Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 64

Step 1: Determine the median of the data. For this, sort the data sample in
ascending order such that x 1 x 2 … x n . Now, for an integer k , the
sample median ( 0.5 x ) is computed as:

x for n 2 k 1

Ï

x =

k1

(41)

0.5

0.5 xx for n k 2

Ð

k k 1

Step 2: Examine all the data of the series to check whether or not it exceeds
the median. A plus sign (+) or minus sign (-) is assigned to every data
of the series according to whether its value is greater than or less than
the median, respectively. If the median coincides with an observed
value ( n is odd), neither a plus nor a minus sign is assigned to such
a value, implying that the total number of observations is reduced
by 1.
Step 3: Count the number of runs (

U of plus signs. A run is defined as a
sequence of the entries of same sign until it is interrupted by opposite
sign.

)

The mean (P U+ ) and the variance (V U+ ) of the statistic U are calculated
by the following formulae:

Step 4:

1

2 n

1

P U+ = (42)

nn
n

2
41

V U+ =

(43)

Step 5: Compute the test-statistic ( z ) using the following formula:

U P 0.5

U

z = (44)

V

U

Step 6: Under the null hypothesis ( H 0 ) that the sequence of (+) and (-) signs
is random, z follows a standard normal distribution. Hence, obtain
the critical value of the standard normal distribution for a given
significance level D and denote it by ± z D/2 .
Step 7: If value of z calculated in Step 5 is greater than its critical value, the
null hypothesis is rejected.

It should be noted that the Wald-Wolfowitz test does not take into account
the length of the runs, and considerable information is ignored. Hence, this
test is not very powerful and not efficient, but can be used to determine
whether the observations of a random variable are independent. If the
observations of a random variable are independent, the time series is said to
have no trend (i.e., trend-free).