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Research Papers

Reliable Estimation of Minimum Embedding Dimension Through Statistical Analysis of Nearest Neighbors

[+] Author and Article Information
David Chelidze

Professor
Mem. ASME
Nonlinear Dynamics Laboratory,
Department of Mechanical, Industrial
and Systems Engineering,
University of Rhode Island,
Kingston, RI 02881
e-mail: chelidze@uri.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 10, 2016; final manuscript received May 4, 2017; published online July 12, 2017. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 12(5), 051024 (Jul 12, 2017) (12 pages) Paper No: CND-16-1488; doi: 10.1115/1.4036814 History: Received October 10, 2016; Revised May 04, 2017

False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay-coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Here, nearest-neighbor statistics are studied for uncorrelated random time series and contrasted with the corresponding deterministic and stochastic statistics. New composite FNN metrics are constructed and their performance is evaluated for deterministic, correlates stochastic, and white random time series. In addition, noise-contaminated deterministic data analysis shows that these composite FNN metrics are robust to noise. All FNN results are also contrasted with surrogate data analysis to show their robustness. The new metrics clearly identify random time series as not having a finite embedding dimension and provide information about the deterministic part of correlated stochastic processes. These metrics can also be used to differentiate between chaotic and random time series.

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References

Figures

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Fig. 1

Linear and nonlinear correlations in chaotic and correlated stochastic time series

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Fig. 2

Reconstructed phase portraits for Lorenz (a), Duffing (b), AR(0.02) (c), AR(0.05) (d), AR(0.2) (e), and AR(0.5) (f)

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Fig. 3

Kennel fraction of FNN using Eq. (7) for Lorenz (a), Duffing (b), Normal (c), Uniform (d), AR(0.02) (e), AR(0.05) (f), AR(0.2) (g), and AR(0.5) (h). The gradual decrease in the FNN fraction is precipitated by incrementing r from zero to 20 by two. Lines with circles reflect the corresponding surrogate data results.

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Fig. 4

Reliable FNN (left plots) and reliable FNS (right plots) fraction of Eq. (12) applied to Lorenz (a), Duffing (b), Normal (c), Uniform (d), AR(0.02) (e), AR(0.05) (f), AR(0.2) (g), and AR(0.5) (h). The gradual decrease in the FNN fraction is precipitated by incrementing s from zero to two by 0.1. Lines with circles reflect the corresponding surrogate data results.

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Fig. 5

Reliable FNN (first column: (a), (c), (e), and (g)) and FNS (second column: (b), (d), (f), and (h)) algorithm applied to Duffing data with 5% ((a) and (b)), 10% ((c) and (d)), 20% ((e) and (f)), and 40% ((g) and (h)) noise. The gradual decrease in the fractions is precipitated by incrementing s from zero to two by 0.1. Lines with circles reflect the corresponding surrogate data results.

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Fig. 6

Expected value of nearest-neighbor distances in d dimensions without (a) and with (b) temporarily decorrelated NNs

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Fig. 7

Expected value of distances between the (d + 1)th coordinates of the NNs in d dimensions without (a) and with (b) temporarily decorrelated NNs

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Fig. 8

Expected NN distance ⟨δi⟩ versus the embedding dimension d for noisy Lorenz (a) and Duffing (b) time-series

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Fig. 9

Expected value of the Kennel ratio in d dimensions without (a) and with (b) temporarily decorrelated NNs

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Fig. 10

Equation (20)-based composite fraction FNN algorithm with s = 0.5 (left plots) and with r = 4 (right plots) for Lorenz (a), Duffing (b), Normal (c), Uniform (d), AR(0.02) (e), AR(0.05) (f), AR(0.2) (g), and AR(0.5) (h). The gradual decrease in the fractions is precipitated by incrementing r from zero to 20 by two and s from zero to two by 0.1. Lines with circles reflect the corresponding surrogate data results.

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Fig. 11

Equation (23)-based composite fraction FNS algorithm with s = 0.5 (left plots) and with r = 10 (right plots) applied to Lorenz (a), Duffing (b), Normal (c), Uniform (d), AR(0.02) (e), AR(0.05) (f), AR(0.2) (g), and AR(0.5) (h). The gradual decrease in the fractions is precipitated by incrementing s from zero to two by 0.1 and r from zero to 20 by two. Lines with circles reflect the corresponding surrogate data results.

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Fig. 12

Equation (22)-based FNN algorithms with k = 1 (left plots) and k = 10 (right plots) for Lorenz (first and third columns) and Duffing (second and fourth columns) and the surrogate data with 5% ((a) and (b)), 10% ((c) and (d)), 20% ((e) and (f)), 40% ((g) and (h)) additive noise levels. Lines with circles reflect the corresponding surrogate data results.

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Fig. 13

Equation (23)-based FNS algorithms for Lorenz (left column) and Duffing (right column) and the surrogate data with 5% ((a) and (b)), 10% ((c) and (d)), 20% ((e) and (f)), 40% ((g) and (h)) additive noise levels. Lines with circles reflect the corresponding surrogate data results.

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