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.

The Chew Bahir Drilling Project (CBDP) aims to test possible linkages between climate and evolution in Africa through the analysis of sediment cores that have recorded environmental changes in the Chew Bahir basin. In this statistical project we consider the Chew Bahir palaeolake to be a dynamical system consisting of interactions between its different components, such as the waterbody, the sediment beneath lake, and the organisms living within and around the lake. Recurrence is a common feature of such dynamical systems, with recurring patterns in the state of the system reflecting typical influences. Identifying and defining these influences contributes significantly to our understanding of the dynamics of the system. Different recurring changes in precipitation, evaporation, and wind speed in the Chew Bahir basin could result in similar (but not identical) conditions in the lake (e.g., depth and area of the lake, alkalinity and salinity of the lake water, species assemblages in the water body, and diagenesis in the sediments). Recurrence plots (RPs) are graphic displays of such recurring states within a system. Measures of complexity were subsequently introduced to complement the visual inspection of recurrence plots, and provide quantitative descriptions for use in recurrence quantification analysis (RQA). We present and discuss herein results from an RQA on the environmental record from six short (<17 m) sediment cores collected during the CBDP, spanning the last 45 kyrs. The different types of variability and transitions in these records were classified to improve our understanding of the response of the biosphere to climate change, and especially the response of humans in the area.

We present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a symbiotic fashion. Using non-uniform delays allows the successful reconstruction of systems inheriting different time scales. In contrast to the established methods, the minimization of an appropriate cost function determines the embedding dimension without using a threshold parameter. Moreover, the method is capable of detecting stochastic time series and, thus, can handle noise contaminated input without adjusting parameters. The superiority of the proposed method is shown on some paradigmatic models and experimental data from chaotic chemical oscillators.

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The development of new approaches to detect motor-related brain activity is key in many aspects of science, especially in brain-computer interface (BCI) applications. Even though some well-known features of motor-related electroencephalograms (EEGs) have been revealed using traditionally applied methods, they still lack a robust classification of motor-related patterns. Here we introduce new features of motor-related brain activity and uncover hidden mechanisms of the underlying neuronal dynamics by considering event-related desynchronization (ERD) of µ-rhythm in the sensorimotor cortex, i.e. tracking the decrease of the power spectral density in the corresponding frequency band. We hypothesize that motor-related ERD is associated with the suppression of random fluctuations of µ-band neuronal activity. This is due to a lowering of the number of active neuronal populations involved in the corresponding oscillation mode. In this case we expect more regular dynamics and a decrease in complexity of the EEG signal recorded over the sensorimotor cortex. In order to support this theses we apply measures of signal complexity by means of recurrence quantification analysis (RQA). In particular, we demonstrate that certain RQA quantifiers are very useful to detect the moment of movement onset and therefore are able to classify the laterality of executed movements.

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.

The use of cyclostratigraphy to reconstruct the timing of deposition of lacustrine deposits requires sophisticated tuning techniques that can accommodate continuous long-term changes in sedimentation rates. However, most tuning methods use stationary filters that are unable to take into account such long-term variations in accumulation rates. To overcome this problem we present herein a new multiband wavelet age modeling (MUBAWA) technique that is particularly suitable for such situations and demonstrate its use on a 293 m composite core from the Chew Bahir basin, southern Ethiopian rift. In contrast to traditional tuning methods, which use a single, defined bandpass filter, the new method uses an adaptive bandpass filter that adapts to changes in continuous spatial frequency evolution paths in a wavelet power spectrum, within which the wavelength varies considerably along the length of the core due to continuous changes in long-term sedimentation rates. We first applied the MUBAWA technique to a synthetic data set before then using it to establish an age model for the approximately 293 m long composite core from the Chew Bahir basin. For this we used the 2nd principal component of color reflectance values from the sediment, which showed distinct cycles with wavelengths of 10–15 and of ∼40 m that were probably a result of the influence of orbital cycles. We used six independent 40Ar/39Ar ages from volcanic ash layers within the core to determine an approximate spatial frequency range for the orbital signal. Our results demonstrate that the new wavelet-based age modeling technique can significantly increase the accuracy of tuned age models.