Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 61

statistical tests for trend detection can be found in Mahé et al. (2001). A brief
description of 15 tests for trend detection is presented in the succeeding
sections.

4.3.1 Regression Test

The most commonly used approach for trend detection is to formulate a linear
model between the data and time in the following form (Hameed et al., 1997):
x t = D + E¹ t + H t (29)
where x t ( t = 1, 2, …, n ) = observed value at time t , D and E = regression
coefficients, and H t = a random error (white noise) with a mean of zero and
variance of S 2 .
The data of n years are substituted in the normal equations obtained by
the least squares technique, and the parameters

ˆ

ˆ anD are estimated. The

sum of squares of the residuals is given by:

n n

ˆ

Ç

2

Ç

2

2

xx E t t

SS res = (30)

t

t 1 t 1

The standard error of regression is calculated as:

@ 1/ 2 >

SS

res /

n 2

s = (31)

The ts -statistic is then computed as:

ˆ

E s E

ts = (32) ˆ

s

s E =

where, (33)

ˆ

1/ 2

n

Ë Û

Ç

2

tt

Ì Ü

Ì Ü

Í Ý

t 1

If the calculated value of the ts -statistic is less than its critical value at 5%
level of significance with n -2 degrees of freedom, the null hypothesis of
trend-free series cannot be rejected.
The main problem with the above approach is that it does not distinguish
between the trend and the persistence (Hameed et al., 1997). This test can be
misleading if seasonal cycles are present, the data are not normally distributed,
and the data are serially correlated (Gilbert, 1987).

4.3.2 Spearman Rank Order Correlation Test
To overcome the problem associated with the linear model for trend detection,
the Spearman rank order correlation (SROC) nonparametric test (McGhee,
1985) is used to check the existence of long-term trend. Among the available
nonparametric trend tests, the World Meteorological Organization (WMO,