Environmental Engineering Reference
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4
b
=
−
1.0
b
= 1.0
2
x
n
0
−
2
−
4
4
b
=
−
0.5
b
= 1.5
2
x
n
0
−
2
−
4
4
b
= 0.0
b
= 2.0
2
x
n
0
2
−
−
4
4
b
= 0.5
b
= 2.5
2
x
n
0
−
2
−
4
4
b
= 1.0
b
= 3.0
2
x
n
0
−
2
−
4
0
128
256
n
384
512
0
128
256
n
384
512
Figure 3.8
Fractional Gaussian noises and Brownian motions. In the left column of this figure, the Fourier coefficients of the
Gaussian white noise (
β
=
0
.
0) have been Fourier filtered (Section 3.5.4) to give fractional Gaussian noises with
β
=−
1
.
0,
−
0
.
5, 0
.
5, 1
.
0. In the right column of this figure, the fractional Gaussian noises on the left, with
β
=−
1
.
0,
−
0
.
5, 0
.
0, 0
.
5,
β
=
.
.
.
.
.
and 1.0 have been summed to give fractional Brownian motions with
1
0, 1
5, 2
0, 2
5, 3
0. This is an extension of Figure 3.4
β
=
β
=
.
β
=
.
where just a Gaussian white noise with
0(pink
noise) can be regarded as either a fractional Gaussian noise or a fractional Browni
a
n motion. Each fractional Gaussian noise and
Brownian motion has
N
0 was summed to give a Brownian Motion
2
0. The transition case
1
=
512 points, and has been rescaled to have zero mean (
x
=
0
.
0) and unit variance (
σ
2
x
=
1
.
0). These are
examples of long-range persistent time series models.
with
β
=+
1
.
0,
+
1
.
5,
+
2
.
5,
+
3
.
0. We could also have
created these time series using Fourier filtering directly.
In Figure 3.8, as
β
=
1
.
0 (also called a pink noise). The time series are
weakly stationary for
1.
The standard deviation of the time series after
n
values,
σ
n
,isgivenby:
β<
1, and nonstationary for
β >
β
becomes larger, adjacent values in the
time series become more strongly correlated and profiles
smoothed. The persistence (in this case long-range) is
increased. For large
β
, the correlations with all lags are
strong; the persistence is long-range with a strength that
is strong. For small
n
Ha
σ
n
∼
(3.15)
where the exponent
Ha
is known as the Hausdorff mea-
sure (Mandelbrot andVanNess, 1968). For any stationary
time series (
β
, the correlations with large lag are
weak but non-zero; the persistence is weak but still long-
range. This difference can be contrasted with time series
that exhibit only short-range persistence, which may be
either strong or weak, but only over a finite set of lags.
The division between fractional noises and motions is
β<
1), by definition of stationarity,
σ
n
is
independent of
n
and
Ha
=
0
.
0. For fractional motions
β
−
1
Ha
=
(3.16)
2
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