SINGULARITY SPECTRUM OF NONPERIODIC TIME SERIES: SURROGATE DATA AND WAVELET TRANSFORM

Fractals ◽  
2000 ◽  
Vol 08 (02) ◽  
pp. 129-137 ◽  
Author(s):  
M. NICOLLET ◽  
A. LEMARCHAND ◽  
G. M. L. DUMAS
Keyword(s):  
Time Series ◽  
Wavelet Transform ◽  
Flame Temperature ◽  
Deterministic Chaos ◽  
Surrogate Data ◽  
Singularity Spectrum ◽  
Cool Flame

We study the singularities of time series giving the evolution of the cool flame temperature of a hydrocarbon. Several tests using surrogate data and wavelet transform are applied to the original signals. The results suggest that the hypothesis of deterministic chaos cannot be ruled out in the original signals. The multifractal organization of their local singularities is proven.

Wavelets Transform of a Multifractal Measure in a Cool Flame Experiment

Fractals ◽  
1997 ◽  
Vol 05 (01) ◽  
pp. 35-46 ◽  
Author(s):  
M. Nicollet ◽  
A. Lemarchand ◽  
G. M. L. Dumas
Keyword(s):  
Wavelet Transform ◽  
Low Temperature ◽  
Flame Propagation ◽  
Flame Temperature ◽  
Singularity Spectrum ◽  
Cool Flame ◽  
Localization Domain ◽  
Wavelets Transform

In order to characterize the chaotic variations of the cool flame temperature observed during the oxidation of a hydrocarbon at low temperature and under non-homogeneous conditions, we perform multifractal analyses of different measures. In the cool flame localization domain, the singularity spectrum obtained for the visited temperature histogram is comparable to the spectrum deduced from a wavelet transform. In the cool flame propagation domain where the temperature histogram is too narrow to be analyzed, the wavelet transform allows us to prove the multifractal character of the chaos observed. The choice of the parameter value retained to perform the wavelet transform is discussed in detail.

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.

scholarly journals Cross-Scale Causality and Information Transfer in Simulated Epileptic Seizures

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 526
Author(s):  
Kajari Gupta ◽  
Milan Paluš
Keyword(s):  
Time Series ◽  
Wavelet Transform ◽  
Time Scales ◽  
Information Transfer ◽  
Epileptic Seizures ◽  
Surrogate Data ◽  
Theoretic Approach ◽  
Phase Amplitude ◽  
Information Theoretic Approach ◽  
Different Time Scales

An information-theoretic approach for detecting causality and information transfer was applied to phases and amplitudes of oscillatory components related to different time scales and obtained using the wavelet transform from a time series generated by the Epileptor model. Three main time scales and their causal interactions were identified in the simulated epileptic seizures, in agreement with the interactions of the model variables. An approach consisting of wavelet transform, conditional mutual information estimation, and surrogate data testing applied to a single time series generated by the model was demonstrated to be successful in the identification of all directional (causal) interactions between the three different time scales described in the model. Thus, the methodology was prepared for the identification of causal cross-frequency phase–phase and phase–amplitude interactions in experimental and clinical neural data.

EVIDENCE CONSISTENT WITH DETERMINISTIC CHAOS IN HUMAN CARDIAC DATA: SURROGATE AND NONLINEAR DYNAMICAL MODELING

2008 ◽  
Vol 18 (01) ◽  
pp. 141-160 ◽  
Author(s):  
YI ZHAO ◽  
JUNFENG SUN ◽  
MICHAEL SMALL
Keyword(s):  
Time Series ◽  
Nonlinear Modeling ◽  
Deterministic Chaos ◽  
Surrogate Data ◽  
Sufficient Evidence ◽  
Nonlinear Dynamical ◽  
Deterministic Processes ◽  
Cardiac System ◽  
Short Term Prediction ◽  
Pressure Propagation

Whether or not the human cardiac system is chaotic has long been a subject of interest in the application of nonlinear time series analysis. The surrogate data method, which identifies an observed time series against three common kinds of hypotheses, does not provide sufficient evidence to confirm the existence of deterministic chaotic dynamics in cardiac time series, such as electrocardiogram data and pulse pressure propagation data. Moreover, these methods fail to exclude all but the most trivial hypothesis of linear noise. We present a recently suggested fourth algorithm for testing the hypotheses of a noise driven periodic orbit to decide whether these signals are consistent with deterministic chaos. Of course, we cannot exclude all other alternatives but our test is certainly stronger than the those applied previously. The algorithmic complexity is used as the discriminating statistic of the surrogate data method. We then perform nonlinear modeling for the short-term prediction between ECG and pulse data to provide further evidence that they conform to deterministic processes. We demonstrate the application of these methods to human electrocardiogram recordings and blood pressure propagation in the fingertip of seven healthy subjects. Our results indicate that bounded aperiodic determinism exists in both ECG and pulse time series. The addition of (the inevitable) dynamic noise means that it is not possible to conclude the underlying system is chaotic.

