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:

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