Introduction - Hydrologic Time Series Analysis: Theory and Practice - page 5

all possible realizations of a process, and it is used in the theory of stochastic
processes and time series analysis in lieu of the well-known statistical term
'population' (Haan, 1977; Shahin et al., 1993). The properties of a time series
can be obtained based on a single realization over a time interval or based on
several realizations at a given time. The properties based on a single realization
are known as time average properties , whereas those based on several
realizations at a particular time are known as ensemble properties (Haan,
1977). If the time average properties and the ensemble properties of a time
series are same, the series is said to be ergodic (Haan, 1977). Ergodicity is the
property by which each realization of a given process is a complete and
independent representative of all possible realizations of the process (Shahin
et al., 1993). Thus, the ergodicity allows the scientists/researchers to determine
the statistical properties of a process from a single realization.
1.3 Time Series Analysis
Time series analysis is the investigation of a temporally distributed sequence
of data or the synthesis of a model for prediction wherein time is an independent
variable. Sometimes, time is not actually used to predict the magnitude of a
random variable such as peak runoff rate, but the data are ordered by time.
The main intent of time series analysis is to detect and describe quantitatively
each of the generating processes underlying a given sequence of observations
(Shahin et al., 1993). Hydrologic time series are analyzed for several reasons.
The main reason as reported in the literature is to detect a trend due to another
random hydrologic variable. Secondly, time series may be analyzed to develop
and calibrate a model that would describe the time-dependent characteristics
of a hydrologic variable. Thirdly, time series models may be used to predict
future values of a time-dependent variable. Besides the time-dependent data
series, there are space-dependent data series of hydrologic systems, which are
known as 'spatial data series'. Thus, in the spatial data series, the data are
location specific instead of depending on time as in the time series. The
examples of spatial data series are: the variability of groundwater levels over
a groundwater basin, spatial variation of aquifer or soil properties, spatial
variation of rainfall in a catchment/basin, and so on. Most of the time series
analysis methods can equally be applied to spatial data series (Shahin et al.,
1993). Therefore, spatial data series is sometimes referred to as time series.
There are four major steps involved in a time series analysis (McCuen,
2003): (i) detection, (ii) analysis, (iii) synthesis, and (iv) verification. In the
detection step, systematic components of the time series such as trends or
periodicity are identified. It is also necessary to decide in this step whether the
systematic effects are physically and statistically significant. In the analysis
step, the systematic components are analyzed to identify their characteristics
including magnitudes, form and their duration over which the effects exist. In
the synthesis step, information from the analysis step is accumulated to develop