Comparative study of embedding methods

Cellucci, Christopher J.; Albano, A. M.; Rapp, Paul E.

doi:10.1103/physreve.67.066210

Cited by 111 publications

(82 citation statements)

References 34 publications

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“…This proposal is supported by Abarbanel (1995). To a limited degree the Fraser-Swinney proposal was confirmed in a recent comparative study of embedding criteria (Cellucci, et al, 2003). It should be noted, however, that this comparison was limited to four criteria and is therefore not definitive.…”

Section: Introductionmentioning

confidence: 84%

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Statistical Validation of Mutual Information Calculations: Comparisons of Alternative Numerical Algorithms

Cellucci¹,

Albano²,

Rapp³

2004

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The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden SPONSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)Bureau of Medicine and Surgery BUMED (Med-02) 2300 E. Street, N.W. SPONSORIMONITOR'S REPORT Washington, DC 20372-5300 NUMBER(S) DN24112612. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited. SUPPLEMENTARY NOTES ABSTRACTGiven two time series X and Y, their mutual information, I(X,Y)=I(Y,X), is the average number of bits of X that can be predicted by measuring Y and vice versa. In the analysis of observational data, calculation of mutual information occurs in three contexts, identification of nonlinear correlation, determination of an optimal sampling interval particularly when embedding data, and in the investigation of causal relationships with directed mutual information. In this report a minimum description length argument is used to determine the optimal number of elements to use when characterizing the distributions of X and Y. However, even when using partitions of the X and Y axis indicated by minimum description length, mutual information calculations performed with a uniform partition of the XY plane can give misleading results. This motivated the construction of an algorithm for calculating mutual information that uses an adaptive partition. This algorithm also incorporates an explicit test of the statistical independence of X and Y in a calculation that returns an assessment of the corresponding null hypothesis. SUBJECT

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Section: Introductionmentioning

confidence: 84%

“…Several candidate criteria for selecting m and Lag have been proposed. An incomplete review of the very large embedding criterion literature is given in Cellucci, et al (2003). Fraser and Swinney (1986) proposed that the best value of Lag to use in an embedding is given by the first minimum of the I(Xi, Xi+Lag) versus Lag function.…”

Section: Introductionmentioning

confidence: 99%

Statistical Validation of Mutual Information Calculations: Comparisons of Alternative Numerical Algorithms

Cellucci¹,

Albano²,

Rapp³

2004

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“…To compare different FNN methods, Cellucci et al [27] have tested them on the Rössler system and the Mackey-Glass equation. Five criteria for selecting embedding parameters have been applied to the observables of the systems.…”

Section: Comparison Of the Fnn Methodsmentioning

confidence: 99%

Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

Krakovská

Mezeiová

Budáčová

2015

Journal of Complex Systems

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If data are generated by a system with a d-dimensional attractor, then Takens' theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in (2 + 1)-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

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“…However, these theorems are existence proofs (Abarbanel et al, 1993) and they do not directly explain how to get a suitable time delay or embedding dimension from a finite time series (Pecora et al, 2007). Implementation of the reconstruction method requires that the time delay and the dimension be determined by using heuristic and/or statistical criteria (for reviews Abarbanel et al, 1993;Schreiber, 1999; see also Cellucci et al, 2003;Letellier et al, 2008). It is important to stress that the embedding dimension required by the embedding theorem is a sufficient condition and could be larger than the necessary embedding dimension (Abarbanel et al, 1993).…”

Section: Introductionmentioning

confidence: 99%

Embedding reconstruction methodology for short time series – application to large El Niño events

Astudillo¹,

Borotto²,

Río

2010

Nonlin. Processes Geophys.

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Abstract.We propose an alternative approach for the embedding space reconstruction method for short time series. An m-dimensional embedding space is reconstructed with a set of time delays including the relevant time scales characterizing the dynamical properties of the system. By using a maximal predictability criterion a d-dimensional subspace is selected with its associated set of time delays, in which a local nonlinear blind forecasting prediction performs the best reconstruction of a particular event of a time series. An locally unfolded d-dimensional embedding space is then obtained. The efficiency of the methodology, which is mathematically consistent with the fundamental definitions of the local nonlinear long time-scale predictability, was tested with a chaotic time series of the Lorenz system. When applied to the Southern Oscillation Index (SOI) (observational data associated with the El Niño-Southern Oscillation phenomena (ENSO)) an optimal set of embedding parameters exists, that allows constructing the main characteristics of the El Niño 1982Niño -1983Niño and 1997Niño -1998 events, directly from measurements up to 3 to 4 years in advance.

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Comparative study of embedding methods

Cited by 111 publications

References 34 publications

Statistical Validation of Mutual Information Calculations: Comparisons of Alternative Numerical Algorithms

Statistical Validation of Mutual Information Calculations: Comparisons of Alternative Numerical Algorithms

Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

Embedding reconstruction methodology for short time series – application to large El Niño events

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