Digital Signal Processing Reference
In-Depth Information
important properties are highlighted below. A fuller discussion of properties
is given in References 4, 6 through 8, and 10.
2.3.3.1 Superordination Property
The fuzzy SR matrix is a superset of the crisp SR matrix, containing information on
the spread of the samples in addition to the crisp SR information.
This property arises from the fact that a fuzzy relation can be mapped to its
crisp counterpart through thresholding:
R
i,
(
j
)
=
R
i,
(
j
)
)
T
1
(
, where
T
δ
(
a
)
is the
thresholding operation that yields 1 if
a
≥
δ
and 0 otherwise. Extending this
R
element-wise to the fuzzy SR matrix yields
R
, showing that the fuzzy
SR matrix can be reduced to it crisp counterpart by thresholding.* In case of
a row (column) normalized fuzzy SR matrix, elements are thresholded with
δ
=
T
1
(
)
set to the maximum row (column) element.
2.3.3.2 Reduction Property
As the membership function spread decreases, fuzzy relations, ranks, and order statis-
tics reduce to their crisp counterparts. Conversely, as the membership function spread
increases to infinity, all fuzzy relations become equal, fuzzy ranks converge to the me-
dian, and fuzzy order statistics converge to the sample mean
.
This property describes the limiting behavior, with respect to membership
function spread, of the fuzzy SR matrix. It is easy to see lim
σ
→
0
R
=
R
. Con-
versely,
indicates infinite spread of the membership function and the
relation between all samples is equal, i.e.,
R
i,
(
j
)
=
σ
→∞
1 for
i, j
=
1
,
2
,
...
,N
. For
R
i,
(
j
)
=
row (column) normalized fuzzy matrix, lim
σ
→∞
1
/
N
. From this, it is
=
=
(
+
)/
2. Also, lim
σ
→
0
x
(
j
)
=
easy to see that lim
σ
→
0
r
r
and lim
σ
→∞
r
i
N
1
N
l
x
(
j
)
and lim
σ
→∞
x
(
j
)
=
x
=
1
/
x
l
,
i, j
=
1
,
2
,
...
,N
.
2.3.3.3 Spread-Sensitivity Property
The intersample spacing is captured in the fuzzy SR matrix.
If the membership function covers the range [0
,x
(
N
)
−
x
(
1
)
], and is continuous
and continuously decreasing (as a function of
|
x
i
−
x
j
|
), then the sample spread
R
. That is,
|=
µ
−
1
R
R
i,
(
r
j
)
)
can be exactly determined from
. Thus, the
observation vector can be recovered from
R
to within a constant. It is easy to
see that the observation vector can be also be recovered from the normalized
SR matrix.
This property explains why the fuzzy SR matrix captures sample spread
information. To illustrate, consider two strictly increasing sequences, one lin-
ear, the other exponential. While their corresponding crisp SR matrices are
diagonal identity matrices, concealing the sample spacing, the fuzzy SR ma-
trices reflect the sample spread. This is illustrated in Figure 2.6 for a sequence
|
x
i
−
x
j
(
* This assumes all observation values are unique. For the case of equally valued samples, stable
sorting is utilized to assign each sample a unique crisp rank index and this rank can be extracted
from the fuzzy SR matrix by simply preserving the original time order of equally valued samples.
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