Civil Engineering Reference
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linearly related in a sample. The population correlation coefficient,
ρ
, is defined for
two variables
x
and
y
by the formula:
(
,
)
ρ
=
σ
Cov
X
Y
Y
σ
X
σ
Y
=
X
,
,
σ
X
σ
Y
where
(
x
i
,
y
i
)
is the
i
th pair observation value,
i
=
1
,
2
,
3
,...,
n
.
x
,
y
are sample mean
for
x
and
y
, respectively.
In this case, the more positive
ρ
is, the more positive the association is. This also
indicates that when
is close to 1, an individual with a high value for one variable
will likely have a high value for the other and an individual with a lower value for
one variable will likely to have a low value for the other. On the other hand, the
more negative
ρ
is, the more negative the association is, this also indicates that an
individual with a high value for one variable will likely have a low value for the
other when
ρ
is close to 0, this means there
is little linear association between two variables. In order to obtain the correlation
coefficient, we need to obtain
ρ
is close to
−
1 and conversely. When
ρ
y
, and the covariance of
x
and
y
. In practice, these
parameters for the population are unknown or difficult to obtain. Thus, we usually
use
r
xy
, which can be obtained from a sample, to estimate the unknown population
parameter. The sample correlation coefficient
r
xy
is expressed as:
x
,
σ
σ
n
i
(
−
)(
−
)
Σ
x
i
x
y
i
y
=
1
r
xy
=
Σ
2
2
.
(1)
n
i
n
i
(
−
)
(
−
)
x
i
x
Σ
y
i
y
=
1
=
1
Pearson correlation coefficient is a straightforward approach to calculate the
relationship between two variables. However, if the variables considered are not real
numbers, but fuzzy data, the formula above is problematic. For example, Mr. Smith
who is a new graduate from college expected salary ranges from [45,000, 50,000]
and his expected working hours are [8, 10]. If we collect this kind of data from
many new graduates, then the correlation between expected salary and working
hours cannot be calculated by us from these data. Suppose IX is the expected salary
for each new graduate and IY is the working hours they desired, then the scatter plot
for these two sets of fuzzy interval numbers would approximate that shown in Fig.
1
.
For the interval-valued fuzzy number, we consider to pick out samples from
population
X
and
Y
. Each fuzzy interval data for the centroids and length of the
sample
X
and sample
Y
will be considered to calculate the correlation coefficient.
In addition, we also employ the maximum value and minimum value of fuzzy
interval data to evaluate the correlation coefficient.
In this paper, there are two kinds of fuzzy correlation which are based on the
Person's correlation as well as the extension principle definitions 1 and 2; the
advantages are that we can compute various samples with fuzzy interval type for
the continuous sample.
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