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111186778Chaos-Toolbox-Ver.2.0.rar （23个子文件）

www.pudn.com.txt 218B

Chaos Toolbox Ver.2.0

Reference

基于混沌理论的太阳黑子时间序列预测.doc 175KB

C-C方法求时间延迟程序设计.doc 46KB

Nonlinear dynamics, delay times, and embedding windows.pdf 384KB

基于Lyapunov指数改进算法的边坡位移预测.caj 238KB

混沌工具箱使用帮助.txt 1KB

Main

reconstitution.m 296B

largest_lyapunov_exponent.m 2KB

lyapunov_rosenstein.m 2KB

C_CMethod_independent.m 1KB

lyapunov_wolf.m 5KB

MainPre_by_jiaquanyijie_n.m 1KB

C_CMethod.m 2KB

MainPre_by_Lya_1.m 899B

G_P.m 1KB

correlation_integral.m 752B

pre_by_lya.m 732B

MainPre_by_Lya_n.m 1KB

nearest_point.m 5KB

heaviside.m 286B

pre_by_lya_new.m 6KB

disjoint.m 283B

MainPre_by_jiaquanyijie_1.m 557B

Physica D 127 (1999) 48–60

Nonlinear dynamics, delay times, and embedding windows

H.S. Kim

1, a

, R. Eykholt

b,∗

, J.D. Salas

Department of Civil Engineering, Colorado State University, Fort Collins, CO 80523, USA

Department of Physics, Colorado State University, Fort Collins, CO 80523, USA

Hydrologic Science and Engineering Program, Department of Civil Engineering, Colorado State University, Fort Collins, CO 80523, USA

Received 11 September 1996; received in revised form 10 January 1998; accepted 12 August 1998

Communicated by A.M. Albano

Abstract

In order to construct an embedding of a nonlinear time series, one must choose an appropriate delay time τ

. Often, τ

estimated using the autocorrelation function; however, this does not treat the nonlinearity appropriately, and it may yield an

incorrect value for τ

. On the other hand, the correct value of τ

can be found from the mutual information, but this process is

rather cumbersome computationally. Here, we suggest a simpler method for estimating τ

using the correlation integral. We

call this the C–C method, and we test it on several nonlinear time series, obtaining estimates of τ

in agreement with those

obtained using the mutual information. Furthermore, some researchers have suggested that one should not choose a ﬁxed

delay time τ

, independent of the embedding dimension m, but, rather, one should choose an appropriate value for the delay

time window τ

= (m −1)τ , which is the total time spanned by the components of each embedded point. Unfortunately, τ

cannot be estimated using the autocorrelation function or the mutual information, and no standard procedure for estimating

has emerged. However, we show that the C–C method can also be used to estimate τ

. Basically τ

is the optimal time

for independence of the data, while τ

is the ﬁrst locally optimal time. As tests, we apply the C–C method to the Lorenz

system, a three-dimensional irrational torus, the Rossler system, and the Rabinovich–Fabrikant system. We also demonstrate

the robustness of this method to the presence of noise.

Keywords: Delay time; Correlation integral; Embedding; Time series

PACS: 05.45. + b;47.52. + j

1. Introduction

Analysis of chaotic time series is common in many ﬁelds of science and engineering, and the method of delays

has become popular for attractor reconstruction from scalar time series. From the attractor dynamics, one can

estimate the correlation dimension and other quantities to see whether the scalar time series is chaotic or stochastic.

Therefore, attractor reconstruction is the ﬁrst stage in chaotic time series analyses. Since the choice of the delay

∗

Corresponding author Tel.: +1-970-491-7366; fax: +1-970-491-7947; e-mail: eykholt@lamar.colostate.edu

Present Address: Department of Construction Engineering, Sun Moon University, Korea

0167-2789/99/$ – see front matter

PII: S0167-2789(98)00240-1

H.S. Kim et al. / Physica D 127 (1999) 48–60 49

time τ

for attractor reconstruction using the method of delays has not been fully developed, many researchers use

the autocorrelation function, which is computationally convenient and does not require large data sets. However, it

has been pointed out [1] that the autocorrelation function is not appropriate for nonlinear systems, and, instead, τ

should be chosen as the ﬁrst local minimum of the mutual information. Unfortunately, this approach is cumbersome

computationally and requires large data sets [2].

