Surrogate Data Preserving All the Properties of Ordinal Patterns up to a Certain Length

A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.

Ordinal methods for a characterization of evolving functional brain networks

Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a resulting loss of information, this approach captures meaningful information about the temporal structure of the underlying system dynamics as well as about properties of interactions between coupled systems. This-together with its conceptual simplicity and robustness against measurement noise-makes ordinal time series analysis well suited to improve characterization of the still poorly understood spatiotemporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni- and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments, which would be necessary to advance characterization of evolving functional brain networks.

Testing for nonlinearity in nonstationary time series: A network-based surrogate data test

The classical surrogate data tests, which are used to differentiate linear noise processes from nonlinear processes, are not suitable for nonstationary time series. In this paper, we propose a surrogate data test that can be applied on both stationary time series as well as nonstationary time series with short-term fluctuations. The method is based on the idea of constructing a network from the time series, employing a generalized symbolic dynamics method introduced in this work, and using any one of the several easily computable network parameters as discriminating statistics. The construction of the network is designed to remove the long-term trends in the data automatically. The network-based test statistics pick up only the short-term variations, unlike the discriminating statistics of the traditional methods, which are influenced by nonstationary trends in the data. The method is tested on several systems generated by linear or nonlinear processes and with deterministic or stochastic trends, and in all cases it is found to be able to differentiate between linear stochastic processes and nonlinear processes quite accurately, especially in cases where the common methods would lead to false rejections of the null hypothesis due to nonstationarity being interpreted as nonlinearity. The method is also found to be robust to the presence of experimental or dynamical noise of a moderate level in an otherwise nonlinear system.

Permutations uniquely identify states and unknown external forces in non-autonomous dynamical systems

It has been shown that a permutation can uniquely identify the joint set of an initial condition and a non-autonomous external force realization added to the deterministic system in given time series data. We demonstrate that our results can be applied to time series forecasting as well as the estimation of common external forces. Thus, permutations provide a convenient description for a time series data set generated by non-autonomous dynamical systems.