<s>
Least-squares	B-Algorithm
support-vector	I-Algorithm
machines	I-Algorithm
(	O
LS-SVM	O
)	O
for	O
statistics	O
and	O
in	O
statistical	O
modeling	O
,	O
are	O
least-squares	B-Algorithm
versions	O
of	O
support-vector	B-Algorithm
machines	I-Algorithm
(	O
SVM	B-Algorithm
)	O
,	O
which	O
are	O
a	O
set	O
of	O
related	O
supervised	B-General_Concept
learning	I-General_Concept
methods	O
that	O
analyze	O
data	O
and	O
recognize	O
patterns	O
,	O
and	O
which	O
are	O
used	O
for	O
classification	B-General_Concept
and	O
regression	O
analysis	O
.	O
</s>
<s>
In	O
this	O
version	O
one	O
finds	O
the	O
solution	O
by	O
solving	O
a	O
set	O
of	O
linear	O
equations	O
instead	O
of	O
a	O
convex	O
quadratic	B-Algorithm
programming	I-Algorithm
(	O
QP	O
)	O
problem	O
for	O
classical	O
SVMs	B-Algorithm
.	O
</s>
<s>
Least-squares	B-Algorithm
SVM	B-Algorithm
classifiers	B-General_Concept
were	O
proposed	O
by	O
Johan	O
Suykens	O
and	O
Joos	O
Vandewalle	O
.	O
</s>
<s>
LS-SVMs	O
are	O
a	O
class	O
of	O
kernel-based	B-Algorithm
learning	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Given	O
a	O
training	O
set	O
with	O
input	O
data	O
and	O
corresponding	O
binary	O
class	O
labels	O
,	O
the	O
SVM	B-Algorithm
classifier	B-Algorithm
,	O
according	O
to	O
Vapnik	O
's	O
original	O
formulation	O
,	O
satisfies	O
the	O
following	O
conditions	O
:	O
</s>
<s>
According	O
to	O
the	O
structural	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
principle	O
,	O
the	O
risk	O
bound	O
is	O
minimized	O
by	O
the	O
following	O
minimization	O
problem	O
:	O
</s>
<s>
By	O
substituting	O
by	O
its	O
expression	O
in	O
the	O
Lagrangian	O
formed	O
from	O
the	O
appropriate	O
objective	O
and	O
constraints	O
,	O
we	O
will	O
get	O
the	O
following	O
quadratic	B-Algorithm
programming	I-Algorithm
problem	O
:	O
</s>
<s>
Solving	O
this	O
QP	O
problem	O
subject	O
to	O
constraints	O
in	O
(	O
8	O
)	O
,	O
we	O
will	O
get	O
the	O
hyperplane	O
in	O
the	O
high-dimensional	O
space	O
and	O
hence	O
the	O
classifier	B-Algorithm
in	O
the	O
original	O
space	O
.	O
</s>
<s>
The	O
least-squares	B-Algorithm
SVM	B-Algorithm
(	O
LS-SVM	O
)	O
classifier	B-Algorithm
formulation	O
above	O
implicitly	O
corresponds	O
to	O
a	O
regression	O
interpretation	O
with	O
binary	O
targets	O
.	O
</s>
<s>
with	O
Notice	O
,	O
that	O
this	O
error	O
would	O
also	O
make	O
sense	O
for	O
least-squares	B-Algorithm
data	O
fitting	O
,	O
so	O
that	O
the	O
same	O
end	O
results	O
holds	O
for	O
the	O
regression	O
case	O
.	O
</s>
<s>
We	O
use	O
both	O
and	O
as	O
parameters	O
in	O
order	O
to	O
provide	O
a	O
Bayesian	O
interpretation	O
to	O
LS-SVM	O
.	O
</s>
<s>
The	O
solution	O
of	O
LS-SVM	O
regressor	O
will	O
be	O
obtained	O
after	O
we	O
construct	O
the	O
Lagrangian	O
function	O
:	O
</s>
<s>
Elimination	O
of	O
and	O
will	O
yield	O
a	O
linear	O
system	O
instead	O
of	O
a	O
quadratic	B-Algorithm
programming	I-Algorithm
problem	O
:	O
</s>
<s>
Here	O
,	O
is	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
is	O
the	O
kernel	O
matrix	O
defined	O
by	O
.	O
</s>
<s>
Radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
RBF	O
kernel	O
:	O
</s>
<s>
This	O
scaling	O
is	O
related	O
to	O
the	O
bandwidth	O
of	O
the	O
kernel	O
in	O
statistics	O
,	O
where	O
it	O
is	O
shown	O
that	O
the	O
bandwidth	O
is	O
an	O
important	O
parameter	O
of	O
the	O
generalization	O
behavior	O
of	O
a	O
kernel	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
Bayesian	O
interpretation	O
of	O
the	O
SVM	B-Algorithm
has	O
been	O
proposed	O
by	O
Smola	O
et	O
al	O
.	O
</s>
<s>
They	O
showed	O
that	O
the	O
use	O
of	O
different	O
kernels	O
in	O
SVM	B-Algorithm
can	O
be	O
regarded	O
as	O
defining	O
different	O
prior	O
probability	O
distributions	O
on	O
the	O
functional	O
space	O
,	O
as	O
.	O
</s>
<s>
A	O
general	O
Bayesian	O
evidence	O
framework	O
was	O
developed	O
by	O
MacKay	O
,	O
and	O
MacKay	O
has	O
used	O
it	O
to	O
the	O
problem	O
of	O
regression	O
,	O
forward	O
neural	B-Architecture
network	I-Architecture
and	O
classification	B-General_Concept
network	O
.	O
</s>
<s>
Kwok	O
used	O
the	O
Bayesian	O
evidence	O
framework	O
to	O
interpret	O
the	O
formulation	O
of	O
SVM	B-Algorithm
and	O
model	O
selection	O
.	O
</s>
