Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 58

1989). The Mann-Whitney test for detecting a shift in the mean or median of
hydrological time series has been applied by McCuen and James (1972),
Lazaro (1976), Lettenmaier (1976), Helsel and Hirsch (1988), Kiely et al.
(1998), Kiely (1999) and Yue and Wang (2002). Also, stationary stochastic
models such as AR (Autoregressive), MA (Moving Average), or ARMA
(Autoregressive Moving Average) models are frequently used to characterize
the standardized time series (Hipel and McLeod, 1994). However, the
standardization procedure does not ensure stationarity in the transformed series
(Salas, 1993). Moreover, some researchers (Appel and Brandt, 1983; Lovell
and Boashash, 1987; Imberger and Ivey, 1991; Chen and Rao, 2002) have
developed segmentation algorithms to determine stationary segments and to
estimate the parameters characterizing each segment in order to establish
piecewise stationary time series models. Two parametric tests and one
nonparametric test for checking stationarity in a time series are described in
the following sections.

4.2.1 Student's t - test

For applying the t -test, the series is divided into a number of subseries, and t -
test is performed to check whether the statistical character of each subseries
is significantly different from that of the original series. The null hypothesis
that the means of each subseries do not significantly differ from the population
mean is examined using the test-statistic, ' ts ', which is defined as follows:

( x P ) N 1

t

ts = (25)

V

where ¯ t = mean of the subseries, P = mean of the series, V = standard
deviation of the series, and N = number of the subseries.
Critical values for the ts -statistic can be obtained from standard texts on
statistics (e.g., Shahin et al., 1993; Haan, 2002). If the calculated value of ' ts '
is found less than its critical value, the null hypothesis cannot be rejected.

4.2.2 Simple t- test

This is a parametric test, which assumes that the annual hydrologic series x t
( t = 1, 2, …, n ) is uncorrelated and normally distributed with mean P and
standard deviation V. The series is divided into two subseries of sizes n 1 and
n 2 such that n = n 1 + n 2 . The first subseries x t ( t = 1, 2, …, n 1 ) has a mean P 1 ,
and standard deviation V and the second subseries x t ( t = n 1 +1, n 1 +2, …, n ),
is assumed to have mean P and standard deviation V. The simple t -test can be
used to examine the null hypothesis P = P when the two subseries have the
same standard deviation. Rejection of the null hypothesis is considered as a
detection of a shift. The test-statistic is defined as (Snedecor and Cochran,
1980):