Breast
cancer
569:
breast
cancer
ranks
as
the
second
leading
cause
of
death
among
w
omen
w
orldwide,
underscoring
the
importance
of
early
detection.
T
o
impro
v
e
predicti
v
e
accurac
y
,
researchers
increasi
ngly
rely
on
adv
anced
machine
learning
algorithms.
A
k
e
y
resource
in
this
ef
fort
is
the
W
isconsin
breast
cancer
dataset
(WBCD)
from
the
UCI
repository
.
This
dataset
contains
569
instances
and
30
features,
with
357
benign
and
212
malignant
cases,
pro
viding
a
comprehensi
v
e
and
rob
ust
foundation
for
assessing
the
ef
fecti
v
eness
of
machine
learning
models
in
predicting
breast
cancer
outcomes
[18].
Fertility:
infertility
carries
a
societal
stigma,
with
30%
of
cas
es
attrib
uted
to
male
f
actors.
Despite
this,
male
infertility
remains
under
diagnosed
and
under
represented.
The
W
orld
Health
Or
g
anization
(WHO)
links
the
decline
in
male
fertility
prim
arily
to
lifestyle
and
en
vironmental
changes.
The
dataset
includes
9
input
features
related
to
these
f
actors,
follo
wing
WHO
criteria.
It
contains
100
samples,
with
88
samples
indicating
normal
seminal
quality
and
12
indicating
altered
seminal
quality
[19].
Immunotherap
y:
human
papilloma
virus
(HPV)
infection
can
cause
w
arts,
particularly
on
the
hands
and
feet,
complicating
treatment
at
later
stages.
A
lite
rature
re
vie
w
sho
ws
v
aried
proposed
treatments,
with
a
k
e
y
challenge
being
the
need
for
personalized
patient
care.
This
v
ariability
complicates
the
identication
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
38,
No.
2,
May
2025:
1383–1391
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J
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&
Comp
Sci
ISSN:
2502-4752
❒
1387
of
optimal
treatments
for
f
aster
reco
v
ery
.
In
their
study
,
the
authors
used
immunothera
p
y
datasets
from
UCI,
which
contain
detailed
information
on
8
attrib
utes
across
90
patient
instances.
These
features
are
crucial
for
analysing
and
understanding
immunotherapeutic
responses
to
HPV
[20].
German
statlog:
the
German
statlog
dataset,
used
for
binary
classication,
cont
ains
1,000
inst
ances
with
20
features
related
to
nancial
history
.
It’
s
commonly
used
to
predict
credit
risk
(good
or
bad)
and
to
compare
the
performance
of
classication
algorithms
in
machine
learning
research
[21].
Ionosphere:
the
Ionosphere
dataset
is
us
ed
for
binary
classication
tasks
in
machine
learning.
It
contains
351
instances
with
34
features,
which
are
radar
signal
attrib
utes
used
to
classify
whether
signals
pass
through
the
ionosphere
(labelled
“good”)
or
are
block
ed
(labelled
“bad”).
This
dataset
is
often
used
to
e
v
aluate
classication
algorithms
in
research
[22].
Sonar:
the
Sonar
dataset
is
used
for
binary
classication
in
machine
learning,
c
on
s
isting
of
208
instances
with
60
features.
Each
feature
represents
the
ener
gy
of
sound
w
a
v
es
reected
of
f
objects
at
v
arious
angles,
and
the
task
is
to
classify
whether
an
object
is
a
rock
or
a
metal
c
ylinder
(mine).
This
dataset
is
commonly
used
to
test
and
compare
the
performance
of
classication
algorithms
[23].
The
purpose
of
our
modelling
is
to
dem
onstrate
the
stability
of
our
model
across
dif
ferent
non-linear
datasets.
Benchmark
datasets
from
v
arious
domains
such
as
biomedical,
nancial,
and
other
sectors
were
tak
en
from
UCI.
