Geoscience Reference
In-Depth Information
where
x
t
= value of variable at the
t
th
location in time;
k
= time lag, P =
population mean and
E
= expectation operator. Furthermore, the
population
autocorrelation
function
(U
k
) is defined as a ratio of the population
autocovariance (U
k
) to the population variance [Var(
x
t
)]. That is,
J
k
U
k
=
(84)
Var(
x
)
t
It should be noted that the
population autocorrelation
(U
k
) can also be
estimated by the
serial autocorrelation
function
(
r
k
) from sample data using
the following expression (Shahin et al., 1993; Haan, 2002):
nk
nk
nk
Ç
Ç Ç
(
xx
¹
)
¹
1 (
n k
)
x
x
t
t k
t
t
k
t
0
t
0
t
0
r
k
=
(85)
1/ 2
1/ 2
2
2
Ë
Û
Ë
Û
nk
nk
nk
nk
È
Ø
È
Ø
Ç
Ç
Ç
Ç
Ì
2
Ü
Ì
2
Ü
x
1(
n
k
)
x
x
1(
n
k
)
x
É
Ù
É
Ù
t
t
t
k
t
k
Ì
Ê
Ú
Ü
Ì
Ê
Ú
Ü
Í
t
0
t
0
Ý
Í
t
0
t
0
Ý
For
k
= 0, Eqns (84 and 85) result in U
0
=
r
0
= 1. As the lag (
k
) increases,
the number of the pairs of elements used in calculating
r
k
decreases. It is a
common practice to set the upper limit of lag between 0.1
n
to 0.25
n
, depending
on the size (
n
) of the series (Matalas, 1967). The detailed information about
the internal structure of the time series can be obtained by examining
autocorrelogram, which is drawn with an array of autocorrelation coefficients
(i.e., U
0
, U
1
, ...) as ordinates and
k
as abscissa.
The upper and lower critical values of autocorrelation function can be
obtained from the Anderson's test as follows (Anderson, 1942):
(
r
k
)
upper
=
^
`
1(
nk
) ( 1
z
nk
1)
(86)
12
D
(
r
k
)
lower
=
^
`
1(
nk
) ( 1
z
nk
1)
(87)
12
D
where
z
1-D/2
= standard normal variate at D significance level. If the value of
r
k
obtained from Eqn. (85) falls within the critical value given by either Eqn.
(86) or Eqn. (87), the null hypothesis that (U
k
)
is zero is rejected. This indicates
that the series is not purely random and some persistence exists.
4.5.2 Spectral Technique
The spectral analysis technique can be considered as an alternative to the
autocorrelation technique where the spectral density function replaces the
Fourier transformation of the autocorrelation function (Shahin et al., 1993).
The spectrum of a time series can be defined by harmonic analysis. The basic
function of the spectrum is to decompose a time series on a frequency basis,
and then frequencies and amplitudes of the series can be estimated, if they are
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