Calculation of correlation dimension of a time series based on enumeration of suboptimal solutions

2003 ◽  
pp. 173-182
Author(s):  
J. V. Romanovsky ◽  
L. A. Evdokimov
Keyword(s):  
Time Series ◽  
Correlation Dimension ◽  
Suboptimal Solutions

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.

Gear Damage Detection Using Chaotic Dynamics Techniques: A Preliminary Study

1995 ◽  
Author(s):  
M. Farid Golnaraghi ◽  
DerChyan Lin ◽  
Paul Fromme
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Damage Detection ◽  
Correlation Dimension ◽  
Chaotic Dynamics ◽  
Crack Detection ◽  
Gear Tooth ◽  
Vibration Signal ◽  
Nonlinear Time Series Analysis ◽  
Preliminary Study

Abstract This paper is a preliminary study applying nonlinear time series analysis to crack detection in gearboxes. Our investigations show that the vibration signal emerging from a gearbox is chaotic. Appearance of a crack in a gear tooth alters this response and hence the chaotic signature. We used correlation dimension and Lyapunov exponents to quantify this change. The main goal of this study is to point out the great potential of these methods in detection of cracks and faults in machinery.

scholarly journals The transient variation in the complexes of the low-latitude ionosphere within the equatorial ionization anomaly region of Nigeria

2015 ◽  
Vol 22 (5) ◽  
pp. 527-543 ◽  
Author(s):  
A. B. Rabiu ◽  
B. O. Ogunsua ◽  
I. A. Fuwape ◽  
J. A. Laoye
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Tsallis Entropy ◽  
Variation Pattern ◽  
Dynamical Complexity ◽  
Equatorial Ionization Anomaly ◽  
Non Linear ◽  
Anomaly Region

Abstract. The quest to find an index for proper characterization and description of the dynamical response of the ionosphere to external influences and its various internal irregularities has led to the study of the day-to-day variations of the chaoticity and dynamical complexity of the ionosphere. This study was conducted using Global Positioning System (GPS) total electron content (TEC) time series, measured in the year 2011, from five GPS receiver stations in Nigeria, which lies within the equatorial ionization anomaly region. The non-linear aspects of the TEC time series were obtained by detrending the data. The detrended TEC time series were subjected to various analyses to obtain the phase space reconstruction and to compute the chaotic quantifiers, which are Lyapunov exponents LE, correlation dimension, and Tsallis entropy, for the study of dynamical complexity. Considering all the days of the year, the daily/transient variations show no definite pattern for each month, but day-to-day values of Lyapunov exponents for the entire year show a wavelike semiannual variation pattern with lower values around March, April, September and October. This can be seen from the correlation dimension with values between 2.7 and 3.2, with lower values occurring mostly during storm periods, demonstrating a phase transition from higher dimension during the quiet periods to lower dimension during storms for most of the stations. The values of Tsallis entropy show a similar variation pattern to that of the Lyapunov exponent, with both quantifiers correlating within the range of 0.79 to 0.82. These results show that both quantifiers can be further used together as indices in the study of the variations of the dynamical complexity of the ionosphere. The presence of chaos and high variations in the dynamical complexity, even in quiet periods in the ionosphere, may be due to the internal dynamics and inherent irregularities of the ionosphere which exhibit non-linear properties. However, this inherent dynamics may be complicated by external factors like geomagnetic storms. This may be the main reason for the drop in the values of the Lyapunov exponent and Tsallis entropy during storms. The dynamical behaviour of the ionosphere throughout the year, as described by these quantifiers, was discussed in this work.

scholarly journals The comparative study of chaoticity and dynamical complexity of the low-latitude ionosphere, over Nigeria, during quiet and disturbed days

2014 ◽  
Vol 21 (1) ◽  
pp. 127-142 ◽  
Author(s):  
B. O. Ogunsua ◽  
J. A. Laoye ◽  
I. A. Fuwape ◽  
A. B. Rabiu
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Chaotic Behavior ◽  
Surrogate Data ◽  
Tsallis Entropy ◽  
Electron Content ◽  
Total Electron ◽  
Dynamical Complexity ◽  
Definite Pattern

