Civil Engineering Reference
In-Depth Information
[
,
]
=
−
=
+
Then
fuzzy correlation
is
r
low
r
up
with
r
low
r
s
r
and
r
up
r
s
r
,where
j
2
j
2
2
Σ
Σ
k
=
1
r
jk
4
Σ
Σ
k
=
1
(
r
jk
−
r
)
=
1
=
1
=
=
.
r
and
s
r
4
1 and 1 which measures
the degree to which two variables are linearly related. If there is perfect linear
relationship with positive slope between the two variables, we have a correlation
coefficient of 1; if there is positive correlation, whenever one variable has a high
value. Thus, based on the measure of evaluation, the degree of the population
correlation coefficient, we will be considered for the correlation of fuzzy interval.
As the correlation of fuzzy interval,
A correlation coefficient is a number between
−
, is computed, then the value of fuzzy
correlation can be evaluated that is defined as:
[
r
low
,
r
up
]
1.
When
[
r
low
,
r
up
]
∈
[
−
0
.
10
,
0
.
10
]
,
the fuzzy correlation is not significant
.
2.
When
[
r
low
,
r
up
]
∈
[
−
0
.
39
,−
0
.
11
]
or
[
0
.
11
,
0
.
39
]
,
the fuzzy correlation is low
.
3.
When
value
[
r
low
,
r
up
]
∈
[
−
0
.
69
,−
0
.
40
]
or
[
0
.
40
,
0
.
69
]
,
the fuzzy correlation is middle
value
.
4.
When
[
r
low
,
r
up
]
∈
[
−
0
.
99
,−
0
.
70
]
or
[
0
.
70
,
0
.
99
]
,
the fuzzy correlation is high
value
.
3
Simulation Studies
In this section, we will employ the Mote Carlo simulation to generate several
sequence of fuzzy interval data set and then compare their correlations coefficient
with different definition as proposed in Sect.
2
. The distribution for the centroid
and area is generated by the normal, uniform, gamma, and Cauchy distribution,
respectively. The procedure to computate correlation coefficient is described below:
Tab le
1
illustrates the result.
Step 1. Generate fuzzy set of sequence
X
with successive 4 points and error term
from the underlying distribution.
Step 2. Let
Y
=
aX
+
e
calculate the fuzzy data set
Y
by the fuzzy data set
X
and
error term.
Step 3. Find the correlation coefficient from the fuzzy data set by the above
definitions.
In Table
1
, there are some results that will be described as follows: (1) when
a
5,
the interval of correlation coefficient is close except the distribution of Cauchy, (3)
when a
=
0
.
2, the interval of the correlation coefficient is very close, (2) when a
=
0
.
8, the estimated interval from definition 3 is bigger than definition 4
did if the center distributions come from gamma, normal, and uniform. While if the
distribution comes from Cauchy distribution, we will get a very odd estimation.
=
0
.
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