Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 63

The test-statistic is represented by the standard normal variate ( z ), and is
given as:

pp
p

z = (39)

var( )

where p is observed number of turning points.
The computed standard normal variate is then compared with the standard
normal variate obtained from the standard table at a given level of significance.
If the calculated value of ' z ' is within the region of acceptance, the hypothesis
of no trend is accepted. If a trend is detected, it can be removed by regression
technique (i.e., fitting a suitable equation).
The turning point test is easy to apply, especially when the time series is
plotted graphically. It is an effective test for checking randomness against
systematic oscillation. But if the turning points tend to bunch together, the
Kendall's phase test is more suitable. However, the difficulty with this test is
that a comparison of observed and theoretical numbers of phases based on the
chi-square criterion is invalidated by the fact that the lengths of phases are not
independent. The distribution of the phase lengths does not tend to normality
for large lengths of a series, but the number of phases does so (Kendall, 1973).

4.3.4 Kendall's Phase Test

The phase is defined as the interval between any two successive turning
points. Let the length of the phase be denoted by ' d '. The expected number of
phases ( n p ) of length ' d ' in a random series of length ' n ' is given as (Kendall,
1973):

2

2 nd 2 d 3 d 1

n p = (40)

d 3!

Once n p is calculated, the observed number of phases and the expected
number of phases for a given length is compared. If this difference is large,
the series is not considered to be random.
Among the above-mentioned trend tests, the superiority of one over other
is mainly associated with the extent of adaptability of a given test to the
structure of the time series to be examined. The turning points and number of
phases tests are practically out-dated due to the availability of more powerful
tests (Shahin et al., 1993), which are described below.

4.3.5 Wald-Wolfowitz Total Number of Runs Test
Let the objective be to test whether the data sample x t ( t = 1, 2, …, n ) is
random based on the runs of the data with respect to the median of the
observation. The step-by-step procedure for using the Wald-Wolfowitz test is
as follows (Shahin et al., 1993):