Daniel F. Mccaffrey, and Douglas W. Nychka for helpful discussions relating to Recently, multiple input, single output, single hidden layer, feedforward neural networks have been shown to be capable of approximating a nonlinear map and its partial derivatives. Specifically, neural nets have been shown to be dense in various Sobolev spaces (Hornik, Stinchcombe and White, 1989). Building upon this result, we show that a net can be trained so that the map and its derivatives are learned. Specifically, we use a result of Gallant (1987b) to show that least squares and similar estimates are strongly consistent in Sobolev norm provided the number of hidden units and the size of the training set increase together. We illustrate these results by an applic~tion to the inverse problem of chaotic dynamics: recovery of a nonlinear map from a time series of iterates. These results extend automatically to nets that embed the single hidden layer, feedforward network as a special case. 1.1 1.

In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (1820-1900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.

: 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...

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)...

: A variant of the method of surrogate data is applied to a single time series from an electroencephalogram (EEG) recording of a patient undergoing an epileptic seizure. The time series is a nearly periodic pattern of spike-and-wave complexes. The surrogate data sets are generated by shuffling the the individual spike-and-wave cycles, and correspond to a null hypothesis that there is no deterministic structure in the cycle-to-cycle variability of the original data. Using estimates of autocorrelation, correlation dimension, and Lyapunov exponent as discriminating statistics, the evidence for dynamical correlations between successive spike-andwave patterns is evaluated both formally and informally. Contents: 1) Introduction 2) The Data 3) The Null Hypothesis 4) Analysis 4.1) Autocorrelation 4.2) Correlation dimension 4.3) Lyapunov exponent 4.4) Inter-spike variation 5) Conclusion 1 Introduction Chaos provides an alluring explanation for erratic behavior, because it can be exhibited by...

Certain deterministic non-linear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus improved predictive capabilities. This paper provides a review of two main key features of chaotic systems, the dimensions of their strange attractors and the Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build non-linear models, which is the topic of the second paper in this series.

Quantitative descriptions of a physical system’s behaviour form the backbone of the scientific method used in the natural sciences. They allow the principled determination of experimental parameters, a clear and unambiguous representation of experiments, and independent replication and verification of experimental results. In mobile robotics to date, quantitative descriptions of robot-environment interaction remain the exception, chiefly due to the lack of those descriptions. Instead, qualitative descriptions of experiments and existence proofs (i.e. unvalidated experimental results) are the norm. This paper discusses this problem, and presents a novel method of describing robot-environment interaction quantitatively — a first step towards scientific mobile robotics. The application of this novel method is illustrated on an example taken from mobile robotics: the comparison between a Nomad 200 mobile robot and its computer model.

We study here a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. We also contrast these with a method based on the definition of topological entropy, using (n; ffl)-spanning sets. 1 Introduction and Preliminaries The topological entropy of a system is a quantitative measure of its orbit complexity. In a certain sense, it is the maximum amount of information lost per unit time by the system using measurements with finite precision. As such, the entropy is an important invariant to know. Since the definition of the entropy requires an exponentially growing number of objects, it is impractical to expect that the definition can effectively be used to estimate it. Fortunately, there are several recent theorems which aid in its estimation. Block and Keesling ...

. The problem of extracting a signal xn from a noise-corrupted time series yn = xn + en is considered. The signal xn is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system F -- in particular, xn = F n (x0 ), where the initial point x0 is assumed to lie in a compact hyperbolic F \Gammainvariant set. It is shown that (1) if the noise sequence en is gaussian then it is impossible to consistently recover the signal xn , but (2) if the noise sequence consists of i.i.d. random vectors uniformly bounded by a constant ffi ? 0, then it is possible to recover the signal xn provided ffi ! 5\Delta, where \Delta is a separation threshold for F . A filtering algorithm for the latter situation is presented. 1. Introduction Physical and numerical experiments carried out over the past 30+ years suggest that the phenomenon of deterministic chaos is ubiquitous in physical systems. Experience has shown that inference of the mathematical objects (the differential equati...

We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. In contrast to the usual method of computing the maximal Lyapunov exponent, our method is applied directly to the time series data and does not require phase space reconstruction. Moreover, the dimension of the dynamical system and the form of the underlying equations is irrelevant. The input is the time series data and the output is 0 or 1 depending on whether the dynamics is non-chaotic or chaotic. The test is universally applicable to any deterministic dynamical system, in particular to ordinary and partial differential equations, and to maps. φ(x(s))cos(θ(s))ds where φ is an observable on the underlying dynamics x(t) and θ(t) = ct+ ∫ t 0 φ(x(s))ds. The constant c> 0 is fixed arbitrarily. We define the mean-square-displacement M(t) for p(t) and set K = limt→ ∞ logM(t) / log t. Using recent developments in ergodic theory, we argue that typically K = 0 signifying nonchaotic dynamics or K = 1 signifying chaotic dynamics. Our diagnostic is the real valued function p(t) = ∫ t