Environmental Engineering Reference
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and discordant if ( X 1 - X 1 ′)( X 2 - X 2 ′) < 0. In mathematics, Kendall's tau is defined as the
probability of concordance minus the probability of discordance between X 1 and X 2 (e.g.,
Nelsen 2006)
τ=
PX
[(
XX
)(
X
′ >− −
)
0
]
P XXXX
[(
)(
′ <
)
0
]
(2.12)
1
1
2
2
1
1
2
2
In Equation 2.12 , the first term on the right-hand side is the probability of concordance
and the second term is the probability of discordance. Like Pearson's rho, Kendall's tau
also produces correlation coefficients between −1 and 1. However, Kendall's tau does not
assume that the relationship between two random variables is linear. Thus, it is invariant
with respect to strictly monotonic linear and nonlinear transformations, which allows for
a unique dependence measure for all transformed variables. Kendall's tau can be further
expressed in terms of a copula function C ( u 1 , u 2 ; θ) as (e.g., Nelsen 2006)
1
1
4
(2.13)
τ
=
Cu uCuu
(, ;) (, ;)
θ
d
θ
1
12
12
0
0
The derivation of Equation 2.13 from Equation 2.12 can be found in Nelsen (2006). It is
clear from Equation 2.13 that the determination of copula parameters using Kendall's tau
depends only on the correlation coefficient. It is independent of the marginal distributions.
This is a critical difference between Pearson's rho and Kendall's tau as demonstrated by Li
et al. (2012a). For a given correlation coefficient τ between X 1 and X 2 , the preceding integral
equation can be solved iteratively to find θ. The sample version of Kendall's tau is given by
sign[(
xxxx
)(
)]
1
i
1
j
2
i
2
j
(2.14)
ij
τ=
05
.
NN
(
1
)
where N is the sample size; sign(.) is calculated by
1
(
xxxx
xxx
)(
)
0
(concordant)
1
i
1
j
2
i
2
j
sign
=
ij
, = 1, 2,
,
N
(2.15)
1
(
)(
x
)
<
0
(discordant)
1
i
1
j
2
i
2
j
The numerator in Equation 2.14 is the difference between the number of concordant pairs
and the number of discordant pairs, whereas the denominator denotes the total number of
observation pairs for a sample size of N . Consider the ( X 1 , X 2 ) samples in Equation 1.43
of Chapter 1 . The total number of pairs for N = 5 is 0.5 × 5 × (5 -1) = 10. The number of
concordant pairs is 9 and 1 is for the number of discordant pairs. Thus, Kendall's tau is
computed as τ = (9 -1)/10 = 0.8. This value can also be obtained using MATLAB ® function
corr( X 1 , X 2 , 'type', 'Kendall').
Generally, Kendall's tau is not the same as Pearson's rho. For the ( X , Y ) data points in
Table 1.6 of Chapter 1 , the Pearson linear correlation coefficient is computed as ρ = 0.906
using corr( X , Y , 'type', 'Pearson'), whereas the Kendall rank correlation coefficient τ = 1 is
obtained by the MATLAB function corr( X , Y , 'type', 'Kendall'). They are not the same; this
is because Pearson's rho measures the degree of linear dependence between X 1 and X 2 and,
in this example, X 1 and X 2 are nonlinear. On the contrary, Kendall's tau measures the con-
cordance between X 1 and X 2 . It is clear that the relationship Y = X 3 is perfectly concordant
and, thus, a Kendall rank correlation coefficient of 1 is expected.
 
 
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