According to Packard et al. [3] and Takens [4], the method of delays can be used to embed a scalar time series

},i = 1, 2,... ,into an m-dimensional space as follows:

= (x

i+1

,... ,x

i+(m−1)t

), x

∈ R

, (1)

where t is the index lag. If the sampling time is τ

, the delay time is τ

= tτ

. Takens’ theorem assumes that we

have an inﬁnite noise-free data set, in which case, we can choose the delay time almost arbitrarily. However, since

real data sets are ﬁnite and noisy, the choice of the delay time plays an important role in the reconstruction of the

attractor from the scalar time series. If τ

is too small, the reconstructed attractor is compressed along the identity

line, and this is called redundance.Ifτ

is too large, the attractor dynamics may become causally disconnected, and

this is called irrelevance [5].

In common practice, the delay time τ

is chosen so as to ensure that the components of x

are independent,

and the same delay time is used for all embedding dimensions m. However, in recent years, it has been suggested

that the delay time window τ

= (m − 1)τ , which is the entire time spanned by the components of x

, should be

independent of m instead [6–11]. In this case, the delay time τ varies with the embedding dimension m. Rosenstein

et al. [6] compared several approaches for estimating τ

and τ

and indicated the critical disadvantages of each, such

as inconsistencies and long computation times. The geometrical concepts of redundance and irrelevance were used

to evaluate the quality of an attractor’s reconstruction. It has been suggested [11] that the delay time window may be

set to τ

≥ τ

, where τ

is the mean orbital period, which can be approximated by examining the oscillations of the

time series. Martinerie et al. [7] examined the delay time window and compared it with the delay times estimated

using the autocorrelation function and the mutual information, but they concluded that τ

cannot be estimaed using

either of these methods. Basically, τ

is the optimal time for independence of the data, but these methods estimate

only the ﬁrst locally optimal time, which is τ

In this work, we develop a technique for choosing either the delay time τ

or the delay time window τ

using

the correlation integral. Although τ

may be a more useful quantity for the estimation of the dimension than τ

many researchers continue to use τ

for attractor reconstruction using the method of delays. Also, the estimation

of both τ

and τ

using this technique helps to illuminate the difference between these two quantities. Brock

et al. [12,13], in their development of a test for nonlinearity in a time series, used the statistic S(m, N, r) =

C(m, N, r) − C

(1,N,r), where C(m, N, r) is the correlation integral (see Eq. (2)). Here, we use the similar

statistic S(m, N, r, t) = C(m, N, r, t) −C

(1,N,r,t), and we examine its dependence on the index lag t. Hence,

we call our technique the C–C method. Compared with the use of the mutual information, the C–C method is easier

to implement, useful for smaller data sets, and less demanding computationally. Also, its estimate of τ

agrees with

the ﬁrst local minimum of the mutual information.

2. Correlation integral and BDS statistic

The correlation dimension introduced by Grassberger and Procaccia [14,15] is widely used in many ﬁelds for

the characterization of strange attractors. The correlation integral for the embedded time series is the following

function:

C(m, N, r, t) =

M(M − 1)

1≤i<j≤M

2(r −kx

−x

k), r > 0, (2)

50 H.S. Kim et al. / Physica D 127 (1999) 48–60

where

2(a) = 0, if a<0, 2(a) = 1, if a ≥ 0,

N is the size of the data set, t is the index lag, M = N − (m − 1)t is the number of embedded points in m-

dimensional space, and k···kdenotes the sup-norm. C(m, N, r, t) measures the fraction of the pairs of points

,i = 1, 2,... ,M,whose sup-norm separation is no greater than r. If the limit of C(m, N, r, t) as N →∞exists

for each r, then the fraction of all state vector points that are within r of each other is denoted by C(m, r, t) =

lim

N→∞

C(m, N, r, t), and the correlation dimension is deﬁned as D

(m, t) = lim

r→0

[logC(m, r, t )/logr]. Since

N remains ﬁnite for real data sets, then we cannot let r go to 0; instead, we look for a linear region of slope D

(m, t)

in the plot of logC(m, N, r, t) vs. log r.

Brock et al. [12,13] studied the BDS statistic, which is based on the correlation integral, to test the null hypothesis

that a given data set is independently and identically distributed (iid). The test has been particularly useful for

chaotic systems and nonlinear stochastic systems. For completeness, we brieﬂy review the BDS statistic. Let F be

the invariant distribution of a variable X in the state space, and deﬁne the spatial correlation integral as follows:

C(m, r) =

2(r −kx

x −y

yk)dF(x

x)dF(y

y), r > 0. (3)

If X is iid, then using 2(r −kx

x −y

yk) =

k=1

2(r −|x

− y

|) yields C(m, r) = C

(1,r), where

C(1,r) =

[F(x + r) − F(x − r)]dF(x) ≡ C. (4)

Denker and Keller [16] showed that C(m, N, r) is a U-statistic estimator. Using the U-statistics theory for an

absolutely regular process, Brock et al. [12,13] proved that, as N →∞,

√

N[C(m, N, r) − C

(1,r)] approaches

a normal distribution with mean zero and variance

(m, r) = 4

− C

+ 2

m−1

i=1

m−i

− C

)

