Geoscience Reference
In-Depth Information
5.1.1 Purely Random Process
A discrete hydrologic process is called a purely random process if the random
data points of a variable x t ( t = 1, 2, ... ) form a sequence of mutually independent,
identically distributed data points of the same variable (Chatfield, 1980). It is
also known as white noise . The definition of a purely random process reflects
that it is strictly stationary. In practice, this type of stochastic process does not
appear. The purely random process has the least practical importance; however,
it is important as a building block for other processes.
5.1.2 Autoregressive (AR) Process
Most time series consist of data points that are serially dependent in the sense
that one can estimate a coefficient or a set of coefficients that describes
consecutive data points of the series from specific, time-lagged (previous)
data points. This can be summarized by the following expression of
autoregressive process (Box and Jenkins, 1976).
x t =
[I
x
I
x
I
x
H
!
(1)
1 t )
2 t )
3 t )
t
where x t = data point of variable x at time t ; x (t - 1) , x (t - 2) and x (t - 3) = data
points of variable x at previous times t - 1, t - 2 and t - 3, respectively;
[ = a constant (intercept or population mean);
II I = autoregressive
model parameters; and H t = random error component or random shock (white
noise).
It is seen from Eqn. (1) that each data point of a time series is made up
of a random error component and a linear combination of prior data points.
For the population of a hydrologic variable, expression given in Eqn. (1)
represents an infinite autoregressive process. However, in practice, population
mean in Eqn. (1) is replaced with sample mean and the order of autoregressive
process is reduced to p . Thus, Eqn. (1) can be rewritten as (Box and Jenkins,
1976):
,
d
12
3
x t =
x
I
x
I
x
I
!
x
H
(2)
1
t1
2
t2
p
tp
t
The order of the autoregressive process is defined by the highest value of
p , for which
I Thus, for p = 1, the autoregressive (AR) process is of the
first order and for p = 2, the process is of the second order. The first and
second order autoregressive processes can be simply denoted as AR(1) and
AR(2), respectively. Similarly, the AR process of order p can be denoted as
AR( p ).
0.
p
Stationarity Requirement: The autoregressive process will be stable only when
the autoregressive model parameters lie within a certain range. Otherwise,
past effects (influence of previous data points) would accumulate and the
successive values of the variable x t would move towards infinity, and therefore,
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