Digital Signal Processing Reference
In-Depth Information
2.3.3.7 Clustering Property
Samples from a finite number of populations yield fuzzy samples (fuzzy ranks)
that are averages of the observation samples (crisp ranks) within the individual
populations.
Let the samples be from
L
populations with
µ(
x
i
,x
k
)
=
1 for
i, k
∈
I
l
,
l
µ(
x
i
,x
k
)
=
0 for
i
∈
I
l
,
k
∈
I
m
,
l
=
m
and
∪
1
I
l
={
1
,
2
,
...
,N
}
. Also, let
=
I
p
i
. In this case, the fuzzy relation
between samples in different populations is 0 and the relation between sam-
ples in the same population
I
l
is, in the normalized case, 1
p
i
be the population index of
x
i
, i.e.,
x
i
∈
I
l
denotes the number of samples in
I
l
. This results in the fuzzy time and rank
samples
/
I
l
, where
1
x
i
=
x
r
i
=
x
k
(2.32)
I
p
i
k
∈
I
p
i
and
1
r
i
=
r
k
.
(2.33)
I
p
i
k
∈
I
p
i
Note that the fuzzy rank indices are obtained by averaging crisp ranks,
which, in the case of equally valued samples, form an integer subsequence,
r
min
,r
min
,r
max
, where
r
min
and
r
max
are the minimum and maximum
rank of samples in a given population. Averaging such a subsequence results
in a value that is always a multiple of
+
1
,
...
1
2
. Thus, fuzzy ranks are typically such
1
,
1
2
,
2
,
2
2
,
that
r
i
∈{
.
Relaxing the membership relations between samples in the above property,
we see that the fuzzy relations introduce averaging among similarly valued
samples, where similarity is determined by the membership function shape
and spread.
...
,N
}
2.4
Fuzzy Filter Definitions
Having established the concepts of affinity and fuzzy SR orderings, these
concepts can now be adopted into filtering structures. One method for ac-
complishing the inclusion is through the simple modification of established
filtering algorithms. Thus, existing algorithms that have proved useful can
be updated to include affinity or fuzzy ordering information. The additional
degrees of freedom introduced by fuzzy methods, and thus the consideration
of sample spread by the filtering algorithm, lead to improved performance.
The fuzzy generalization of two broad classes is considered here. First, the
general class of affine filters is established.
11
Affine filters are realized as a
simple extension of the broad class of weighted sum filters in which affinity,
Search WWH ::
Custom Search