Environmental Engineering Reference
In-Depth Information
10
3
10
3
10
3
10
2
10
2
10
2
10
1
10
1
10
1
10
0
10
0
10
-1
10
-1
10
0
10
0
10
2
0
2
Y
1
=LI
4
0
2
Y
1
=LI
4
exp(Y
3
)=
S
t
Figure 1.30
Simulated multivariate dataset of (Y
1
, Y
2
, Y
3
) by forcing
δ
12
=
δ
13
=
δ
23
=
0.
are <1. For instance, the Pearson correlation between (Y
1
, Y
2
) is only 0.979. This is because
the Pearson correlation is a measure of “linear” correlation. Even if Y
1
is related to Y
2
in a
deterministic way, the Pearson correlation is <1 if this relationship is not linear.
It is worthwhile to point out that the Pearson correlation between (Y
1
, Y
2
) cannot exceed
0.979. The reason is that the correlation coefficient in nonnormal space is monotonically
related to the correlation coefficient in the normal space in the CDF transform approach
and it is not possible for the Pearson correlation coefficient between (X
1
, X
2
) to exceed 1.
One may rightfully wonder as an engineer if this limitation is inconsequential to practice.
It is rare for physical variables to produce correlation coefficients near to 1 in practice and,
in any case, a correlation coefficient of 0.979 is as good as 1 in the presence of statistical
uncertainty. Unfortunately, this theoretical limitation associated with the CDF transform
approach can be practically important as illustrated below.
First, we state without proof here that the exact relation between these correlation coef-
ficients is given by the integral equation below:
∞
∞
∫
∫
F
−
1
[( )]
Φ
x
×
F
−
1
[( )]
Φ
x
×
f
(, ;
xx
δ
)
d
xx
d
−×
µ
µ
1
1
2
2
1
2 2
1
2
Y
Y
(1.108)
1
2
−∞
−∞
δ
=
YY
σσ
Y
12
Y
1
2
10
3
10
3
10
3
Pearson correlation
= 0.979
Pearson correlation
= 0.988
Pearson correlation
= 0.998
10
2
10
2
10
2
10
1
10
1
10
1
10
0
10
0
10
-1
10
-1
10
0
0
2
Y
1
=LI
4
0
2
Y
1
=LI
4
10
0
10
2
exp(Y
3
)=
S
t
Figure 1.31
Simulated multivariate dataset of (Y
1
, Y
2
, Y
3
) by forcing
δ
12
=
δ
13
=
δ
23
=
1.
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