Physica D 158 (2001) 32–44 Testing for nonlinearity in high-dimensional time series from continuous dynamics A. Galka a,∗, T. Ozaki b a Institute of Experimental and Applied Physics, University of Kiel, 24098 Kiel, Germany b Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan Received 22 September 2000; received in revised form 12 June 2001; accepted 2 July 2001 Communicated by F.H. Busse Abstract We address the issue of testing for nonlinearity in time series from continuous dynamics and propose a quantitative measure for nonlinearity which is based on discrete parametric modelling. The well-known problems of modelling contin- uous dynamical systems by discrete models are addressed by a subsampling approach. This measure should preferably be combined with conventional surrogate data testing. The performance of the test is demonstrated by application to simu- lated, heavily noise-contaminated time series from high-dimensional Lorenz systems, and to experimental time series from a high-dimensional mode of Taylor–Couette flow. We also discuss the discrimination power of the test under surrogate data testing, when compared with other well-tried test statistics. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Time series analysis; Nonlinearity; Autoregressive modelling; Surrogate data testing 1. Introduction While for many systems the assumption of nonlin- earity may be correct in principle, it has for specific In the last two decades the analysis of time series cases to be shown explicitly that employing nonlinear data obtained from experiments or field observations tools and models is justified and useful. If an exper- has been a field of active research. If such time series imental time series of limited length and finite pre- are generated by complicated systems for which it is cision is given, it may be impossible to distinguish impossible to solve or even set up the equations gov- between nonlinear dynamics and linear dynamics in- erning the dynamics, it is quite commonly assumed volving stochastic components. This has been demon- without further proof that such time series display sig- strated by Casdagli et al. [1] for the closely related nificant nonlinearity. Consequently the analysis is car- case of high-dimensional determinism. ried out by advanced numerical algorithms borrowed It is for this reason that tests for nonlinearity are im- from nonlinear dynamics, whereas the repertory of portant tools in time series analysis. Currently the tech- more traditional linear tools is largely neglected. nique of surrogate data testing [2] is one of the most popular approaches to nonlinearity testing. Being mo- tivated by statistical hypothesis testing, this technique ∗ Corresponding author. presents an indirect way of detecting nonlinearity; as E-mail address: [email protected] (A. Galka). a consequence of this a failure to detect nonlinear- 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S0167-2789(01)00318-9 A. Galka, T. Ozaki / Physica D 158 (2001) 32–44 33 ity does not disprove nonlinearity, but may also result was possibly distorted by a monotonic static nonlin- from an inappropriate choice of the test statistic. Re- ear transform (i.e. a filter changing the distribution of cently more weak points of surrogate data testing were amplitudes, but not involving previous values of the observed, first and foremost the effect that a rejection time series). Then a set of surrogate time series is of the null hypothesis does not necessarily prove non- generated which corresponds to this null hypothesis. linearity, but still may be a consequence of other prop- In this paper we use the iterative amplitude-adjusted erties of the time series such as non-stationarity [3]. phase randomisation (IAAPR) algorithm proposed by There are also considerable problems with artefacts Schreiber and Schmitz [8] for the generation of surro- occurring in the process of generating the surrogate gate time series. This algorithm employs the Fourier data sets [4,5]. expansion as a non-parametric linear model of the For these reasons it would be desirable to design original time series, and aims at simultaneously pre- alternative, possibly more direct, tests for nonlinearity, serving the power spectrum and the amplitude dis- which avoid the generation of surrogate data sets. Such tribution of the original time series, while nonlinear tests exist, e.g. based on the bispectrum [6], but when properties are destroyed by phase randomisation. Fol- applied to nonlinear deterministic systems they tend lowing the results of Theiler and Prichard [9] the pe- to display rather poor performance. riodogram is employed as an estimator of the power In this paper we design a simple but efficient