Geoscience Reference
In-Depth Information
For 5
n
40, critical values of the test-statistic can be obtained from
Kanji (2001). For
n
> 40, '
K
' may be assumed to follow a normal distribution
with the mean (P
k
) and variance (V
2
k
) given as:
21
3
n
P
k
=
(53)
16
n
29
V
2
k
=
(54)
90
Now, the test-statistic (
z
) for the case when
n
> 40 is computed by the
following expression:
K
P
V
0.5
k
k
z
=
(55)
Critical values of '
z
' can be obtained from the standard normal distribution
tables available in statistics topics. For both the cases of time series sizes, if
the computed test-statistic values are less than its critical values, the null
hypothesis cannot be rejected. That is, the time series is considered to be
random.
4.3.10 Wilcoxon-Mann-Whitney Rank Sum Test
This is a nonparametric test (i.e., distribution-free), which is applicable if the
observations are random and independent. It is used to examine whether the
occurrence of increasing or decreasing successive values of a time series is
random. Consider a time series
x
t
(
t
= 1, 2, …,
n
). First, successive observations
in the sequence are coded with a '+' or '-' sign by comparing two successive
values, and the ranks (1, 2, 3, …,
n
) are assigned to all the observations of the
series. Thereafter, the number of '+' and '-' signs are counted and the larger
of two numbers is noted down (say
n
1
). If
n
2
be the number of opposite signs,
then
n
=
n
1
+
n
2
. Now, from the integers describing the natural order of signs,
the rank sum '
R
1
' of '
n
2
' signs is determined. Finally, the value of '
R
2
' statistic
is calculated by the following expression:
R
2
=
nn
1
R
(56)
2
1
The smaller of '
R
1
' and '
R
2
' is used as the test-statistic. The critical values
of the test-statistic can be found in Natrella (1963). If the computed value of
the test-statistic is greater than its critical value, the null hypothesis of no
trend is rejected.
4.3.11 Inversions Test
This test is used to examine the linear trend in a time series. It is almost
similar to the Kendall's rank correlation test (described ahead). The number of
x
j
-values (
j
>
i
) each smaller than a chosen
x
i
-value is counted for all
i
's
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