Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 59

xx 2 1

ts = (26)

11

S

nn

1 2

2 2

n 1 s n 1 s

1 1 2 2

and S = (27)

n 2

2 2

where xx s s are the estimated means and variances of the first and

the second subseries, respectively. Critical values of this test-statistic can be
taken from the Student's t -distribution standard tables for ' n -2' degrees of
freedom and 5% level of significance. If the computed value of the test-
statistic is greater than its critical value, the null hypothesis that both the
subseries belong to the same population is rejected.

, , d

121 2

4.2.3 Mann-Whitney Test
For applying this nonparametric test, the annual hydrologic series x t ( t =1, 2,
…, n ) is divided into two subseries x 1 , x 2 , …, x n1 and x n1+1 , x n1+2 , …, x n of
sizes n 1 and n 2 , respectively such that n 1 + n 2 = n . A new series, z t ( t = 1, 2, …,
n ) is defined by arranging the original data ( x t ) in increasing order of magnitude.
The test-statistic to test the null hypothesis that the mean of the first subseries
is equal to the mean of the second subseries is defined as (Snedecor and
Cochran, 1980):

n 1

Ç

Rx () n n ( n /2

t 1 1 2

t 1

u c = (28)

12

> @

nn n n 1/12

12 1 2

where R ( x t ) is the rank of the observation x t in ordered series z t . The null
hypothesis of equal means is rejected, if the computed value of this test-
statistic is greater than its critical value obtained from the tables of standard
normal distribution at 5% significance level.

4.3 Methods for Detecting Trend

A common deterministic component in a time series is a trend. A trend is a
tendency for successive values to be increasing or decreasing over time (Haan,
2002). Changes in hydrologic conditions by natural and/or artificial factors
can introduce linear or nonlinear trends into a hydrologic time series. The
trend in a time series can be expressed by a suitable linear or nonlinear model;
the linear model is widely used in hydrology (Shahin et al., 1993). The simplest
of linear trend detection models is Student's t -test (Hameed et al., 1997),
which requires that the series under testing should be normally distributed.
Thus, whether or not the sample data follow a normal distribution has to be