Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 65

4.3.6 Sum of Squared Lengths Test

This test considers the length of runs while testing a series for trend. The runs
of different lengths are counted. A run consists of a sequence of like signs as
defined in the Wald-Wolfowitz test. The test-statistic ( N ), the sum of the
squares of the run-lengths, is given by

Ç

2

jn

N =

(45)

j

j

where j = length of the run, and n j = number of runs of length j .
Critical values of the test-statistic at 5% level of significance and n /2
degrees of freedom can be obtained from Himmelblau (1969). If the calculated
test-statistic values are greater than its critical values, the null hypothesis of
trend-free series is rejected. Note that the Sum of Squared Lengths test is
more powerful than the Wald-Wolfowitz test (Himmelblau, 1969).

4.3.7 Adjacency Test

The adjacency test is applied to test the null hypothesis that the fluctuations in
a series are random in nature. The limitation of this test is the assumption that
the observations are obtained independently of each other and under similar
conditions (Kanji, 2001). For a time series x t ( t = 1, 2, …, n ), the test-statistic
' z ' for n > 25 is computed as follows:

L
V

z = (46)

wherein L for n > 25 is given as:

n 1

Ç

2

x x

t1 t

t 1

1

L = (47)

n

Ç

2

2 xx

t

t 1

It should be noted that L for n > 25 follows a normal distribution with zero
mean and variance (V 2 ) as:

n 2

V 2 = (48)

n 1 n 1

Critical values of ' z ' can be obtained from the tables for standard normal
distribution available in the textbooks on statistics.
For n 25, the test-statistic ' z ' is computed with L as: