Predicting Spatio-temporal Time Series Using Dimension Reduced Local States

Isensee, Jonas; Datseris, George; Parlitz, Ulrich

doi:10.1007/s00332-019-09588-7

Cited by 16 publications

(13 citation statements)

References 37 publications

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“…In the following, a state space reconstruction v(t) of a single time series s(t) is used to further predict its course. Besides a very recent idea [90] to train neural ordinary differential equations on a reconstructed trajectory, which then allows prediction, several attempts have been published [10][11][12][13][14][15][16] which more or less rely on the same basic idea. For the last vector of the reconstructed trajectory, denoted with a time-index l, v(t l ), a nearest neighbor search is performed.…”

Section: Short Time Prediction Of the Hénon Map Time Seriesmentioning

confidence: 99%

“…The famous embedding theorems of Whitney [1], Mañé [2], and Takens [3] together with their enhancement by Sauer et al [4] allow a high dimensional state space reconstruction from (observed) uni-or multivariate time series. Computing dynamical invariants [5][6][7][8][9] from the observed system, making meaningful predictions even for chaotic or stochastic systems [10][11][12][13][14][15][16], detecting causal interactions [17][18][19] or nonlinear noise reduction algorithms [20,21] all rely explicitly or implicitly on (time delay) embedding [22] the data into a reconstructed state space. Other ideas rather than time delay embedding (TDE) are also possible [22][23][24][25][26][27], but due to its simple use and its proficient outcomes in a range of situations, TDE is by far the most common reconstruction technique.…”

Section: Introductionmentioning

confidence: 99%

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Optimal state space reconstruction via Monte Carlo decision tree search

et al. 2022

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A novel idea for an optimal time delay state space reconstruction from uni- and multivariate time series is presented. The entire embedding process is considered as a game, in which each move corresponds to an embedding cycle and is subject to an evaluation through an objective function. This way the embedding procedure can be modeled as a tree, in which each leaf holds a specific value of the objective function. By using a Monte Carlo ansatz, the proposed algorithm populates the tree with many leafs by computing different possible embedding paths and the final embedding is chosen as that particular path, which ends at the leaf with the lowest achieved value of the objective function. The method aims to prevent getting stuck in a local minimum of the objective function and can be used in a modular way, enabling practitioners to choose a statistic for possible delays in each embedding cycle as well as a suitable objective function themselves. The proposed method guarantees the optimization of the chosen objective function over the parameter space of the delay embedding as long as the tree is sampled sufficiently. As a proof of concept, we demonstrate the superiority of the proposed method over the classical time delay embedding methods using a variety of application examples. We compare recurrence plot-based statistics inferred from reconstructions of a Lorenz-96 system and highlight an improved forecast accuracy for map-like model data as well as for palaeoclimate isotope time series. Finally, we utilize state space reconstruction for the detection of causality and its strength between observables of a gas turbine type thermoacoustic combustor.

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Section: Short Time Prediction Of the Hénon Map Time Seriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal state space reconstruction via Monte Carlo decision tree search

et al. 2022

View full text Add to dashboard Cite

show abstract

“…In the following, a state space reconstruction v(t) of a single time series s(t) is used to further predict its course. Besides a very recent idea [20] to train neural ordinary differential equations on a reconstructed trajectory, which then allows prediction, several attempts have been published [23,15,85,79,50,74,45] which more or less rely on the same basic idea. For the last vector of the reconstructed trajectory, denoted with a time-index l, v(t l ), a nearest neighbor search is performed.…”

Section: Short Time Prediction Of the Hénon Map Time Seriesmentioning

confidence: 99%

“…The famous embedding theorems of Whitney [97], Mañé [64], and Takens [88] together with their enhancement by Sauer et al [81] allow a high dimensional state space reconstruction from (observed) uni-or multivariate time series. Computing dynamical invariants [35,36,41,49,51] from the observed system, making meaningful predictions even for chaotic or stochastic systems [23,15,85,79,50,74,45], detecting causal interactions [86,24,98] or non-linear noise reduction algorithms [52,68] all rely explicitly or implicitly on (time delay) embedding [72] the data into a reconstructed state space. Other ideas rather than time delay embedding (TDE) are also possible [72,8,32,73,65,63], but due to its simple use and its proficient outcomes in a range of situations, TDE is by far the most common reconstruction technique.…”

Section: Introductionmentioning

confidence: 99%

Optimal State Space Reconstruction via Monte Carlo Decision Tree Search

Kraemer

Gelbrecht

Pavithran

et al. 2021

Preprint

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A novel idea for an optimal time delay state space reconstruction from uni- and multivariate time series is presented. The entire embedding process is considered as a game, in which each move corresponds to an embedding cycle and is subject to an evaluation through an objective function. This way the embedding procedure can be modeled as a tree, in which each leaf holds a specific value of the objective function. By using a Monte Carlo ansatz the proposed algorithm populates the tree with many leafs by computing different possible embedding paths and the final embedding is chosen as that particular path, which ends at the leaf with the lowest achieved value of the objective function. The method aims to prevent getting stuck in a local minimum of the objective function and can be used in a modular way, enabling practitioners to choose a statistic for possible delays in each embedding cycle as well as a suitable objective function themselves. As a proof of concept, we demonstrate the superiority of the proposed method over the classical time delay embedding methods using a variety of application examples. We compare recurrence plot based statistics inferred from reconstructions of a Lorenz-96 system and highlight an improved forecast accuracy for map-like model data as well as for paleoclimate isotope time series. Finally we utilize state space reconstruction for the detection of causality and its strength between observables of a gas turbine type thermoacoustic combustor.

show abstract

“…In these "big data" days, a signi cant number of researchers are focusing on developing novel methods for time series forecasting based on machine learning algorithms. [10][11][12][13][14] Delay embedding and recurrent neural networks have been used to predict the evolution of chaotic systems such as the Lorenz system and the Mackey-Glass system. 15 Locally linear neurofuzzy models 16 and support vector machine 17 have also been used to forecast chaotic signals.…”

Section: Introductionmentioning

confidence: 99%