Practical implementation of nonlinear time series methods: The TISEAN package

Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at http://www.mpipks-dresden.mpg.de/~tisea

The human ECG - nonlinear deterministic versus stochastic aspects

We discuss aspects of randomness and of determinism in electrocardiographic signals. In particular, we take a critical look at attempts to apply methods of nonlinear time series analysis derived from the theory of deterministic dynamical systems. We will argue that deterministic chaos is not a likely explanation for the short time variablity of the inter-beat interval times, except for certain pathologies. Conversely, densely sampled full ECG recordings possess properties typical of deterministic signals. In the latter case, methods of deterministic nonlinear time series analysis can yield new insights.Comment: 6 pages, 9 PS figure

Nonlinear projective filtering in a data stream

We introduce a modified algorithm to perform nonlinear filtering of a time series by locally linear phase space projections. Unlike previous implementations, the algorithm can be used not only for a posteriori processing but includes the possibility to perform real time filtering in a data stream. The data base that represents the phase space structure generated by the data is updated dynamically. This also allows filtering of non-stationary signals and dynamic parameter adjustment. We discuss exemplary applications, including the real time extraction of the fetal electrocardiogram from abdominal recordings.Comment: 8 page

Training Echo State Networks with Regularization through Dimensionality Reduction

In this paper we introduce a new framework to train an Echo State Network to predict real valued time-series. The method consists in projecting the output of the internal layer of the network on a space with lower dimensionality, before training the output layer to learn the target task. Notably, we enforce a regularization constraint that leads to better generalization capabilities. We evaluate the performances of our approach on several benchmark tests, using different techniques to train the readout of the network, achieving superior predictive performance when using the proposed framework. Finally, we provide an insight on the effectiveness of the implemented mechanics through a visualization of the trajectory in the phase space and relying on the methodologies of nonlinear time-series analysis. By applying our method on well known chaotic systems, we provide evidence that the lower dimensional embedding retains the dynamical properties of the underlying system better than the full-dimensional internal states of the network

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Mathematical tools for identifying the fetal response to physical exercise during pregnancy

In the applied mathematics literature there exist a significant number of tools that can reveal the interaction between mother and fetus during rest and also during and after exercise. These tools are based on techniques from a number of areas such as signal processing, time series analysis, neural networks, heart rate variability as well as dynamical systems and chaos. We will briefly review here some of these methods, concentrating on a method of extracting the fetal heart rate from the mixed maternal-fetal heart rate signal, that is based on phase space reconstructio