T
o
demonstrate
the
stability
of
the
proposed
model,
their
a
ccurac
y
,
F-mean,
recall,
and
elapsed
time
were
compared
with
t
hose
of
other
e
xisting
SVM
models.
T
able
1
sho
ws
the
measure
of
imbalance
ratio
and
their
descriptions
about
dataset.
T
able
1.
Class
imbalance
f
actor
Dataset
Instances
Attrib
utes
IR
Breast
cancer
699
10
1.90041
Breast
cancer
569
32
1.68396
Fertility
100
10
7.3333
Immunotherap
y
90
8
3.5
German
statlog
1,000
20
2.3333
Ionosphere
351
34
1.7857
Sonar
208
61
1.1224
Figure
1
depicts
the
scatter
plots
which
are
useful
for
visualizing
the
relationship
or
correlation
be-
tween
tw
o
continuous
v
ariables.
Each
Figures
1(a)
to
1(g)
represents
the
dispersion
of
the
respecti
v
e
dataset
along
the
x
and
y
ax
es,
respecti
v
ely
.
It
allo
w
us
to
observ
e
patterns,
trends,
and
potential
outliers
in
the
data.
The
position
of
each
point
on
the
plot
pro
vides
information
about
the
v
alues
of
both
v
ariables
x
1
,
x
2
for
that
specic
data
point.
Performance
of
proposed
method
is
computed
by
the
follo
wing
metrics
as
sensiti
vity
,
accurac
y
,
F1
score,
and
its
elapsed
time.
Numerical
analysis
carry
through
linear
k
(
x
i
,
x
j
)
=
x
t
i
x
j
and
the
non-linear
k
ernel
such
as
g
aussian
k
ernel
k
(
x
i
,
x
j
)
=
exp
(
−
∥
x
i
−
x
j
∥
2
2
µ
2
)
with
µ
where
the
h
yperparameter
c
1
and
c
2
are
tuned
using
Chebyshe
v
distance
whereas
xing
rectangular
k
ernel
for
more
precise
results.
Performance
of
the
indi
vidual
benchmark
dataset
and
e
xperimental
results
signies
the
rob
ustness
and
adv
antage
of
CB-TWSVM
approach.
T
able
2
sho
ws
the
performance
of
non-linear
CB-TWSVM
of
bench-
mark
breast
cancer
datasets
and
it
is
compared
with
v
arious
SVM’
s
lik
e
PSVM,
CPSVM,
CPSVM,
TWSVM,
LSTSVM,
CPTWSVM,
CB-TWSVM
with
their
elapsed
time
[2].
T
able
3
sho
ws
the
performance
of
non-linear
CB-TWSVM
of
fertility
and
immunotherap
y
datasets.
Also,
the
comparison
with
other
e
v
olv
ed
SVM
models,
e
v
aluation
metrics
and
computational
time
are
discussed
[24].
T
able
4
sho
ws
the
performance
of
non-linear
CB-TWSVM
of
v
arious
benchmark
datasets.
Also,
the
comparison
with
other
e
v
olv
ed
SVM
models
lik
e
TWSVM,
TBSVM,
F
-TWSVM,
γ
-TWSVM,
WLSTSVM,
and
LSTSVM’
s
e
v
aluation
metrics
and
computational
time
are
discussed
[25],
[26].
The
proposed
CB-TWSVM
outperforms
all
the
other
e
xisting
SVM
models
in
accurac
y
as
well
as
other
metrics
for
the
benchmark
datasets
tak
en
for
numerical
e
xperiments.
Figure
2
with
2(a)
to
2(g)
in
Appendix
sho
ws
the
three
dimensional
surf
ace
plot
for
representing
data
with
multiple
components
or
for
visualizing
uncertainties
or
error
.
The
benchmark
dataset
is
analysed
with
the
tar
get
v
alues
represented
along
the
x-axis,
actual
v
alues
along
the
y-axis,
and
the
corresponding
errors
depicted
along
the
z-axis.
The
proposed
method
is
further
applied
to
lar
ge
datasets,
including
those
for
image
and
te
xt
recognition.
