Geoscience Reference
In-Depth Information
1988) recommends the SROC test for detecting trend in flow volumes. The
procedures for applying the SROC test are given below (McGhee, 1985).
Let the data series
x
t
(
t
= 1, 2, …,
n
) be observed in time
t
.
Step 1: Assign ranks
R
xt
to
x
t
such that the largest
x
t
has
R
xt
= 1 and the least
x
t
has a rank =
n
. If there are ties in the
x
t
, then assign to each of the
ties a rank equal to the mean of the ranks that would have been used
had there been no ties.
Step 2: Compute the difference,
d
t
=
R
xt
-
t
(34)
Step 3: Compute the coefficient of trend (
r
s
) using the following expression:
n
Ç
t
6
d
t
1
1
r
s
=
(35)
2
nn
1
Under the null hypothesis (
H
0
) that the time series has no trend, it can
be shown that the statistic,
ts
has a Student's
t
-distribution with
n
-2
degrees of freedom. Here,
ts
is defined as:
n
2
r
ts
=
(36)
s
s
1
r
Step 4: Calculate the value of
ts
from Eqn. (36) and get the critical value of
the
t
-distribution for the chosen significance level, D and
n
-2 degrees
of freedom. For a two-tailed test, denote the critical value by
Step 5: Finally, compare the computed value of '
ts
' with its critical value.
Reject null hypothesis (
H
0
) if
ts
>
ts
D
/2,
n
2
ts
D
or
ts < -
ts
D
2
.
/2,
n
2
/2,
n
4.3.3 Turning Point Test
Let's assume that a turning point occurs in the series
x
t
(
t
= 1, 2, …,
n
) at any
time
t
(
t
= 2, 3, …,
n
-1) if
x
t
is larger than each of
x
t-1
and
x
t+1
or
x
t
is smaller
than
x
t-1
and
x
t+1
. This situation has four chances of occurrence in six different
possibilities of the occurrence of
x
t-1
,
x
t
and
x
t+1
, assuming that all three
elements have different values. Accordingly, the chance of having a turning
point in a sequence of three values is 4/6 or 2/3, for all the values of '
t
' except
for
t
= 1 and
t
=
n
. In other words, the expected number of turning points ( ¯)
in the given random series can be expressed as:
¯ =
(37)
22/3
n
For the same random series, variance is given by (Kendall, 1973):
var (
p
¯) =
16
n
29 /90
(38)
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