Abstract
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.
Highlights
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 uni- or multivariate time series
We propose a Monte Carlo Decision Tree Search (MCDTS) idea to ensure the reach of a global minimum of a freely selectable objective function Γ, e.g., the L- or false nearest neighbors (FNN)-statistic or any other suitable statistic, which evaluates the goodness of the reconstruction with respect to the task
A novel perspective of the embedding process has been proposed, in which the state space reconstruction from single time series can be treated as a game, in which each move corresponds to an embedding cycle and is subject to an evaluation through an objective function
Summary
N. Marwan Institute of Geosciences, University of Potsdam, 14476 Potsdam, Germany 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 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. 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. We utilize state space reconstruction for the detection of causality and its strength between observables of a gas turbine type thermoacoustic combustor. Keywords State space reconstruction · Embedding · Optimization · Time series analysis · Causality · Prediction · Recurrence analysis
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