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 article, we will describe (yet) another source of difficulty that arises in the analysis of time series data. The particular problem of detecting nonlinear structure --- either by comparison of the data to linear surrogate data, or by comparing linear and nonlinear predictors --- is seen to be complicated when the data exhibits long coherence times. In this section we define some terms and discuss linear modeling of time series. Section 2 describes the method of surrogate data, and compares two approaches to generating surrogate data. We find that both have difficulties trying to mimic data with long coherence time. We illustrate these problems with real and computer-generated time series in Section 3, including the time series E.dat from the the SFI competition. In the last section, we discuss what it is about the analysis or the data that is problematic.

Both academic and applied researchers studying nancial markets and other economic series have become interested in the topic of chaotic dynamics. The possibility ofchaos in nancial markets opens important questions for both economic theorists as well as nancial market participants. This paper will clarify the empirical evidence for chaos in nancial markets and macroeconomic series. It will also compare these two concepts from a nancial market perspective contrasting the objectives of the practitioner with those of economic researchers. Finally, the paper will speculate on the impact of chaos and nonlinear modeling on future economic research. The author is grateful to the Alfred P. Sloan Foundation and the University of Wisconsin Graduate School for It has now been almost ten years since economists began searching for chaotic dynamics in economic time series. This search has yielded deeper understandings of the dynamics of many di erent series, and has led to the development of several useful tests for nonlinear structure. However, the direct evidence for deterministic chaos in many economic series remains weak. This paper will survey the

Copyright c○2003 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher bepress. Determinism in Financial Time Series The attractive possibility that financial indices may be chaotic has been the subject of much study. In this paper we address two specific questions: “Masked by stochasticity, do financial data exhibit deterministic nonlinearity?”, and “If so, so what?”. We examine daily returns from three financial indicators: the Dow Jones Industrial Average, the London gold fixings, and the USD-JPY exchange rate. For each data set we apply surrogate data methods and nonlinearity tests to quantify determinism over a wide range of time scales (from 100 to 20,000 days). We find that all three time series are distinct from linear noise or conditional heteroskedastic models and that there therefore exists detectable deterministic nonlinearity that can potentially be exploited for prediction.

Empirical time series in the life sciences are often non-stationary and have small signalto-noise ratios, making it difficult to accurately detect and characterize dynamical structure. The usual response to high noise is averaging, but time domain averaging is inappropriate, especially when the dynamics are nonlinear. We review alternative delayspace averaging methods based on the topology and short-term predictability of nonlinear dynamics and illustrate their application using the TISEAN software (Hegger, Kantz & Schreiber, 1999). The methods were applied to a Lorenz series, which resembles the dynamics found by Kelly, Heathcote, Heath and Longstaff (2001) in series of decision times. The Lorenz series was corrupted with up to 80 % additive Gaussian noise, a lower signal-to-noise ratio than has been used in any previous test of these methods, but consistent with Kelly et al.’s data. Prediction methods performed the best for detecting nonstationarity and nonlinear dynamics, and optimal predictability provided an objective criterion for setting the parameters required by the analyses. Local linear filtering methods preformed best for characterization, producing informative plots that revealed the nature of the underlying dynamics. These results suggest that a methodology based on

A general method for testing nonlinearity in time series is described and applied to measurements of different pressure data inside the draft tube surge of a real Francis turbine. Comparing the current original time series to an ensemble of surrogates time series, suitably constructed to mimic the linear properties of the original one, we was able to distinguish a linear stochastic from a nonlinear deterministic behaviour and, moreover, to quantify the degree of nonlinearity present in the related dynamics. The problem of detecting nonlinear structure in real data is quite complicated by the influence of various contaminations, like broadband noise and/or long coherence times. These difficulties have been overcame using the combination of a suitable nonlinear filtering technique and a qualitative redundancy statistic analysis. The above investigations allow a quantitative characterization of different dynamical regimes of motion of gas cavities inside real turbines and, moreover, allow to support the reliability of some related mathematical modelizations.