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used this test and Kendall (1975) subsequently derived the test-statistic
distribution. This test has been found to be an excellent tool for trend detection
(e.g., Hirsch et al., 1982; Gan, 1992). Considering the time series
x
t
(
t
= 1, 2,
…,
n
), each value of the series (
x
t
) is compared with all subsequent values
(
x
t+1
)
and a new series
z
k
is generated as follows (Salas, 1993):
z
1f r
0f r
1for
x
!
x
k
t
t
'
z
x
x
k
t
t
'
(63)
z
x
x
k
t
t
'
where
k
is given by:
k
=
t
1(2
n
t
)/2
t
t
(64)
The Mann-Kendall statistic (
S
) is defined as follows (Hirsch et al., 1982):
n
1
n
ÇÇ
S
=
z
(65)
k
t
1
t
t
1
Thus, this statistic represents the number of positive differences minus
the number of negative differences for all the differences under consideration.
Moreover, the above test-statistic for
n
> 40 may be written as (Hirsch et
al., 1982):
Sm
VS
u
c
=
(66)
()
g
Ë
Û
1
Ç
nn
12 5
n
e e
12
e
5
where
V
(
S
) =
Ì
Ü
(67)
i
i
i
18
Ì
Ü
Í
Ý
i
1
In Eqns (66) and (67),
m
= 1 for
S
< 0 and
m
= -1 for
S
> 0,
g
is the
number of tied groups, and
e
i
is the number of data in the
i
th
tied group. The
value of the test-statistic
u
c
is taken as zero for
S
= 0. Now, if the computed
absolute value of
u
c
is greater than the critical value of the standard normal
distribution, the hypothesis of an upward or downward trend cannot be rejected
at the D significance level. It should be noted that Kendall (1975) suggested
for using the Mann-Kendall test even for
n
values as low as 10 provided there
are not too many tied values. Hirsch et al. (1982) reported the application of
this test to seasonal time series.
4.3.14 Sen's Slope Estimation Test
Kendall slope (E), initially given by Sen (1968) and later extended by Hirsch
et al. (1982), is a useful index to quantify monotone trend in the hydrologic
time series (Hirsch et al., 1982; Gan, 1998). Sen's test for the estimation of
slope requires a time series of equally spaced data. The slope is estimated by:
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