Chebyshe
v
distance-embedded
twin
support
vector
mac
hine
for
...
(Sai
Lakshmi
Balasubr
amanian)
Evaluation Warning : The document was created with Spire.PDF for Python.
1388
❒
ISSN:
2502-4752
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure
1.
Dataset
dispersion
of:
(a)
breast
cancer
(699),
(b)
breast
cancer
(569),
(c)
fertility
,
(d)
immunotherap
y
,
(e)
German
statlog,
(f)
ionosphere,
and
(g)
sonar
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
38,
No.
2,
May
2025:
1383–1391
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1389
T
able
2.
Performance
analysis
of
v
arious
SVMs
for
breast
cancer
dataset
in
non-linear
case
Dataset
SVM
PSVM
CPSVM
TWSVM
LSTSVM
CPTSVM
CB-TWSVM
Acc
Acc
Acc
Acc
Acc
Acc
Acc
F-mean
F-mean
F-mean
F-mean
F-mean
F-mean
F-mean
Recall
Recall
Recall
Recall
Recall
Recall
Recall
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
BC
(683)
97.06
96.88
96.54
96.91
95.98
97.69
99.51
-
-
-
-
-
-
0.9976
-
-
-
-
-
-
0.9951
0.0036
0.1852
0.7261
0.0233
0.0126
0.2346
0.3745
BC
(569)
97.93
98.01
97.97
96.34
96.38
97.56
99.12
-
-
-
-
-
-
0.9956
-
-
-
-
-
-
0.9912
0.18
0.1054
0.4479
0.0539
0.0086
0.3759
0.5193
T
able
3.
Performance
analysis
of
v
arious
SVMs
for
the
fertility
and
immunotherap
y
in
non-linear
case
Dataset
FSSVM
FS-CTBSVM
Chi-TBSVM
TBSVM
TWSVM
NN
CB-TWSVM
Acc
Acc
Acc
Acc
Acc
Acc
Acc
F-mean
F-mean
F-mean
F-mean
F-mean
F-mean
F-mean
Recall
Recall
Recall
Recall
Recall
Recall
Recall
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
Fertility
91.89
88.89
89.88
88.98
88.14
87.00
96.67
-
-
-
-
-
-
0.9831
-
-
-
-
-
-
0.9667
0.02
0.004
0.009
0.005
0.74
0.48
0.04283
Immunotherap
y
87.89
87.89
86.77
83.41
78.93
87.80
96.30
-
-
-
-
-
-
0.9811
-
-
-
-
-
-
0.963
0.04
0.06
0.03
0.03
0.41
0.56
0.02086
T
able
4.
Performance
analysis
of
v
arious
SVMs
for
the
benchmark
datasets
in
non-linear
case
Dataset
TWSVM
TBSVM
F-TWSVM
γ
-TWSVM
WLSTSVM
LSTSVM
CB-TWSVM
Acc
Acc
Acc
Acc
Acc
Acc
Acc
F-mean
F-mean
F-mean
F-mean
F-mean
F-mean
F-mean
Recall
Recall
Recall
Recall
Recall
Recall
Recall
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
T
ime(sec)
German
Statlog
-
-
78.20
76.25
76.08
74.96
98
-
-
-
-
-
-
0.9899
-
-
-
-
-
0-
0.98
-
-
14.98
13.89
10.713
9.8172
2.4528
Ionosphere
87.46
87.75
-
94.72
95.68
91.82
94.03
-
-
-
-
-
-
0.9692
-
-
-
-
-
-
0.9403
0.0064
0.088
-
3.24
0.6141
0.5438
0.14326
Sonar
89.64
90.0
-
91.71
-
-
95.16
-
-
-
-
-
-
0.9752
-
-
-
-
-
-
0.9516
0.014
0.008
-
0.52
-
-
0.0615
5.
CONCLUSION
This
paper
introduces
a
no
v
el
e
xtension
to
TWSVM
called
the
CB-TWSVM,
designed
to
enhance
the
performance
of
the
traditional
TWSVM.