, (5)

where

K ≡

[(F (x +r) − F(x − r)]

dF(x) (6)

(assuming that K>C

). Thus, the BDS statistic deﬁned by

BDS(m, N, r) =

√

σ (m, r)

[C(m, N, r) − C

(1,r)] (7)

approaches a standard normal distribution. However, since the distribution F is generally unknown, then we can-

not obtain the values of C and K and the variance σ

(m, r) from the above deﬁnitions. Instead, the correlation

integral C(1,r)and the variance σ

(m, r) must be estimated from the sample data. Thus, C(1,r)is estimated by

C(1,N,r,t), and σ

(m, r) is estimated by

ˆσ

(

m(m−1)

2(m−1)

(

K−

−

+ 2

m−1

i=1

[

(

m−i

−

2(m−i)

)−m

2(m−i)

(

K−

)]

)

, (8)

where

C = C(m, N, r, t) and

K =

M(M − 1)(M −2)

1≤i<j<k≤M

2(r −kx

−x

k)2(r −kx

−x

k), (9)

H.S. Kim et al. / Physica D 127 (1999) 48–60 51

with M = N − (m − l)t. Then, under the iid hypothesis, if

and m>1, the BDS statistic becomes

BDS(m, N, r) =

√

ˆσ

[C(m, N, r, t) − C

(1,N,r,t)], (10)

and this converges to a standard normal distribution as N →∞.

The BDS statistic originatesfrom the statistical properties of the correlation integral, and it measures the statistical

signiﬁcance of calculations ofthe correlation dimension. Even though theBDS statistic cannot be used todistinguish

between a nonlinear deterministic system and a nonlinear stochastic system, it is a powerful tool for distinguishing

random time series from chaotic or nonlinear stochastic time series. Further statistical properties and proofs can be

found in [12,13].

3. Measure of nonlinear dependence

3.1. C–C method

This study is concerned with the properties of S(m, N, r, t) = C(m, N, r, t) − C

(1,N,r,t). We refer to a

comment by Brock et al. [12]: “If the stochastic process {x

} is iid, it will be shown that C(m, r) = C

(l,r) for

all m and r. That is to say, the correlation integral behaves much like the characteristic function of a serial string

in that the correlation integral of a serial string of independent random variables is the product of the correlation

integrals of component substrings.” This leads us to interpret the statistic S(m, N, r, t) as the serial correlation of a

nonlinear time series. Therefore, it can be regarded as a dimensionless measure of nonlinear dependence. For ﬁxed

m, N, and r the plot of S(m, N, r, t) vs. t is a nonlinear analog of the plot of the autocorrelation function vs. t.

In order to study the nonlinear dependence and eliminate spurious temporal correlations, we must subdivide the

time series {x

},i = 1, 2,... ,N, into t disjoint time series, where t is the index lag. S(m, N, r, t) is then computed

from these disjoint time series as follows:

For t = 1, we have the single time series {x

,... ,x

}, and

S(m, N, r, 1) = C(m, N, r, 1) − C

(1,N,r,1). (11)

For t = 2, we have the two disjoint time series {x

,... ,x

N−1

} and {x

,... ,x

}, each of length N/2, and

we average the values of S(m, N/2,r,2) for these two series:

S(m, N, r, 2) =

{[C

(m, N/2,r,2) − C

(1,N/2,r,2)] + [C

(m, N/2,r,2) − C

(1,N/2,r,2)]}. (12)

For general t, this becomes

S(m, N, r, t) =

s=1

(m, N/t, r, t) − C

(1,N/t,r,t)]. (13)

Finally, as N →∞, we can write

S(m, r, t ) =

s=1

(m, r, t) − C

(1,r,t)],m= 2, 3,... (14)

For ﬁxed m and t , S(m, r, t) will be identically equal to 0 for all r if the data are iid and N →∞. However, real

data sets are ﬁnite, and the data may be serially correlated; so, in general, we will have S(m, r, t) 6= 0. Thus, the

locally optimal times may be either the zero crossings of S(m, r, t) or the times at which S(m, r, t) shows the least

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赵广陆

2020-03-19

还可以，值得参考
zenghuishizhutou

2012-12-20

有的可以用，但是里面的方法不全，没陆振波工具箱全面
xzwxqnuaa

2013-05-02

不是很好用吧..关键的函数代码没有给出..
xxxxghost

2014-02-12

工具箱是好，但对于初学者来说还有点压力能看懂
wxsnopy

2013-03-28

这个工具箱有助于初学者学习

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