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where
x
t
=hydrologic variable cons
ti
tuting the sequence in time,
n
= total
number of hydrologic records, and
x
= average of
x
t
.
Under the null hypothesis of constant mean, i.e., homogenous time series,
the expected value of the von Neumann ratio is 2. However, it tends to be < 2
for the non-homogenous time series. The values of von Neumann ratio for
normally distributed samples can be found in Owen (1962).
4.1.2 Cumulative Deviations Test
Tests for homogeneity are based on the adjusted partial sums or cumulative
deviations from the mean, which are expressed as (Buishand, 1982):
k
Ç
xx
,
k
S
=
k
= 1, 2, ......,
n
(2)
t
t1
**
k
*
Rescaled adjusted partial sums (
S
) are obtained by dividing
S
k
's
by
the sample standard deviation (
D
x
).
*
k
*
k
,
=
k
= 1, 2, ......,
n
(3)
S
SD
n
1
Ç
2
D
2
x
=
xx
with
(4)
t
n
t1
The values of
*
S
's are not dependent on the unit of the hydrologic
variable, and hence homogeneity tests are based on the rescaled adjusted
partial sums.
Sensitivity to the departures from homogeneity is defined by the following
statistic:
*
k
Ma
kn
S
Q
=
(5)
0
High values of
Q
are an indication for non-homogeneity in the time
series. Another statistic which can be used for testing homogeneity is the
range (
R
). It is defined as:
**
**
Max
S
Mi
kn
S
R
=
(6)
k
k
0
kn
0
Critical values of
Q
for some specified values of
n
are given by Buishand
(1982), which are based on the 19,999 synthetic sequences of Gaussian random
numbers. For
n
the critical values of
Q
can be obtained from the
Kolmogorov-Smirnov goodness-of-fit statistic table (Doob, 1949). The critical
values of the distribution of
R
under the null hypothesis are given by Wallis
and O'Connell (1973) in a graphical form, while Buishand (1982) presents
salient critical values in a tabular form.
,
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