H. Y. HU, H. M. HUANG 47
attempting to reconstruct the dynamics in phase-space.
Monthly runoff series can be decided by many factors,
and a number of variables will be required when we use
equations to describe the process of its evolution. Gener-
ally, it is believed that the time series of a variable is th e
overall performance of the system, so the chaotic behav-
ior of the system can be studied by investigating the sin-
gle variable time series which can decide the long-term
evolution of the system.
The nature of dynamics of a real-world system may be
stochastic, deterministic or in between. This can be iden-
tified, at least as a preliminary indicator, by using the
phase space concept. A method for reconstructing a
phase-space from a single time series has been presented
by Takens [6]. The Takens theorem states that the un-
derlying dynamics can be fully recovered by building an
m-dimensional space. The dynamics of a scalar time se-
ries
123
,,,,
n
xx x are embedded in the
m-dimensional phase-space. The phase-space is defined
by:
,,,1 1,2,Yixi xiximiN
,
(1)
where
1Nn m
, τ is the time delay, m is the
embedding dimension, and
t123
=,,,,
n
xx xx with
N-observed values.
To determine the fractal dimension of a dynamical
system, one should first decide the time delay and the
embedding dimension for the correlation dimension
analysis. Usually, the time delay is made with the help of
the autocorrelation function and the embedding dimen-
sion can be obtained by the G-P method [7]. Correlation
dimension can be used to estimate the sufficient embed-
ding dimension and the estimated fractal dimension. A
strange attractor could also be revealed in a chaotic sys-
tem under a phase space reconstruction environment.
For a monthly runoff time series 123
,,,,
n
xx x
, the
reconstruction of the phase-space is defined by:
111 1(1)
222 2(1)
(1)
,,,
,,,
,,,
m
m
lll lm
Yxx x
Yxx x
Yxx x
(2)
where,
1ln m
.
3.2. Autocorrelation Function
The purpose of using autocorrelation function in chaotic
analysis is to help select a proper time delay. The time
delay can’t be too small, or too large. The choice of a too
small time delay causes information overlap among sub-
sequent datum. On the other hand, the choice of a too
large time delay can lead to loss of all relevant informa-
tion in phase space reconstruction because neighboring
trajectories diverge. Therefore, the selection of a suitable
time delay would allow unfolding of the attractor in the
phase space while the components of any state vector
must be as uncorrelated as possible.
A good choice of the time delay is essential for geo-
metrical and numerical analysis of the phase space re-
construction. For the time series 123
,,,,
n
xx x
, the
Autocorrelation function is defined as follows:
12
1
n
ii
i
xx
Cnx
(3)
where, τ is the time delay,
and
is the mean and
standard deviation of the time series, respectively. It may
be chosen as the lag time at which the auto-correction
becomes zero. However, considering various values of τ
demonstrates that the results do not show a strong de-
pendence on the actual value chosen.
3.3. G-P Method
There are few distinct methods for computing fractal
dimensions: relative dispersion analysis, correlation
analysis, Fourier analysis and rescaled range analysis. To
estimate the fractal dimension of a time series, the con-
cept of correction dimension is useful. Correlation di-
mension is a nonlinear measure of the correlation be-
tween pairs lying on the attractor. Correlation dimension
estimation is related to the relativ e frequency with which
the attractor visits each covering element. Correlation
dimension is generally a lower bound measure of the
fractal dimension.
The correlation dimension can be measured by the G-P
method suggested by Grassberger and Procaccia [7]. The
main steps of the G-P method are:
1) For a time series 123
,,,,
n
xx x , selecting a proper
time delay τ and embedding dimension m and recon-
structing the phase space.
2) The correlation integral is then calculated,
which is given by
Cr
2,1
1N
ij
ij
ij
Crr Y Y
N
(4)
where
is the Heaviside step function, ij
urYY
with
u=1
for , and , for 0u
u=0
0u
. N is
the number of points on the reconstructed attractor, r is
the radius of the sphere centered on i or Y
Y. ij
YY
is the Euclidean distance between the elements of
and i
Y
Y.
3) For a large number of points, the correlation inte-
gral follows the power law:
Copyright © 2013 SciRes. EPE