Lyapunov Spectral Analysis on Random Data

1997 ◽  
Vol 07 (06) ◽  
pp. 1267-1282 ◽  
Author(s):  
Tohru Ikeguchi ◽  
Kazuyuki Aihara
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Nonlinear Dynamical Systems ◽  
Deterministic Chaos ◽  
Surrogate Data ◽  
Random Time ◽  
Statistical Hypothesis ◽  
Largest Lyapunov Exponent ◽  
Lyapunov Spectra

In this paper, an algorithm for estimating all Lyapunov exponents, or Lyapunov spectra of deterministic dynamical systems, is applied to random time series, that is, the time intervals of gamma ray emissions of cobalt. The algorithm used in this paper is that proposed by Sano and Sawada, which estimates Jacobian matrices of an assumed dynamical system from the data points on a possible attractor reconstructed. As a result, the largest Lyapunov exponent of the cobalt data is estimated to be positive, the same as the case of deterministic chaos. Although the sum of all Lyapunov exponents is positive in lower reconstructed dimensions, the values are estimated to be negative when the reconstructed dimension becomes higher. This result is also the same as deterministic chaos in dissipative dynamical systems. It is an indication that naive application of estimating algorithm for Lyapunov spectra to real data does not necessarily lead us to correct results. In order to check the appearance of positive Lyapunov exponents, firstly an analysis by the local versus global (LVG) plots is adopted. The results by LVG plots on the cobalt data are clearly different from those on typical nonlinear dynamical systems in lower dimensional reconstructed state space. Secondly, a direct approach for estimating largest Lyapunov exponents is applied. The results show that estimated Lyapunov exponents on cobalt data do not necessarily indicate existence of positive values. Lastly, a test of statistical hypothesis with surrogate data is also applied to the cobalt data under the null hypotheses that the cobalt data is produced from a linear stochastic system. The results show that statistical significances between the original and surrogate data are so small that some null hypotheses on the cobalt data cannot be rejected; the results imply that an analysis with surrogate data sets could avoid spurious interpretation that random time series, such as the cobalt data, has low-dimensional nonlinear deterministic dynamics with positive Lyapunov exponents.

scholarly journals Superiority of Hybrid Soft Computing Models in Daily Suspended Sediment Estimation in Highly Dynamic Rivers

2021 ◽  
Vol 13 (2) ◽  
pp. 542
Author(s):  
Tarate Suryakant Bajirao ◽  
Pravendra Kumar ◽  
Manish Kumar ◽  
Ahmed Elbeltagi ◽  
Alban Kuriqi
Keyword(s):  
Time Series ◽  
Wavelet Transform ◽  
Suspended Sediment ◽  
River Basin ◽  
Time Series Data ◽  
Series Data ◽  
Discrete Wavelet ◽  
Original Time Series ◽  
Inference System ◽  
Original Time

Estimating sediment flow rate from a drainage area plays an essential role in better watershed planning and management. In this study, the validity of simple and wavelet-coupled Artificial Intelligence (AI) models was analyzed for daily Suspended Sediment (SSC) estimation of highly dynamic Koyna River basin of India. Simple AI models such as the Artificial Neural Network (ANN) and Adaptive Neuro-Fuzzy Inference System (ANFIS) were developed by supplying the original time series data as an input without pre-processing through a Wavelet (W) transform. The hybrid wavelet-coupled W-ANN and W-ANFIS models were developed by supplying the decomposed time series sub-signals using Discrete Wavelet Transform (DWT). In total, three mother wavelets, namely Haar, Daubechies, and Coiflets were employed to decompose original time series data into different multi-frequency sub-signals at an appropriate decomposition level. Quantitative and qualitative performance evaluation criteria were used to select the best model for daily SSC estimation. The reliability of the developed models was also assessed using uncertainty analysis. Finally, it was revealed that the data pre-processing using wavelet transform improves the model’s predictive efficiency and reliability significantly. In this study, it was observed that the performance of the Coiflet wavelet-coupled ANFIS model is superior to other models and can be applied for daily SSC estimation of the highly dynamic rivers. As per sensitivity analysis, previous one-day SSC (St-1) is the most crucial input variable for daily SSC estimation of the Koyna River basin.

scholarly journals Nonlinear analysis of magnetospheric data Part I. Geometric characteristics of the AE index time series and comparison with nonlinear surrogate data

1999 ◽  
Vol 6 (1) ◽  
pp. 51-65 ◽  
Author(s):  
G. P. Pavlos ◽  
M. A. Athanasiu ◽  
D. Kugiumtzis ◽  
N. Hatzigeorgiu ◽  
A. G. Rigas ◽  
...  
Keyword(s):  
Time Series ◽  
State Space ◽  
Null Hypothesis ◽  
Surrogate Data ◽  
Statistical Testing ◽  
Geometrical Parameters ◽  
Geometric Characteristics ◽  
Geometrical Characteristics ◽  
Ae Index ◽  
Index Time Series

Abstract. A long AE index time series is used as a crucial magnetospheric quantity in order to study the underlying dynainics. For this purpose we utilize methods of nonlinear and chaotic analysis of time series. Two basic components of this analysis are the reconstruction of the experimental tiine series state space trajectory of the underlying process and the statistical testing of an null hypothesis. The null hypothesis against which the experimental time series are tested is that the observed AE index signal is generated by a linear stochastic signal possibly perturbed by a static nonlinear distortion. As dis ' ' ating statistics we use geometrical characteristics of the reconstructed state space (Part I, which is the work of this paper) and dynamical characteristics (Part II, which is the work a separate paper), and "nonlinear" surrogate data, generated by two different techniques which can mimic the original (AE index) signal. lie null hypothesis is tested for geometrical characteristics which are the dimension of the reconstructed trajectory and some new geometrical parameters introduced in this work for the efficient discrimination between the nonlinear stochastic surrogate data and the AE index. Finally, the estimated geometric characteristics of the magnetospheric AE index present new evidence about the nonlinear and low dimensional character of the underlying magnetospheric dynamics for the AE index.

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