Before we apply nonlinear techniques, for example those inspired by chaos theory, to dynamical phenomena occurring in nature, it is necessary to first ask if the use of such advanced techniques is justified by the data. While many processes in nature seem very unlikely a priori to be linear, the possible nonlinear nature might not be evident in specific aspects of their dynamics. The method of surrogate data has become a very popular tool to address such a question. However, while it was meant to provide a statistically rigorous, foolproof framework, some limitations and caveats have shown up in its practical use. In this paper, recent efforts to understand the caveats, avoid the pitfalls, and to overcome some of the limitations, are reviewed and augmented by new material. In particular, we will discuss specific as well as more general approaches to constrained randomisation, providing a full range of examples. New algorithms will be introduced for unevenly sampled and multivariate da...

: This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.

It is important to remain uncertain, of observation, model and law. For the Fermi Summer School, Criticisms Requested email : lenny@maths.ox.ac.uk, Contents 1

A method for testing nonlinearity in time series is described based on information-theoretic functionals -- redundancies, linear and nonlinear forms of which allow either qualitative, or, after incorporating the surrogate data technique, quantitative evaluation of dynamical properties of scrutinized data. An interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated. Evaluation of redundancies and redundancy-based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study. Keywords: time series, nonlinearity, mutual information, redundancy, surrogate data 1 Introduction The problem of inferring the dynamics of a system from measured data is a perpetual challenge for time series analysts. Ideas and concepts from nonlinear dynamics and theory of deterministic chaos have led to a num...

We present techniques of symbolic time-series analysis which are useful for analyzing temporal patterns in dynamic measurements of engine combustion variables. We focus primarily on techniques that characterize predictability and the occurrence of repeating temporal patterns. These methods can be applied to standard, cycle-resolved engine combustion measurements, such as IMEP and heat release. The techniques are especially useful in cases with high levels of measurement and/or dynamic noise. We illustrate their application to experimental data from a production V8 engine and a laboratory single-cylinder engine. MOTIVATION Recent studies have demonstrated that cyclic combustion variations in spark-ignition engines under lean fueling exhibit patterns that can be explained as the result of noisy nonlinear combustion instabilities [1, 2, 3, 4]. These instabilities are dominated by the effects of residual cylinder gas (prior-cycle effects) and noisy perturbations of engine parameters. Beca...

This paper concerns the application of new nonlinear time-series modeling methods to recordings of infant respiratory patterns. The techniques used combine the concept of minimum description length modeling with radial basis models. Our first application of the methods produced results that were not entirely satisfactory, particularly with respect to accurately modeling long term quantitative and qualitative features of respiration patterns. This paper describes a number of modifications of the original methods and makes a comparison of the improvements the various modifications gave. The modifications made were increasing the class of basis function, broadening the range of possible embedding strategies, improving the optimization of the likelihood of the model parameters and calculating a closer approximation to description length. The criteria used in the comparisons were description length, root mean square prediction error, model size, free run behavior and amplitude size and vari...

This report has a much more modest goal: to describe newly developed techniques using "Surrogate Data" to detect nonlinearities and nonstationarities in data. Detecting nonlinearities --- or failing to detect them --- allows us to know when linear analysis techniques are and are not capturing all of the information in the time series. Detecting nonstationarities allows us to make informed decisions about issues such as whether collecting longer runs of data provides better estimates of physiological variables, or about which are the best analysis techniques that can allow us to track changes in the physiological system without unnecessarily increasing the variance of the estimates. 1.1 Nonlinearity

The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than 2 degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.

Currently surrogate data analysis can be used to determine if data is consistent with various linear systems, or something else (a nonlinear system). In this paper we propose an extension of these methods in an attempt to make more specific classifications within the the class of nonlinear systems. In the method of surrogate data one estimates the probability distribution of values of a test statistic for a set of experimental data under the assumption that the data is consistent with a given hypothesis. If the probability distribution of the test statistic is different for different dynamical systems consistent with the hypothesis one must ensure that the surrogate generation technique generates surrogate data that are a good approximation to the data. This is often achieved with a careful choice of surrogate generation method and for noise driven linear surrogates such methods are commonly used. This paper argues that, in many cases (particularly for nonlinear hypotheses), it is ea...

We review a relatively new statistical test that may be applied to determine whether an observed time series is inconsistent with a specific class of dynamical systems. These surrogate data methods may test an observed time series against the hypotheses of: i) independent and identically distributed noise; ii) linearly filtered noise; and iii) a monotonic nonlinear transformation of linearly filtered noise. A recently suggested fourth algorithm for testing the hypothesis of a periodic orbit with uncorrelated noise is also described. We propose several novel applications of these methods for various engineering problems, including: identifying a deterministic (message) signal in a noisy time series; and separating deterministic and stochastic components. When employed to separate deterministic and noise components, we show that the application of surrogate methods to the residuals of nonlinear models is equivalent to fitting that model subject to an information theoretic model selection criteria.