TY - JOUR A1 - Krämer, Hauke Kai A1 - Gelbrecht, Maximilian A1 - Pavithran, Induja A1 - Sujith, Ravindran A1 - Marwan, Norbert T1 - Optimal state space reconstruction via Monte Carlo decision tree search JF - Nonlinear Dynamics N2 - 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. KW - State space reconstruction KW - Embedding KW - Optimization KW - Time series analysis KW - Causality KW - Prediction KW - Recurrence analysis Y1 - 2022 U6 - https://doi.org/10.1007/s11071-022-07280-2 SN - 0924-090X SN - 1573-269X VL - 108 IS - 2 SP - 1525 EP - 1545 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Krämer, Hauke Kai A1 - Marwan, Norbert T1 - Border effect corrections for diagonal line based recurrence quantification analysis measures JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - Recurrence Quantification Analysis (RQA) defines a number of quantifiers, which base upon diagonal line structures in the recurrence plot (RP). Due to the finite size of an RP, these lines can be cut by the borders of the RP and, thus, bias the length distribution of diagonal lines and, consequently, the line based RQA measures. In this letter we investigate the impact of the mentioned border effects and of the thickening of diagonal lines in an RP (caused by tangential motion) on the estimation of the diagonal line length distribution, quantified by its entropy. Although a relation to the Lyapunov spectrum is theoretically expected, the mentioned entropy yields contradictory results in many studies. Here we summarize correction schemes for both, the border effects and the tangential motion and systematically compare them to methods from the literature. We show that these corrections lead to the expected behavior of the diagonal line length entropy, in particular meaning zero values in case of a regular motion and positive values for chaotic motion. Moreover, we test these methods under noisy conditions, in order to supply practical tools for applied statistical research. KW - Recurrence plots KW - Recurrence quantification analysis KW - Shannon entropy KW - Dynamical invariants Y1 - 2019 U6 - https://doi.org/10.1016/j.physleta.2019.125977 SN - 0375-9601 SN - 1873-2429 VL - 383 IS - 34 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Krämer, Hauke Kai A1 - Donner, Reik Volker A1 - Heitzig, Jobst A1 - Marwan, Norbert T1 - Recurrence threshold selection for obtaining robust recurrence characteristics in different embedding dimensions JF - Chaos : an interdisciplinary journal of nonlinear science N2 - The appropriate selection of recurrence thresholds is a key problem in applications of recurrence quantification analysis and related methods across disciplines. Here, we discuss the distribution of pairwise distances between state vectors in the studied system’s state space reconstructed by means of time-delay embedding as the key characteristic that should guide the corresponding choice for obtaining an adequate resolution of a recurrence plot. Specifically, we present an empirical description of the distance distribution, focusing on characteristic changes of its shape with increasing embedding dimension. Our results suggest that selecting the recurrence threshold according to a fixed percentile of this distribution reduces the dependence of recurrence characteristics on the embedding dimension in comparison with other commonly used threshold selection methods. Numerical investigations on some paradigmatic model systems with time-dependent parameters support these empirical findings. Recurrence plots (RPs) provide an intuitive tool for visualizing the (potentially multi-dimensional) trajectory of a dynamical system in state space. In case only univariate observations of the system’s overall state are available, time-delay embedding has become a standard procedure for qualitatively reconstructing the dynamics in state space. The selection of a threshold distance 𝜀 , which distinguishes close from distant pairs of (reconstructed) state vectors, is known to have a substantial impact on the recurrence plot and its quantitative characteristics, but its corresponding interplay with the embedding dimension has not yet been explicitly addressed. Here, we point out that the results of recurrence quantification analysis (RQA) and related methods are qualitatively robust under changes of the (sufficiently high) embedding dimension only if the full distribution of pairwise distances between state vectors is considered for selecting 𝜀, which is achieved by consideration of a fixed recurrence rate. Y1 - 2018 U6 - https://doi.org/10.1063/1.5024914 SN - 1054-1500 SN - 1089-7682 VL - 28 IS - 8 PB - American Institute of Physics CY - Melville ER -