Characterising nonlinear determinism from experimental time series data

Abstract

This thesis includes two parts. In the first part, we will mainly introduce Takens' embedding theorem and a few nonlinear statistics which will be utilized throughout this thesis. Time delay embedding reconstruction based on Takens' embedding theorem is a powerful and vital tool to reconstruct the underlying system from a scalar time series, based on which estimation of invariant measures such as correlation dimension can be performed. In practical situations, however, we need to choose two suitable parameters, i.e., embedding dimension and time delay, to properly apply this technique. We will review some popular criteria on the choice of these two parameters, and describe a new method we have developed to choose suitable time delay for a continuous dynamical system. We compare our algorithm with several available algorithms (for example, the average mutual information criterion) and found that our algorithm has a satisfactory performance, while the implementation is simple and its computational cost is fairly low. Hence this algorithm is suitable to apply to situations such as surrogate tests, where a large amount of data might be used. In the second part, we will introduce the technique of surrogate tests to detect possible nonlinear determinism (or nonlinearity). An irregular time series in practice can be produced either from a stochastic process or from a nonlinear deterministic system. To understand the underlying system of the time series, the first step shall be to investigate whether the irregularity is brought by stochasticity or by nonlinearity (often chaos), only then can corresponding strategies for further analysis be properly applied. In this part, we will mainly introduce surrogate algorithms to detect nonlinear determinism for time series from unknown dynamical systems. To apply surrogate techniques on a time series for nonlinearity detection, we need to adopt a null hypothesis, which usually supposes the time series is generated by a linear stochastic process and potentially filtered by a nonlinear filter. Based on this null hypothesis, a large number of data sets (surrogates) are to be produced from the original time series, which in principle keeps the linearity of the original time series while destroying all other structures. We then calculate some nonlinear statistics (discriminating statistics), for example, correlation dimension, of both the original time series and the surrogates. If the discriminating statistics of the original time series deviate from those of the surrogates, then we can reject the null hypothesis we proposed and claim that the original time series is deterministic with a certain confidence level (depending on how many surrogates we have generated). After the detection of nonlinearity, we are also interested in examining whether the time series is pseudoperiodic or chaotic (This distinction is important in some situations, for example, cardiac disease diagnosis, where heart rate data are believed to be chaotic for healthy patients but indicate regularity for those with congestive heart failure ). We propose a further null hypothesis, i.e., we assume the time series is pseudoperiodic rather than chaotic. We will present a new algorithm to generate surrogates for pseudoperiodic time series, then by choosing the correlation dimension as the discriminating statistic, we can distinguish chaotic time series from pseudoperiodic time series. As an application of the surrogate tests, we will apply this technique to some experimental data sets.