1-C-C算法(延迟时间确定).pdf

HYSICA D
ELSEVIER
Physica I)127(1999)48-60
Nonlinear dynamics, delay times, and embedding windows
H.S. Kim, a, R. Eykholt D,. J D. Salas o
Department of Civil Engineering. Colorado State University, Hort Collins, CO 80523, US.
Hdrlogic Science d ngineering gram, Deparimen of Civil Engineering, Colorado Stae Universit HOT Collins,o 80523, S
ccived II September 1996: reecived in revised form 10 January 1998; accepted 12 Mugust 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 d. Often, td is
estimated using the autocorrelation fumction; however, this does not treat the nonlinearity appropriately, and it may yield an
incorrect value for ta. On the other hand, the correct value of rd can be found from the mutual information, but this process is
rathcr cumbersome computationally Hcrc, wc suggcst a simpler mcthod for estimating d using thc corrclation integral. W
call this the C-C method and we test it on several nonlinear time series, obtaining estimates of td in agreement with those
obtaincd using the mutual information. Furthcrmorc, some rcscarchcrs havc suggested that onc should not choose a fixcd
delay time a, independent of the embedding dimension m, but, rather, one should choose an appropriate value for the delay
time window w =(m-1), which is the total time spanned by the components of each embedded point. Unfortunately, w
cannot be estimated using the autocorrelation function or the mutual information, and
Dard proccdure for
tw has emerged. However, we show that the C-C method can also be used to estimate tw. Basically tw is the optimal time
for independence of the data, while td is the first locally optimal time. As tests, we apply the C-C method to the Lorenz
systcm, a thrcc-dimcnsional irrational torus, thc Rosslcr systcm, and thc Rabinovich-fabrikant systcm. Wc also demonstrate
the robustness of this mcthod to the prcscncc of noisc ( @1999 Elscvicr Scicncc B V. All rights rcscrvcd
Keywords: Delay time; Corrclation integral; I bedding; Time serics
CS:05.45.+b,47.52.+j
1. ntroduction
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; [ ax: +-1-970-491-7947; c-mail cykholl( lamar. colostatc cdu
Present Address: Departmcnt o! Construction Engincering, Sun Moon University, Korea.
0167-278999/s-see front matter(@1999 Flsevier Science B V. All rights reserved
PI:S0167-2789(98)00240-1