Digital Signal Processing Reference
In-Depth Information
Tabl e 7 . 1
Membership functions and interference functions for the FALVQ1, FALVQ2, and
FALVQ3 families of algorithms
Algorithm
u(z)
w(z)
n(z)
FA LVQ 1 (0
<α<∞
)
z
(1 +
αz
)
−1
(1 +
αz
)
−2
αz
2
(1 +
αz
)
−2
FA LVQ 2 (0
<β<∞
z
exp (
−βz
)
(1
− βz
)exp(
−βz
)
βz
2
exp (
−βz
)
γz
2
FA LVQ 3 (0
<γ<
1)
z
(1
− γz
)
1
−
2
γz
be determined based on minimizing the loss function.
The winning prototype
L
i
is adapted iteratively, based on the fol-
lowing rule:
⎛
⎞
c
η
∂D
x
∂
L
i
⎝
1+
⎠
,
Δ
L
i
=
−
=
η
(
x
−
L
i
)
w
ir
(7.83)
i=r
where
w
ir
=
u
||
=
w
||
.
L
i
||
2
L
i
||
2
x
−
x
−
(7.84)
L
r
||
2
L
r
||
2
||
x
−
||
x
−
The nonwinning prototype
L
j
=
L
i
is also adapted iteratively, based on
the following rule:
η
∂D
x
∂
L
j
Δ
L
j
=
−
=
η
(
x
−
L
j
)
n
ij
(7.85)
where
n
ij
=
n
||
=
u
ij
−
||
L
i
||
2
L
i
||
2
x
−
x
−
w
ij
||
x
−
L
j
||
2
||
x
−
L
j
||
2
It is very important to mention that the fuzzyness in FALVQ is employed
in the learning rate and update strategies, and is not used for creating
fuzzy outputs.
The above-presented mathematical framework forms the basis of the
three fuzzy learning vector quantization algorithms presented in [131].
Table 7.1 shows the membership functions and interference functions
w
(
·
)and
n
(
) that generated the three distinct fuzzy LVQ algorithms.
An algorithmic description of the FALVQ is given below.
·
1. Initialization:
Choose the number
c
of prototypes and a fixed learning
Search WWH ::
Custom Search