Environmental Engineering Reference
In-Depth Information
function of frequency. But, as the length of the time
series
N
He showed that the river flow could be modelled with a
persistence that had a power-law dependence on the lag.
A new model for the generation of this type of long-
range persistent behaviour was given by Mandelbrot and
Wallis (1968). This model was based on the filtering of
the spectral coefficients of a white noise time series to
generate noises that are well approximated by the power-
law dependence of
S
m
on
f
m
, given in Equation 3.13, for
a range of values of
=
T
/δ
approaches infinity (see Equation 3.11),
2
diverges. It is common practice to
convert the energy to a power - that is we divide
the total energy
|
X
m
|
2
by
|
X
m
|
the length of time considered,
T
. This is the basis
for the introduction of the
power-spectral density
, defined
as (Priestley, 1981):
=
N
δ
|
|
2
2
X
m
(
f
m
,
N
)
,
m
=
1, 2, 3,
...
,
N
2
β
. These are known as fractional
S
m
(
f
m
)
=
(3.12)
δ
N
Gaussian noises.
The
Fourier filtering technique
(e.g. see Theiler
et al
.,
1992; Malamud and Turcotte, 1999) is one method used
to generate a fractional Gaussian noise, and consists of
the following steps:
A plot of
S
m
against
f
m
is known as a
periodogram
,and
is a useful way for visualizing the dominant frequencies in
a time series. It is often convenient to view a periodogram
on log-log axes. As examples of periodograms, we consider
the Gaussian white noise and Brownian motions given
in Figures 3.4a and 3.4b, and extend them both to 4096
values. The resultant periodograms of these time series are
given in Figures 3.7a and 3.7b, respectively. For the white
noise, the periodogram (Figure 3.7a) is flat (in addition to
noise), indicating the power is equally distributed over all
frequencies. For the Brownian motion, we see a power-
law decay for the periodogram (a straight line on log-log
axes). In both cases, we find good agreement with the
power-law relation:
1. Choose the desired
, strength of long-range persis-
tence, and
N
, length of the noise.
2. Begin with a Gaussian distributed white noise with 2
N
elements.
3. Apply a discrete Fourier transform (Equation 3.11) to
the mean-corrected white noise. This transformation
results in the (complex-valued) Fourier coefficients
X
m
,
m
β
,
N
, where the amplitude of the
X
m
values are approximately equal.
4. These coefficients are filtered using the relation:
=
1,
...
S
m
(
f
m
)
∼
f
−
m
,
m
=
1, 2, 3,
...
,
N
2
m
N
−
β/
2
X
m
(3.13)
X
m
(
f
m
)
=
(3.14)
For the white noise, we have
β
=
0
.
003. Averages of
5. An inverse discrete Fourier transform is applied to
the
X
m
(
f
m
) coefficients. The result is a fractional noise
with power-law exponent
large numbers of simulations give
00 for white
noises. For the Brownian motion, we have
β
=
0
.
β
=
1
.
993.
β
.
Averages of large numbers of simulations give
00
for Brownian motions. As we will show (in the next
section)
β
=
2
.
Examples of fractional Gaussian noises generated using
the Fourier filtering technique are shown in the left col-
umn in Figure 3.8 for
is a measure of the strength of long-range
persistence (
β
β
>
0) or antipersistence (
β<
0), and an
5 and 1.0. As the
value of
β
increased from
−
1
.
0to
+
1
.
0, the contribution
of the high-frequency (short-period) terms is reduced.
With
β
=−
1
.
0and
−
0
.
5, the high-frequency contri-
butions dominate over the low-frequency contributions.
These time series exhibit
antipersistence
; adjacent values
are anti-correlated relative to a Gaussian white noise
(
β
=−
1
.
0,
−
0
.
5, 0
.
uncorrelated time series has
β
=
0.
3.5.4 Models for long-rangepersistence
Long-range persistence implies that all values in a time
series are correlated with one another. The models intro-
duced in Section 3.5.2 for short-range persistence can
be used to produce time series that approach long-range
persistence, if considered for very large lags. However,
in these short-range persistence models, there are no
constraints on the decay of persistence with lag, i.e. the
decay could be exponential, power-law or other. Hurst
et al
. (1965) introduced an alternative approach to the
quantification of correlations in stationary time series.
Henry Hurst spent his life studying the hydrology of the
Nile River, specifically the river flow as a time series.
0). For these antipersistent time series, values larger
than the mean tend to be followed by a value smaller than
the mean. With
β
=
0, the high-frequency contributions
are equal to the low-frequency contributions. The results,
as we have seen earlier (Section 3.5.1), form an uncorre-
lated time series; adjacent values are not correlated with
one another. With
β
=
5 and 1.0, the low-frequency
contributions dominate over the high-frequency contri-
butions. These time series exhibit
persistence
; adjacent
β
=
0
.
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