Environmental Engineering Reference
In-Depth Information
TABLE 10.7. Values of Shapiro-Francia Statistic,
W
′
tions rather than their actual values, and has the desir-
able properties that it is not affected by the actual
distribution of the data, is less sensitive to outliers, and
missing values are allowed. In contrast, parametric trend
tests, although more powerful, require the data to be
normally distributed and are more sensitive
Confidence level,
α
Sample Size,
N
0.01
0.05
50
0.935
0.953
to
55
0.940
0.958
outliers.
The Mann-Kendall trend test is based on the correla-
tion between the ranks of a time series and their time
order. For a time series
X
= {
x
1
,
x
2
, . . . ,
x
N
}, the test
statistic is given by
65
0.948
0.965
75
0.956
0.969
85
0.961
0.972
95
0.965
0.974
Source of data
: Shapiro and Francia (1972).
TABLE 10.8. Data Transformations
−
∑
sign(
N
1
N
S
=
x
−
x
)
(10.112)
j
i
Arithmetic
y
=
x
+
c
i
=
1
j
= +
i
1
reciprocal
y
=
x
−1
logarithm
y
= log
x
or
y
= ln
x
where
1
2
Square root
y
=
x
1
3
+
1
0
1
x
<
=
x
i
j
Cube root
y
=
x
sign
(
x
−
x
)
=
sign
(
R R
−
)
=
x
x
(10.113)
j
i
j
i
i
j
−
x
>
x
i
j
10.9.4.3 Data Transformations to Achieve Normal-
ity.
Since many statistical tests and analysis methods
are based on the assumption that the underlying popu-
lation distribution has a normal distribution, in cases
where the data is not normally distributed, it might be
possible to transform the data such that it fits a normal
distribution. under these circumstances, statistical anal-
yses are performed on the transformed data and conclu-
sions are drawn regarding the transformed data, and
then these conclusions are related to the statistical prop-
erties of the untransformed data. A list of common data
transformations are given in Table 10.8, where the trans-
formed value is denoted by
y
and the sample value is
denoted by
x
.
Confidence intervals that are symmetric on the trans-
formed scale will not be symmetric when transformed
back to the original scale.
and
R
i
and
R
j
are the ranks of observations
x
i
and
x
j
of
the time series, respectively. under the assumption that
the data are independent and identically distributed, the
mean and variance of the
S
statistic in Equation (10.112)
are given by
µ
S
= 0
(10.114)
1
18
σ
S
2
=
N N
(
−
1 2
)(
N
+
5
)
(10.115)
where
N
is the number of observations. The existence
of tied ranks (equal observations) in the data results in
a reduction of the variance of
S
to become
m
1
18
1
18
∑
ˆ
σ
S
2
=
N N
(
−
1 2
)(
N
+
5
)
−
t
(
t
−
1 2
)(
t
+
5
)
j
j
j
10.9.5 Trends
j
=
1
(10.116)
In many cases, an assessment of whether there is a sta-
tistically significant trend in the data with time is to be
determined. The purpose of trend analysis might be to
assess whether there is increased pollution resulting
from changing land use practices or to determine if
levels of pollution have declined following the initiation
of pollution control programs. Commonly used statisti-
cal tests for trend analysis are given below.
where
m
is the number of groups of tied ranks, each with
t
j
tied observations. The distribution of
S
tends to nor-
mality as the number of observations becomes large
(
N
> 10). The significance of trends are tested using the
standardized variable,
Z
, given by
S
−
1
S
>
0
σ
S
10.9.5.1 Mann-Kendall Test.
The Mann-Kendall
trend test (Kendall, 1975; Mann, 1945) is one of the most
widely used nonparametric tests to detect significant
trends in time series. This test uses the ranks of observa-
(10.117)
Z
=
0
S
=
0
S
+
1
S
<
0
σ
S
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