Weighted recurrence network for characterizing continuous dynamical systems

Recurrence network (RN) is a powerful tool for the analysis of complex dynamical systems. It integrates complex network theory with the idea of recurrence of a trajectory, i.e. whether two state vectors are close neighbors in a phase space. However, the differences in proximity between connected state vectors are not considered in the RN construction. Here, we propose a weighted state vector recurrence network method which assigns weights to network links based on the proximity of the two connected state vectors. On the basis, we further propose a weighted data segment recurrence network that takes continuous data segments as nodes for the analysis of noisy time series. The feasibility of the proposed methods is illustrated based on the Lorenz system. Finally, an application to five types of EEG recordings is conducted to demonstrate the potentials of the proposed methods in the study of real-world data.

Abrupt climate change as rate-dependent cascading tipping point

We propose a conceptual model comprising a cascade of tipping points as a mechanism for past abrupt climate changes. In the model, changes in a control parameter, which could for instance be related to changes in the atmospheric circulation, induce sequential tipping of sea-ice cover and the ocean's meridional overturning circulation. The ocean component, represented by the well-known Stommel box model, is shown to display so-called rate-induced tipping. Here, an abrupt resurgence of the overturning circulation is induced before a bifurcation point is reached due to the fast rate of change of the sea-ice. During the rate-induced transition, the system is attracted by the stable manifold of a saddle. This results in a significant delay of the tipping if the system spends longer periods of time in the vicinity of the saddle before escaping towards the alternative state of a vigorous overturning circulation. The delay opens up the possibility for an early warning of the impending abrupt transition by detecting the change in linear stability. We propose early warning by estimating properties of the Jacobian from the noisy time series, which are shown to be useful as a generic precursor to detect rate-induced tipping.

A Comprehensive Framework for Uncovering Non-Linearity and Chaos in Financial Markets: Empirical Evidence for Four Major Stock Market Indices

The presence of chaos in the financial markets has been the subject of a great number of studies, but the results have been contradictory and inconclusive. This research tests for the existence of nonlinear patterns and chaotic nature in four major stock market indices: namely Dow Jones Industrial Average, Ibex 35, Nasdaq-100 and Nikkei 225. To this end, a comprehensive framework has been adopted encompassing a wide range of techniques and the most suitable methods for the analysis of noisy time series. By using daily closing values from January 1992 to July 2013, this study employs twelve techniques and tools of which five are specific to detecting chaos. The findings show no clear evidence of chaos, suggesting that the behavior of financial markets is nonlinear and stochastic.

Reconstructions of model equations of time-delay system from short experimental time series

This study aims to investigate the scope of methods for the reconstruction of time-delay systems. We consider an approach to the reconstruction of time-delay systems based on the synchronous response of the driven system with the structure similar to the structure of the studied object. This approach is used for the recovery of the parameters of time-delay systems from short and noisy time series. To show the operational performance and capabilities of this approach, the parameters were reconstructed for a radiophysical chaotic generator with quadratic nonlinearity and for the model of a system for the baroreflectory regulation of the mean arterial pressure.

Clustering Noisy Time Series

One of the urgent tasks of machine learning is the problem of clustering objects. Clustering time series is used as an independent research technique, as well as part of more complex data mining methods, such as rule detection, classification, anomaly detection, etc.A comparative analysis of clustering noisy time series is carried out. The clustering sample contained time series of various types, among which there were atypical objects. Clustering was performed by k-means and DBSCAN methods using various distance functions for time series.A numerical experiment was conducted to investigate the application of the k-means and DBSCAN methods to model time series with additive white noise. The sample on which clustering was carried out consisted of m time series of various types: harmonic realizations, parabolic realizations, and “bursts”.The work was carried out clustering noisy time series of various types.DBSCAN and k-means methods with different distance functions were used. The best results were shown by the DBSCAN method with the Euclidean metric and the CID function.Analysis of the results of the clustering of time series allows determining the key differences between the methods: if you can determine the number of clusters and you do not need to separate atypical time series, the k-means method shows fairly good results; if there is no information on the number of clusters and there is a problem of isolating non-typical rows, it is advisable to use the DBSCAN method.