Environmental Engineering Reference
In-Depth Information
+∞
+∞
x
−
µ
x
−
µ
∫
∫
1
1
2
2
ρ
=
DF xFxf
(( ),
( ;) (()()
θ
xf xxx
12 2
dd
(2.10)
11 22
1
1
2
σ
σ
1
2
−∞
−∞
It is evident from
Equation 2.10
that the determination of copula parameters using
Pearson's rho depends on both the correlation coefficient and the marginal distributions.
For given marginal distributions
F
1
(
x
1
) and
F
2
(
x
2
) of
X
1
and
X
2
, and correlation coefficient
ρ between
X
1
and
X
2
, the preceding integral equation can be solved iteratively to obtain θ.
For a Gaussian copula,
Equation 2.10
i
is reduced to Equation 1.108 of
Chapter 1
. The obser-
vation that the absolute values of Pearson's rho between non-normal variables are smaller
than those between the transformed standard normal variables (see Figure 1.32 in
Chapter
1
) can be readily explained by
Equation 2.10
.
In practical application, Pearson's rho is often
estimated from the measured data. The sample version of Pearson's rho is expressed as (e.g.,
Mari and Kozt 2001)
(
)
(
)
∑
N
xxxx
−
−
1
if
1
2
if
2
if
=
1
ρ=
(2.11)
(
)
(
)
∑
N
2
∑
N
2
xx
−
xx
−
1
if
1
2
if
2
if
=
1
if
=
1
where (
x
1
if
,
x
2
if
) denotes a pair of
X
1
and
X
2
values;
N
is the sample size; and
x
1
and
x
2
are the sample means of
X
1
and
X
2
, respectively. It is noted that this equation is the same
as Equation 1.40 of
Chapter 1
. Since Pearson's rho is a measure of linear dependence, it
is valid only when the joint CDF is Gaussian. Therefore, it is not adequate for character-
izing a nonlinear dependence between two random variables. Furthermore, Pearson's rho
is invariant only under strictly monotonic linear transformations; thus, the dependence
measure needs to be evaluated for each nonlinear transformation. If random variables are
not jointly Gaussian or some nonlinear transformations are used, Pearson's rho is not an
effective dependence measure. Sometimes, it may produce misleading results (Tang et al.
2013a).
2.2.2.2 Kendall's tau
To remove the aforementioned limitations underlying Pearson's rho, rank correlation can be
used. The commonly used rank correlation includes Spearman rank correlation coefficient
and Kendall rank correlation coefficient. The former has been introduced in
Chapter 1
. In
this chapter, the latter is adopted to determine the copula parameters due to its intrinsic
relation to a copula function. As a rank correlation, Kendall's tau depends only on the ranks
underlying the sample for each variable of interest rather than the actual numerical values.
The idea of a “rank” is quite simple. Consider a list of 10 numbers, containing realizations
of a random variable
X
1
. The smallest number is assigned a rank = 1, the second smallest
number is assigned a rank = 2, and so forth. The list of 10 numbers (
x
1
if
) is replaced by a
list of ranks (
r
1
if
), which is a permutation of integers from 1 to 10. Numerical examples to
illustrate a rank are referred to Equations 1.43 and 1.44 in
Chapter 1
.
Unlike Pearson's rho, Kendall's tau measures the degree of concordance between
X
1
and
X
2
. The concept of concordance is straightforward:
X
1
and
X
2
are concordant if “large”
values of
X
1
tend to be with “large” values of
X
2
or “small” values of
X
1
tend to be with
“small” values of
X
2
. Let (
X
1
′,
X
2
′) be an independent copy of (
X
1
,
X
2
). Then, a formulation
of concordance is as follows: (
X
1
,
X
2
) and (
X
1
′,
X
2
′) are concordant if (
X
1
-
X
1
′)(
X
2
-
X
2
′) > 0
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