: We compare two theoretically distinct approaches to generating artificial (or "surrogate") data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single "best" model to the original data, and then to generate surrogate data sets that are "typical realizations" of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches...

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

Extensions to various information-theoretic quantities (such as entropy, redundancy, and mutual information) are discussed in the context of their role in nonlinear time series analysis. We also discuss "linearized" versions of these quantities and their use as benchmarks in tests for nonlinearity. Many of these quantities can be expressed in terms of the generalized correlation integral, and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics (such as the BDS and Green-Savit statistics). Further, numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation. Finally, we consider several "local" versions of these quantities, including a local Kolmogorov-Sinai entropy, which gives an estimate of variability of the short-term predictability. 1 Introduction In Shaw's influential (and prize-winning)...

Several authors have recently investigated the predictability of exchange rates by fitting a sequence of long-horizon error-correction/regressions.//We/show/that/in/small/to medium/samples/such/a/procedure/gives/rise/to/spurious/evidence/of/predictive/power./A simulation/study/demonstrates/that/even/when/using/this/technique/on/two/independent/series, estimates/and/diagnostic/statistics/suggest/a/high/degree/of/predictability/of/the/dependent variable./We/apply/a/simple/modification/of/the/long-horizon/regression/due/to/Jegadeesh (1991),/which/may/provide/more/accurate/inference/for/researchers/interested/in/comparing short/and/long-run/predictability of U.S. dollar exchange rates.

In this article we provide a review of the literature with respect to the e cient markets hypothesis and chaos. In doing so, we contrast the martingale behavior of asset prices to nonlinear chaotic dynamics, discuss some recent techniques used in distinguishing between probabilistic and deterministic behavior in asset prices, and report some evidence. Moreover, we look at the controversies that have arisen about the available tests and results, and raise the issue of whether dynamical systems theory is practical in nance.

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

A novel `Delay Vector Variance' (DVV) method for detecting the presence of determinism and nonlinearity in a time series is introduced. The method is based upon the examination of local predictability of a signal. Additionally, it spans the complete range of local linear models due to the standardisation to the distribution of pairwise distances between delay vectors. This provides consistent and easy-tointerpret diagrams, which convey information about the nature of a time series. In Preprint submitted to Elsevier Science 3 April 2002 order to gain further insight into the technique, a DVV scatter diagram is introduced, which plots the DVV curve against that for a globally linear model (surrogate data). This way, the deviation from the bisector line represents a qualitative measure of nonlinearity, which can be used additionally for constructing a quantitative measure for statistical testing. The proposed method is compared to existing methods on synthetic, as well as standard real-world signals, and is shown to provide more consistent results overall, compared to other, established nonlinearity analysis methods.

A single statistical test for nonlinearity can indicate whether or not the generating mechanism of a time series is or is not linear. However, if the null hypothesis of linearity is rejected, the test result conveys little information as to what kind of nonlinear model is appropriate. Here we show that a battery of different nonlinearity tests, in contrast, can yield valuable model identification information. Applying such a battery of tests to data on U.S. real GNP, we are able to conclude that the commonly held notion that this time series is generated by some sort of two-state regime switching process is most likely incorrect.

We show that one can use recently introduced statistics for continuity and differentiability to show the effect of filters of infinite extent in time on a chaotic time series. The statistics point to a discontinuous or nondifferentiable function between the unfiltered attractor and the filtered attractor as the origin of attractor dimension increase when the filtering is severe. The density of discontinuities as a function of resolution follows a scaling relation. We present direct visualization of this effect in the filtered Henon attractor where the origin of dimension increase becomes obvious. PACS NO. 05.45.+b, 47.52.+j, 06.50.-x, 02.40.-k, 02.50.-r Final version submitted to CHAOS 2 Apr 1996. Reference for this paper: L. Pecora and T. Carroll, CHAOS 6, 432-439 (1996). 1. 10:11 AM, January 28, 1998 L. Pecora and , Naval Research Laboratory I. Introduction In this paper we show that recently devised statistics for testing continuity and differentiability [1] can be useful in test...

In this article, we introduce a methodology for comparing the nonlinearities present in sets of time series using four different nonlinearity measures, one of which, the `Delay Vector Variance' method, is a novel approach to the characterisation of a time series. It is then applied to examine the difference in nonlinearity between fMRI signals that have been recorded using different contrast agents. Recently, an exogenous contrast agent (MION) has been introduced for fMRI, which has been shown to increase the functional sensitivity compared to the traditional BOLD technique.