Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 73

4.5 Methods for Persistence Testing

Persistence can sometimes be treated as periodicity. In many hydrologic time
series studies, no distinction is made between persistence and randomness
(McMohan and Mein, 1986; Aksoy, 2007). Therefore, the tests to examine the
randomness of a hydrologic time series are used for detecting both trend and
persistence (Machiwal and Jha, 2006). Generally, randomness or non-
persistence is defined as the independence among data in a time series. On the
contrary, the series is called persistent if the data in the series are dependent
on each other. Practically, persistence is a tendency of the successive values of
a time series to 'remember' their antecedent values and to be influenced by
them (Giles and Flocas, 1984). Mathematically, persistence is defined as the
correlational dependency of order k between each i th element and the (i-k) th
element of the series (Kendall, 1973), and is measured by autocorrelation (i.e.,
correlation between two terms of the same time series). Here, ' k ' is usually
called time lag . The detection of persistence can be made by autocorrelation
technique (time domain) and/or spectral technique (frequency domain).
However, the autocorrelation technique has been applied in several studies
such as Mirza et al. (1998), Maidment and Parzen (1984), Schwankl et al.
(2000), etc. Here, it is worth mentioning that some researchers (e.g.,
Jayawardena and Lai, 1989) have used the autocorrelation technique for testing
periodicity in hydrologic time series. Such a misconception is quite common
in the analysis of hydrologic time series (Machiwal and Jha, 2006).
The persistence test of a time series can be performed in two ways:
(i) time domain (autocorrelation technique), and (ii) frequency domain (spectral
technique). However, some investigators (e.g., Quimpo, 1968) suggest the
application of autocorrelation technique only because the spectral technique
alone cannot be used without knowing the autocorrelation in a series. This is
due to the fact that the spectral density is a Fourier transform of the
autocorrelation function. The autocorrelation and spectral techniques for
examining the persistence in a time series are described below.

4.5.1 Autocorrelation Technique

The autocorrelation function in essence expresses the degree of temporal
dependency among observations. It is actually a process of self-comparison,
expressing the linear correlation between an equally-spaced series and the
same series at a specified time lag or separation (Jenkins and Watts, 1968). If
x 0 , x 1 , x 2 , ..., x n-1 is a realization of a stationary stochastic process, the covariance
between x t and its value x t+k , separated by a time interval k , is known as the
population autocovariance (J k ) and is mathematically expressed as follows
(Box and Jenkins, 1976):

J k = Ex [( P ) ( x P )] (83)

t t + k