CB-TWSVM
incorporates
Chebyshe
v
distance
as
re
gularized
h
y-
perparameters,
le
v
eraging
the
distance
between
positi
v
e
and
ne
g
ati
v
e
classes
in
the
training
data
to
construct
non-parallel
h
yperplanes.
This
is
achie
v
ed
by
inte
grating
the
standard
de
viation
of
Chebyshe
v
distances
into
the
model.
One
of
the
k
e
y
adv
antages
of
CB-TWSVM
is
its
ability
to
naturally
generalize
from
TWSVM,
strate
gically
considering
the
distance
between
binary
classes
rather
than
relying
on
random
selection.
This
approach
is
particularly
benecial
in
real-w
orld
scenarios
where
the
distance
measures
between
positi
v
e
and
ne
g
ati
v
e
classes
dif
fer
signicantly
.
Experimental
results
demonstrate
that
CB-TWSVM
of
fers
superior
gener
-
alization
performance
compared
to
other
classiers,
outperforming
e
xisting
models
in
terms
of
accurac
y
and
Chebyshe
v
distance-embedded
twin
support
vector
mac
hine
for
...
(Sai
Lakshmi
Balasubr
amanian)
Evaluation Warning : The document was created with Spire.PDF for Python.
1390
❒
ISSN:
2502-4752
other
statistical
measures.
Specically
,
CB-TWSVM
achie
v
es
the
highest
accurac
y
rates
of
99.51%,
99.12%,
96.67%,
96.29%,
98%,
94.03%,
and
95.16%
on
the
breast
cancer
699,
breast
cancer
569,
fertility
,
immunother
-
ap
y
,
German
statlog,
ionosphere,
and
sonar
datasets
respecti
v
ely
.
APPENDIX
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure
2.
Surf
ace
plot
for
misclassication:
(a)
breast
cancer
(699),
(b)
breast
cancer
(569),
(c)
fertility
,
(d)immunotherap
y
,
(e)
German
statlog(f)
ionosphere,
and
(g)
sonar
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BIOGRAPHIES
OF
A
UTHORS
Sai
Lakshmi
Balasubramanian
earned
her
M.Sc.
de
gree
in
Mathematics
from
DG
V
aishna
v
a
Colle
ge,
Chennai,
T
amil
Nadu,
India,
in
2011,
follo
wed
by
an
M.Phil.
de
gree
from
PRIST
Uni
v
ersity
,
Thanja
vur
,
T
amil
Nadu,
India,
in
2013.
She
is
currently
pursuing
her
Ph.D.
in
the
De-
partment
of
Mathematics
at
the
Colle
ge
of
Engineering
and
T
echnology
,
SRM
Institute
of
Science
and
T
echnology
,
Kattankulathur
,
Chennai,
T
amil
Nadu,
India.
Her
research
interests
include
neural
netw
orks,
support
v
ector
machines,
genetic
algorithms,
and
con
v
olution
neural
netw
orks.
She
can
be
contacted
at
email:
sb7831@srmist.edu.in.
Gajendran
Ganesan
is
an
assistant
professor
(Sr
.Gr)
at
the
Department
of
Mathematics,
Colle
ge
of
Engineering
and
T
echnology
,
SRM
Institut
e
of
Science
and
T
echnology
,
Kattankulathur
,
Chennai,
T
amil
Nadu,
India.
He
is
more
than
30
years
e
xperience
in
teaching
as
well
as
in
the
research.
His
interests
include
machine
learning:
articial
neural
netw
orks
with
back
propag
ation,
dynamic
neural
netw
orks,
radial
basis
function
and
support
v
ector
machine.
He
can
be
contacted
at
email:
g
ajendr
g@srmist.edu.in.
Chebyshe
v
distance-embedded
twin
support
vector
mac
hine
for
...
(Sai
Lakshmi
Balasubr
amanian)
Evaluation Warning : The document was created with Spire.PDF for Python.