Information Technology Reference
In-Depth Information
8.5 Kernel Function
Using different functions for convolution of the inner products kernel function,
one can construct learning machines with different types of nonlinear decision
surfaces in input space. At present, prevalent kernel functions are polynomial
kernel function, radial basis function, multi-layer perceptron and dynamic kernel
function.
8.5.1
Polynomial kernel function
Polynomial kernel function:
d
K
(
x
,
x
)
=
[(
x
,
x
)
+
1
(8.38)
i
i
We construct a d-dimensional polynomials decision function of the form
Ä
Ô
Ã
vector
Å
d
Õ
f
(
x
,
α
)
=
sign
y
α
[
x
•
x
)
+
1
−
b
i
i
i
Æ
Ö
sup
port
8.5.2
Radial Basis Function
Classical radial basis function (RBF) machines use the following set of decision
rules:
Ä
l
Ô
Ã
=
f
(
x
)
=
sign
Å
Æ
α
K
(
x
−
x
)
−
b
Õ
Ö
(8.39)
i
γ
i
i
1
K
γ
(
x
−
x
)
x
−
x
where
depends on the distance
between two vectors. For
i
i
γ −
is a nonnegative monotonic function. It
tends to zero as training sample's total goes to infinity. The most popular
function of this type is
K
(
x
x
)
any fixed γ, the function
i
Ê
2
Ú
x
−
x
Ë
Û
(
)
i
K
x
−
x
=
exp
−
(8.40)
γ
i
2
σ
Ì
Ü
To construct the decision rule (8.39) one has to estimate
(1) the value of the parameter γ,
(2) the number
N
of the centers
x
i
,
x
i
, describing the centers,
(4) the value of the parameters α
i
.
(3) the vectors
