OPTIMAL RECONSTRUCTION SPACE FOR ESTIMATING CORRELATION DIMENSION

1996 ◽  
Vol 06 (02) ◽  
pp. 377-381 ◽  
Author(s):  
ROBERT C. HILBORN ◽  
MINGZHOU DING
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Time Series ◽  
Largest Lyapunov Exponent ◽  
Correlation Integral ◽  
Past Work ◽  
Optimal Reconstruction ◽  
Scaling Region ◽  
Delay Coordinates

In this paper we consider the estimation of the correlation dimension from a scalar chaotic time series using delay coordinates. Past work has shown that there appears to be a reconstruction space for which the correlation integral has the longest scaling region. We give a firmer foundation to this idea by developing a theory that estimates the dimension of this “optimal” reconstruction space in terms of dynamical quantities such as the largest Lyapunov exponent.

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

The Lyapunov Exponent Variance of an Electronic Manufacturing Enterprise’s Daily Qualified Rate Time Series by Improved Small Data Sets Approach

2012 ◽  
Vol 197 ◽  
pp. 271-277
Author(s):  
Zhu Ping Gong
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Time Series ◽  
Quality System ◽  
Quality Level ◽  
Largest Lyapunov Exponent ◽  
Small Data ◽  
Data Set ◽  
Electronic Manufacturing ◽  
Small Data Set

Small data set approach is used for the estimation of Largest Lyapunov Exponent (LLE). Primarily, the mean period drawback of Small data set was corrected. On this base, the LLEs of daily qualified rate time series of HZ, an electronic manufacturing enterprise, were estimated and all positive LLEs were taken which indicate that this time series is a chaotic time series and the corresponding produce process is a chaotic process. The variance of the LLEs revealed the struggle between the divergence nature of quality system and quality control effort. LLEs showed sharp increase in getting worse quality level coincide with the company shutdown. HZ’s daily qualified rate, a chaotic time series, shows us the predictable nature of quality system in a short-run.

AN INTEGRATED MODEL FOR THE CORRELATION INTEGRAL

2007 ◽  
Vol 07 (01) ◽  
pp. L31-L37
Author(s):  
ALEXANDROS LEONTITSIS ◽  
JENNY PANGE
Keyword(s):  
Time Series ◽  
Linear Part ◽  
Main Tool ◽  
Integrated Model ◽  
Chaotic Time Series ◽  
Artificial Data ◽  
Correlation Integral ◽  
Scaling Region

The correlation integral is the main tool for the analysis of the chaotic time series. Its linear part (i.e. the scaling region) gives the attractor's dimension. In this work we propose a model that explains all the possible neighborhood ranges of the correlation integral. Empirical examples are given on artificial data.

Local Piecewise-Linearity Prediction Method of Chaotic Time Series

2013 ◽  
Vol 712-715 ◽  
pp. 2415-2418
Author(s):  
Juan Liu ◽  
Xue Wei Bai ◽  
Dao Cai Chi
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Prediction Method ◽  
Chaotic Time Series ◽  
Local Region ◽  
Liaoning Province ◽  
Largest Lyapunov Exponent ◽  
Region Method ◽  
Local Prediction ◽  
Piecewise Linearity

A Local Piecewise-Linearity Prediction method is presented, Based on the advantages and limitations of local prediction of chaotic time series. Taking time series of rainfall as example for prediction the rainfall of one city in Liaoning province, which includes the application of the largest Lyapunov exponent, Local-region method and Local Piecewise-Linearity method. The method proposed is proved practical in comparison with the observed data.

Correction of the Correlation Dimension for Noisy Time Series

1997 ◽  
Vol 07 (06) ◽  
pp. 1283-1294 ◽  
Author(s):  
D. Kugiumtzis
Keyword(s):  
Time Series ◽  
Noise Level ◽  
Correlation Dimension ◽  
Epileptic Seizures ◽  
Chaotic Time Series ◽  
Simple Method ◽  
Correlation Integral ◽  
Free Data ◽  
Correction Scheme ◽  
Eeg Data

In the computation of the correlation dimension of chaotic time series corrupted with observational noise, the scaling region is often masked resulting in deteriorated estimates. Here a simple method is proposed to correct the corrupted correlation integral based on the statistical properties of the Euclidean norm used to compute the noisy point interdistances. When the noise level is known, the corrected slope from the noisy data is very close to the slope for the noise-free data. Thus if the scaling property of the noise-free attractor holds for distances around the noise amplitude then the correct dimension can be inferred. The problem of estimating the correct noise level is discussed and a simple approach is proposed. Simulations with synthetic noisy chaotic time series demonstrate the efficiency of the correction scheme. Furthermore, the correction scheme is used to enhance correlation dimension estimates for the Taylor–Couette chaotic data and for EEG data from epileptic seizures.

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