Abstract. The deterministic chaotic behavior and dynamical complexity of the space plasma dynamical system over Nigeria are analyzed in this study and characterized. The study was carried out using GPS (Global Positioning System) TEC (Total Electron Content) time series, measured in the year 2011 at three GPS receiver stations within Nigeria, which lies within the equatorial ionization anomaly region. The TEC time series for the five quietest and five most disturbed days of each month of the year were selected for the study. The nonlinear aspect of the TEC time series was obtained by detrending the data. The detrended TEC time series were subjected to various analyses for phase space reconstruction and to obtain the values of chaotic quantifiers like Lyapunov exponents, correlation dimension and also Tsallis entropy for the measurement of dynamical complexity. The observations made show positive Lyapunov exponents (LE) for both quiet and disturbed days, which indicates chaoticity, and for different days the chaoticity of the ionosphere exhibits no definite pattern for either quiet or disturbed days. However, values of LE were lower for the storm period compared with its nearest relative quiet periods for all the stations. The monthly averages of LE and entropy also show no definite pattern for the month of the year. The values of the correlation dimension computed range from 2.8 to 3.5, with the lowest values recorded at the storm period of October 2011. The surrogate data test shows a significance of difference greater than 2 for all the quantifiers. The entropy values remain relatively close, with slight changes in these values during storm periods. The values of Tsallis entropy show similar variation patterns to those of Lyapunov exponents, with a lot of agreement in their comparison, with all computed values of Lyapunov exponents correlating with values of Tsallis entropy within the range of 0.79 to 0.81. These results show that both quantifiers can be used together as indices in the study of the variation of the dynamical complexity of the ionosphere. The results also show a strong play between determinism and stochasticity. The behavior of the ionosphere during these storm and quiet periods for the seasons of the year are discussed based on the results obtained from the chaotic quantifiers.

IDENTIFICATION OF DYNAMIC PATTERNS OF BODY SWAY DURING QUIET STANDING: IS IT A NONLINEAR PROCESS?

2010 ◽  
Vol 20 (04) ◽  
pp. 1269-1278 ◽  
Author(s):  
HAMED GHOMASHCHI ◽  
ALI ESTEKI ◽  
JULIEN CLINTON SPROTT ◽  
ALI MOTIE NASRABADI
Keyword(s):  
Time Series ◽  
Postural Control ◽  
Correlation Dimension ◽  
Postural Sway ◽  
Convincing Evidence ◽  
Temporal Organization ◽  
Quiet Standing ◽  
Body Sway ◽  
Linear Measure ◽  
Control Mechanisms

During quiet standing, the human body continuously moves about an upright posture in an erratic fashion. Many researchers characterize postural fluctuations as a stochastic process while some others suggest chaotic dynamics for postural sway. In this study, first we examined these assumptions using principles of chaos theory in normal healthy and in patients with deteriorated postural control mechanisms. Next, we compared the ability of a nonlinear dynamics quantifier correlation dimension to that of a linear measure standard deviation to describe variability of healthy and deteriorated postural control mechanisms during quiet standing. Our findings did not provide convincing evidence for existence of low dimensional chaos within normal and abnormal sway dynamics but support the notion that postural fluctuations time series are distinguishable from these generated by a random process. The results indicated that although linear variability measures discriminated well between groups, they did not provide any information about the structure of postural fluctuations. Calculated correlation dimension as a complexity measure which describes spatio temporal organization of time series may be useful in this regard.

scholarly journals Nonlinear time series analysis of geomagnetic pulsations

1994 ◽  
Vol 1 (2/3) ◽  
pp. 145-155 ◽  
Author(s):  
Z. Vörös ◽  
J. Verö ◽  
J. Kristek
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Correlation Dimension ◽  
Deterministic Chaos ◽  
Nonlinear Time Series ◽  
Surrogate Data ◽  
Nonlinear Time Series Analysis ◽  
Geomagnetic Pulsations ◽  
Series Analysis ◽  
Low Dimensional

Abstract. A detailed nonlinear time series analysis has been made of two daytime geomagnetic pulsation events being recorded at L'Aquila (Italy, L ≈ 1.6) and Niemegk (Germany, L ≈ 2.3). Grassberger and Procaccia algorithm has been used to investigate the dimensionality of physical processes. Surrogate data test and self affinity (fractal) test have been used to exclude coloured noise with power law spectra. Largest Lyapunow exponents have been estimated using the methods of Wolf et al. The problems of embedding, stability of estimations, spurious correlations and nonlinear noise reduction have also been discussed. The main conclusions of this work, which include some new results on the geomagnetic pulsations, are (1) that the April 26, 1991 event, represented by two observatory time series LAQ1 and NGK1 is probably due to incoherent waves; no finite correlation dimension was found in this case, and (2) that the June 18, 1991 event represented by observatory time series LAQ2 and NGK2, is due to low dimensional nonlinear dynamics, which include deterministic chaos with correlation dimension D2(NGK2) = 2.25 ± 0.05 and D2(NDK2) = 2.02 ± 0.03, and with positive Lyapunov exponents λmax (LAQ2) = 0.055 ± 0.003 bits/s and λmax (NGK2) = 0.052 ± 0.003 bits/s; the predictability time in both cases is ≈ 13 s.

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