Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 66

n 1

Ç

2

x x

t1 t

t 1

L = (49)

n

Ç

2

xx

t

t 1

Critical values of the test-statistic ' z ' for n 25 can be obtained from Hart
(1942). If the computed values of ' z ' are less than its critical values, there is
no reason to reject the null hypothesis.

4.3.8 Difference Sign Test

The difference sign test is used to examine whether fluctuations in a time
series observations are independent of the order in the sequence. This test is
applied with the assumption that the number of observations is large and that
the observations are obtained under similar conditions (Kanji, 2001). In order
to apply this test, first of all, a sequence of successive differences ( x t+1 - x t ) is
formed from the sequence of ' n ' observations ( x 1 , x 2 , …, x n ). Thereafter, the
number of '+' signs ( n + ) in this derived sequence is counted. For large values
of ' n ', n + may be assumed to follow a normal distribution with the mean (P n+ )
and variance (

n V ) given as:

n 1

P n+ = (50)

2

n 1

n V =

(51)

12

The test-statistic or standard normal variate ( z ) is defined as:

n P

V

0.5

n
n+

z = (52)

The test-statistic is computed using Eqn. (52) and is compared with 1.64
(i.e., critical value of ' z ' at 5% level of significance). If the computed value of
' z ' is greater than its critical value, the null hypothesis is rejected, which
suggests that the time series has a trend.

4.3.9 Run Test on Successive Differences
The necessary condition for applying this test is that the observations in the
sample are obtained under similar conditions. Null hypothesis ( H 0 ) is made
that the observations in a time series are independent of the order in the
sequence, which is tested by the run test on successive differences. From the
sequence of observations x t ( t = 1, 2, …, n ), a sequence of successive differences
( x t+1 - x t ) is formed (i.e., each observation has the preceding one subtracted
from it). The test-statistic ( K ) is defined as the number of runs of '+' and
'-' signs in the sequence of differences.