Environmental Engineering Reference
In-Depth Information
and
Z
is compared with the standard normal variate at
the desired significance level
α
. Positive values of
Z
indicate increasing trends, while negative values of
Z
indicate decreasing trends.
10.9.5.2 Sen's Slope Estimator.
If a linear trend is
present in a time series, the slope can be estimated by
using a simple nonparametric procedure developed by
Sen (1968). The slope estimates of
M
pairs of data are
first computed by
x
x
j k
−
−
EXAMPLE 10.22
j
k
Q
=
for
i
=
1,
,
…
M
(10.118)
i
over a period of 10 years, concentrations in the ground-
water downstream of a hazardous waste landfill are
measured annually, and the results are as follows:
where
x
j
and
x
k
are data values at times
j
and
k
(where
j
>
k
), respectively. The calculated values of
Q
i
are then
ranked and the median of these
M
values of
Q
i
is Sen's
estimator of slope given by
year,
i
c
i
(mg/l)
year,
i
c
i
(mg/l)
1
3.53
6
1.56
2
0.24
7
5.51
Q
M
odd
[(
M
+
1 2
)/
]
3
1.34
8
5.73
Q
=
(10.119)
1
2
med
(
)
Q
+
Q
M
even
4
2.26
9
5.78
[
M
/
2
]
[(
M
+
2 2
)/
]
5
4.56
10
6.97
The estimated slope of the data is
Q
med
. The 100(1 −
α
)
confidence interval of the estimated slope can be
determined using the following procedure (Gilbert,
1987):
use the Mann-Kendall test to assess whether there
is a trend in the data at the 5% significance level.
Solution
Step 1.
Choose the significance level,
α
, and deter-
mine the standard normal deviate
z
α
/2
.
Step 2.
Determine
σ
S
using either Equation (10.115)
(no ties) or Equation (10.116) (with ties). Calcu-
lated the parameter
C
α
where
From the measured data set,
N
= 10, and using the given
sequence of concentration measurements and the sign
convention given by Equation (10.113) yields
i
∑
10
sign
(
c
−
c
)
i
∑
10
sign
(
c
−
c
)
j
= 1
i
+
j
i
j
= 1
i
+
j
i
C
=
z
α
σ
α
/
2
S
1
1
6
4
2
8
7
3
Step 3.
Compute the ranks
M
1
and
M
2
where
3
7
8
2
4
4
9
1
1
2
5
3
(
)
M
=
M C
−
1
α
Based on these results, the statistics of the Mann-
Kendall test are:
1
2
(
)
α
M
=
M C
+
2
S
= + + + + + + + + =
1 8 7 4 3 4 3 2 1
33
where
M
is the number of slopes used in calculat-
ing the Sen estimate.
Step 4.
Determine the values of the ranked slopes,
Q
i
, corresponding to ranks
M
1
and
M
2
; these are
the lower and upper
α
confidence limits, respec-
tively, of the estimated slope.
1
18
1
18
σ
S
=
N N
[
−
1 2
][
N
+
5
]
=
(
10 10 1 2 10
)[
−
][ (
)
+
5
]
=
11.18
S
−
1
33 1
11 18
−
Z
=
=
=
2 862
.
σ
.
S
EXAMPLE 10.23
The CDF of the standard normal deviate shows that
Z
= 2.862 has an exceedance probability of 0.002 or
0.2%. Therefore, it can be asserted that the measured
concentration data has a significant positive (upward)
slope at a significance level of 5%.
using the concentration measurements given in
Example 10.22, use Sen's slope estimator to estimate the
slope of the time-series measurements and the 95%
confidence interval of the slope.
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