Determining the Minimum Embedding Dimensions of Input–Output Time Series Data

Cao, Liangyue; Mees, A. I.; Judd, Kevin; Froyland, Gary

doi:10.1142/s0218127498001145

Cited by 31 publications

(14 citation statements)

References 64 publications

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“…In some situations when the predictability of the system is very low, but the system is still deterministic (with a high Lyapunov exponent), the method may find it difficult to distinguish such a system from a stochastic one (CAO et al, 1998). However, for our purposes, the very low predictability is already an important result.…”

Section: Results Of the Nonlinear Time Series Analysismentioning

confidence: 99%

“…However, the typical problems of this approach are that it is often very data intensive, certainly subjective, and time-consuming for computation (CAO, 1997). Therefore, we choose to apply the method of false neighbors (KENNEL et al, 1992), in an improved form (CAO, 1997, CAO et al, 1998. The method (CAO, 1997) first defines: aði; dÞ ¼ y i ðd þ 1Þ À y nði;dÞ ðd þ 1Þ y i ðdÞ À y nði;dÞ ðdÞ ; i ¼ 1; 2; .…”

Section: Methods Of Analysismentioning

confidence: 99%

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Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

Enescu

Itō

Rădulian

et al. 2005

Pure appl. geophys.

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The Vrancea seismic region contains an isolated cluster of events beneath the Carpathian Arc Bend in Romania, dipping to about 200 km depth. Seismic activity mainly occurs at intermediate depths (h > 60 km). The main goal of the paper is to perform an in-depth, complex analysis of the occurrence times of these intermediate-depth events. We also try to show the versatility of the methods used to characterize different aspects of the seismicity evolution and to offer a user-friendly software toolbox to do most of the related computations. The earthquake catalog used in this study spans from 1974 to 2002 and includes only the intermediate-depth events. In the first part of the paper, we analyze the multifractal characteristics of the temporal distribution of earthquakes. The study reveals two distinct scaling regimes. At small scales we found a clear nonhomogeneous, multifractal pattern, while at large scales the temporal distribution of events shows a monofractal, and close to Poissonian (random), behavior. The multifractal behavior at small scales (minutes-hours) is shown to be clearly an effect of the ''short'' aftershock sequences that occurred after some major Vrancea earthquakes. In the second part of the paper we analyze whether our temporal series shows a persistent (or anti-persistent) long-term behavior, by using the Detrended Fluctuation Analysis (DFA) method. The results suggest that the analyzed temporal series of Vrancea earthquakes is a non-correlated process. In part three of the paper we seek to determine whether the dynamics of our earthquake system (described by the occurrence time of Vrancea earthquakes) is deterministically chaotic, deriving from a rather simple evolution law, or whether it is stochastic and is generated by a system that possesses many degrees of freedom. The results suggest that our signal is stochastic (probably does not possess an attractor). The limited time-span of the catalog and the analysis performed in this paper cannot rule out the emergence of an interesting, quasideterministic and low-dimensional structure in the case of major Vrancea earthquakes.

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Section: Results Of the Nonlinear Time Series Analysismentioning

confidence: 99%

Section: Methods Of Analysismentioning

confidence: 99%

Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

Enescu

Itō

Rădulian

et al. 2005

Pure appl. geophys.

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“…Equivalently, h i → h i+1 defines the dynamics in an input-output time delay space (see [9,10]). The correlation entropy K 2 is a lower bound for the Kolmogorov-Sinai entropy of a dynamical system [11], which can be calculated from correlation sums.…”

Section: Correlation Entropy Estimated From Input-output Time Seriesmentioning

confidence: 99%

Correlation entropy of synaptic input-output dynamics

Kleppe

Robinson

2006

Phys. Rev. E

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The responses of synapses in the neocortex show highly stochastic and nonlinear behavior. The microscopic dynamics underlying this behavior, and its computational consequences during natural patterns of synaptic input, are not explained by conventional macroscopic models of deterministic ensemble mean dynamics. Here, we introduce the correlation entropy of the synaptic input-output map as a measure of synaptic reliability which explicitly includes the microscopic dynamics. Applying this to experimental data, we find that cortical synapses show a low-dimensional chaos driven by the natural input pattern.

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“…Later, a number of works discussed the theoretical foundations of the delay embedding of the input-output time series (Casdagli, 1993;Stark et al, 1996). This led to the generalizations of the existing method for the case of non-autonomous dynamical systems (Rhodes and Morari, 1997;Cao et al, 1998). Recently, considerable attention was drawn to the embedding analyses of the time series generated by random dynamical systems (Muldoon et al, 1998;Stark, 2001).…”

Section: Estimating the Embedding Dimensionmentioning

confidence: 99%

“…In the original false nearest neighbors method (Kennel et al, 1992), as well as in its generalized versions (Cao, 1997;Rhodes and Morari, 1997;Cao et al, 1998), this is done by examining whether a given state of the system and the state which was identified as its closest neighbor are nearest neighbors by virtue of the projection into a low dimensional space that was too low. The minimum dimension is assessed by examining all points of the attractor in dimension one, then dimension two, etc., until there are no incorrect or false neighbors remaining.…”

Section: Estimating the Embedding Dimensionmentioning

confidence: 99%

Combining global and multi-scale features in a description of the solar wind-magnetosphere coupling

et al. 2003

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Abstract. The solar wind-magnetosphere coupling during substorms exhibits dynamical features in a wide range of spatial and temporal scales. The goal of our work is to combine the global and multi-scale description of magnetospheric dynamics in a unified data-derived model. For this purpose we use deterministic methods of nonlinear dynamics, together with a probabilistic approach of statistical physics. In this paper we discuss the mathematical aspects of such a combined analysis. In particular we introduce a new method of embedding analysis based on the notion of a mean-field dimension. For a given level of averaging in the system the mean-filed dimension determines the minimum dimension of the embedding space in which the averaged dynamical system approximates the actual dynamics with the given accuracy. This new technique is first tested on a number of wellknown autonomous and open dynamical systems with and without noise contamination. Then, the dimension analysis is carried out for the correlated solar wind-magnetosphere database using vB S time series as the input and AL index as the output of the system. It is found that the minimum embedding dimension of vB S − AL time series is a function of the level of ensemble averaging and the specified accuracy of the method. To extract the global component from the observed time series the ensemble averaging is carried out over the range of scales populated by a high dimensional multiscale constituent. The wider the range of scales which are smoothed away, the smaller the mean-field dimension of the system. The method also yields a probability density function in the reconstructed phase space which provides the basis for the probabilistic modeling of the multi-scale dynamical features, and is also used to visualize the global portion of the solar wind-magnetosphere coupling. The structure of its input-output phase portrait reveals the existence of two energy levels in the system with non-equilibrium dynamical features such as hysteresis which are typical for non-equilibrium phase transitions. Further improvements in space weather forecasting tools may be achieved by a combiCorrespondence to: A. Y. Ukhorskiy (sasha@astro.umd.edu) nation of the dynamical description for the global component and a statistical approach for the multi-scale component.

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Determining the Minimum Embedding Dimensions of Input–Output Time Series Data

Cited by 31 publications

References 64 publications

Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

Correlation entropy of synaptic input-output dynamics

Combining global and multi-scale features in a description of the solar wind-magnetosphere coupling

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