On the Characterisation of the Deterministic/Stochastic and Linear/Nonlinear Nature of Time Series

Most statistical signal nonlinearity analyses adopt the Monte-Carlo approach proposed by Theiler and co-workers, namely the ‘surrogate data’ method. A surrogate time series, or ‘surrogate’ for short, is generated as a realisation of the null hypothesis of linearity. A measure (‘test statistic’) is computed for the original time series and it is compared to those computed for an ensemble of surrogates. If the test statistic computed for the original is significantly different from that computed for the surrogates, the null hypothesis is rejected, and the original time series is judged nonlinear. One of the key issues in signal nonlinearity analysis is the definition of a linear signal. The standard definition is that such a signal is generated by a Gaussian linear stochastic process. This definition, however, is very stringent. Indeed, if a linear signal were to be measured via a zero-memory, nonlinear observation function, or if the driving noise were not Gaussian, the test for linearity would fail, and the signal would be interpreted as nonlinear. Therefore, we extend the definition of linearity to incorporate these uninteresting (Theiler et al., 1992) deviations in the null hypothesis and, consequently, the method for generating the surrogate data. We propose a novel method for characterising a time series, the ‘Delay Vector Variance’ (DVV) method, from which a novel test statistic can be derived. It is shown that, in the context of surrogate data testing, it outperforms a number of established nonlinearity measures. It is based upon the local unpredictability of a time series, which is analysed in a standardised manner, and allows both for a straightforward visualisation, and for a quantitative measure of the nonlinearities present in a time series. A comprehensive comparative study is performed with other nonlinearity analysis methods in a systematic manner and for a wide variety of benchmark time series.