Geoscience Reference
In-Depth Information
than two variables are observed simultaneously at a time, the series is known
as a 'multivariate time series'. This topic deals with univariate time series
only.
1.5 Structure of Time Series
A time series is often adequately described as a function of four components:
trend
,
seasonality
,
dependent stochastic component
and
independent residual
component
. In general, a time series can be mathematically expressed as
(Shahin et al., 1993):
x
t
=
T
t
+
S
t
+ H
t
+ K
t
(1)
where
T
t
= trend component,
S
t
= seasonality, H
t
= dependent stochastic
component, and K
t
= independent residual component.
In the time series analysis, it is assumed that the data (observations)
consist of a
systematic pattern
and
random noise
(error); the latter usually
makes the pattern difficult to be identified. The
systematic pattern
is represented
by the first two components of Eqn. (1), which are deterministic in nature,
whereas the stochastic component accounts for the random error. Generally,
the stochastic component contains a dependent part which may be represented
by an ARMA(
p
,
q
) model, where '
p
' and '
q
' are the orders of the autoregressive
and moving-average models, respectively, and an independent part that
can only be described by some sort of probability distribution function. When
p
= 0, the ARMA(
p
,
q
) represents an MA(
q
) model, and when
q
= 0, it represents
an AR(
p
) model.
Thus, the process of hydrologic time series analysis should be viewed as
a process of identifying and separating the total variation in measured data
into above-mentioned four components. When a time series has been analyzed
and the components accurately characterized, each component can then be
modelled. Methods for identifying trends in time series are described in Chapter
4 and the methods for identifying stochastic component are described in
Chapter 5.
1.6 Salient Characteristics of Time Series
Most statistical analyses of hydrologic time series at the usual time scale
encountered in water resources studies are based on a set of fundamental
assumptions, which are: the series is homogenous, stationary, free from trends
and shifts, non-periodic with no persistence (Adeloye and Montaseri, 2002).
The term 'homogeneity' implies that the data in the series belong to one
population, and therefore have a time invariant mean. Non-homogeneity arises
due to changes in the method of data collection and/or the environment in
which it is done (Fernando and Jayawardena, 1994). On the other hand,
'stationarity' implies that the statistical parameters of the series computed
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