test spectrum (constrained-realisation approach) instead statistic for nonlinearity based on discrete paramet- of a consistent estimator. ric modelling, which provides in itself a meaningful Then a suitable numerical algorithm (the test statis- measure of nonlinearity. The technique of surrogate tic) is applied to each time series from the set of all data testing is retained only as a tool for investigating surrogate time series and the original time series; see the significance of the results of the test statistic, and [10,11] for discussions on the issue of choosing a suit- for excluding the possibility of static nonlinear trans- able test statistic. It should be stressed that the test forms. The new test is applied both to simulated time statistic needs not be designed specifically for detect- series from the Lorenz system and to experimental ing nonlinearity, in general any numerical measure time series from Taylor–Couette flow. which can be computed from time series data could We also present results of comparing three dif- be employed. If the result for the original time series ferent test statistics to high-dimensional time series, differs significantly from the distribution of results for contaminated by strong noise components, under the surrogate time series, the null hypothesis can be conventional surrogate data testing. It is shown that rejected. We will usually evaluate the results by sim- depending on the dimension of the underlying dynam- ple ranking, i.e. we will reject the null hypothesis, ical system and the amount and spectral properties of if the result for the original is larger than any result the noise, different test statistics provide the highest from the surrogates. This will work since we employ power of discrimination. only one-sided tests [12]. The more surrogates are em- ployed, the smaller will be the size (i.e. the probability of wrong rejection of the null) of the test. 2. Surrogate data testing In this section we give a very brief summary of 3. Nonlinear autoregressive modelling surrogate data testing; for more detailed presentations see [2,7] and references therein. In order to design a quantitative measure for non- In conventional surrogate data testing a null hy- linearity we start from the assumption that nonlinear pothesis is stated, such as: the time series in ques- time series can be modelled better by nonlinear mod- tion was generated by a stationary linear gaussian els than by linear models. Now the problem arises that stochastic process; in the process of measurement it nonlinearity itself is not a property, but rather the 34 A. Galka, T. Ozaki / Physica D 158 (2001) 32–44 absence of a property. Hence we have to be more spe- The bandwidth parameter h should be chosen such (−x2 /h) cific as to our interpretation of nonlinearity, i.e. we that exp i−1 approaches zero for the largest oc- have to specify a well-defined class of nonlinear mod- curring values of xi−1; therefore for each time series els. Such classes of models can be based on recon- h is estimated by structed state spaces [10,11], but here we prefer to max(x2 ) explore direct parametric modelling of the dynamics, h =− i−1 , c (5) i.e., given the (scalar) time series xi, i = 1,... ,N log (which we assume to have zero mean and unit vari- where c is a small number selected in advance. It has ance), we look for an autoregressive model, been demonstrated that this approach yields consis- tently good estimates for h [15]. We choose log c = xi = f(xi−1,...,xi−p) + i, (1) −30.0 (which is a reasonable choice for unit variance where p is the model order and i represents dynam- time series). ical noise. If f(·) is to be chosen as a linear function, we obtain a classical AR(p) model, p 4. Design of the test statistic xi = aj xi−j + i =: xˆi + i, (2) j=1 If for given model orders p and q the optimal AR(p) or ExpAR(q) models are estimated there will still re- where xˆi denotes the prediction (or conditional mean) main prediction errors giving rise to a non-zero resid- of xi. As an example of nonlinear choices for f(·) ual variance polynomial models have been used for the detection of nonlinearity [5,13]. However, here we choose to N ν = 1 (x −ˆx )2, employ exponential autoregressive (ExpAR) models p N − p i i (6) i=p+ as introduced by Ozaki and Oda [14]: 1 q q −x2 and similarly for ExpAR( ). Based on this residual x = a + b i−1 x + variance, the Akaike Information Criterion (AIC) i j j exp h i−j i j=1 [16,17] is defined as =: xˆi + i, (3) AIC = N log νp + 2(P + 1), (7) where h is a suitably chosen constant bandwidth pa- where P denotes the number of the data-adaptive pa- rameter. In comparison to polynomial models these rameters aj and bj in the model (excluding the mean).