Information Technology Reference
In-Depth Information
From the
α
i
obtained by the solution of (4), we can recover the separating
surface and the scalar
b
determined by the support vectors denoted by
#SV
(for which
α
i
>
0). By changing the kernel function
K
as a linear inner prod-
uct, a polynomial, a radial basis function or a sigmoid neural network, we can
get different classification model. The classification of a new data point
x
is
based on:
SV
f
(
x
)=
sign
(
y
i
α
i
K
x, x
i
−
b
)
i
=1
5.3.2
Novelty Detection with Support Vector Machines (One-Class
SVM)
For one-class (novelty detection), the SVM algorithm has to find an hypersphere
with a minimal radius
R
and center
c
which contains most of the data and then
we test if novel points lie outside the boundary of the hypersphere. One-class
SVMs pursue these goals with the quadratic program (5):
m
m
m
min
α
i
α
j
K
x
i
,x
j
−
α
i
K
x
i
,x
i
i
=1
j
=1
i
=1
m
s.t.
α
i
=1
(5.5)
i
=1
0(
i
=1
,...,m
)
where a constant
ν ∈
(0
,
1) is used to tune the number of support vectors and
outliers.
1
/mν
≥
α
i
≥
Anewdatapoint
x
is novel if:
SV
SV
SV
R
2
f
(
x
)=
K
(
x, x
)
−
2
α
i
K
x, x
i
+
α
i
α
j
K
x
i
,x
j
−
≥
0
i
=1
i
=1
j
=1
where
R
2
is computed by a training datapoint which is non bound and setting
decision function
f(x)
to zero.
5.3.3
Support Vector Regression (SVR)
SVM can also be applied to regression problem by the introduction of an alterna-
tive loss function. By using an
insensitive
loss function proposed by Vapnik,
Support vector regression (SVR) aims to find a predictive function
f(x)
that
has at most
deviation from the actual value
y
i
. This task is also accomplished
through the quadratic program (6):
−
m
m
m
m
(
α
i
− α
i
∗
)(
α
j
− α
j
∗
)
Kx
i
,x
j
−
(
α
i
− α
i
∗
)
y
i
+
(
α
i
+
α
i
∗
)
min (1
/
2)
i
=1
j
=1
i
=1
i
=1
