Chaos places no a priori restrictions on predictability: any uncertainty in the initial condition can be evolved and then quanti ed as a function of forecast time. If a speci ed accuracy at a given future time is desired, a perfect model can specify the initial accuracy required to obtain it, and accountable ensemble forecasts can be obtained for each unknown initial condition. Statistics which reect the global properties of in nitesimals, such as Lyapunov exponents which de ne \chaos", limit predictability only in the simplest mathematical examples. Model error, on the other hand, makes forecasting a dubious endeavor. Forecasting with uncertain initial conditions in the perfect model scenario is contrasted with the case where a perfect model is unavailable, perhaps nonexistent. Applications to both low (2 to 400) dimensional models and high (10 7 ) dimensional models are discussed. For real physical systems no perfect model exists; the limitations of nearperfect models are consider...

The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing certain new results on generalised Fourier series and the asymptotic properties of Toeplitz-like matrices in such a way that the new variance expressions presented here encompass pre-existing ones as special cases. Via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification. Namely, that orthonormal bases are much more than an implementational option offering improved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, or...

We present a mobile, GPS-based multimodal navigation system, equipped with inertial control that allows users to explore and navigate through an augmented physical space, incorporating and displaying the uncertainty resulting from inaccurate sensing and unknown user intentions. The system propagates uncertainty appropriately via Monte Carlo sampling and predicts at a usercontrollable time horizon. Control of the Monte Carlo exploration is entirely tilt-based. The system output is displayed both visually and in audio. Audio is rendered via granular synthesis to accurately display the probability of the user reaching targets in the space. We also demonstrate the use of uncertain prediction in a trajectory following task, where a section of music is modulated according to the changing predictions of user position with respect to the target trajectory. We show that appropriate display of the full distribution of potential future users positions with respect to sites-of-interest can improve the quality of interaction over a simplistic interpretation of the sensed data. H5.m. Information interfaces and presentation (e.g., HCI): Miscellaneous,

Statistical properties of chaotic dynamical systems are difficult to estimate reliably. Using long trajectories as data sets sometimes produces misleading results. It has been recognised for some time that statistical properties are often stable under the addition of a small amount of noise. Rather than analysing the dynamical system directly, we slightly perturb it to create a Markov model. The analogous statistical properties of the Markov model often have "closed forms" and are easily computed numerically. The Markov construction is observed to provide extremely robust estimates and has the theoretical advantage of allowing one to prove convergence in the noise → 0 limit and produce rigorous error bounds for statistical quantities. We review the latest results and techniques in this area.

This paper investigates the accuracy of the linear component that forms part of an overall Hammerstein model-structure estimate, and a key finding is that the process of estimating the non-linear element can have a strong effect on the associated estimate of the linear dynamics. Furthermore, this effect is not explained simply by way of considering how the input spectrum is changed by the non-linearity. Instead, it arises that the linear model-estimate variability may be dominated by a term that depends on the frequency response of the linear system itself. Amongst other things, the main results derived here have experiment design implications for Hammerstein system estimation. Technical Report EE9933, Department of Electrical and Computer Engineering, University of Newcastle, AUSTRALIA 1

Inasmuch as Lyapunov exponents provide a necessary condition for chaos in a dynamical system, confidence bounds on estimated Lyapunov exponents are of great interest. Estimates derived either from observations or from numerical integrations are limited to trajectories of finite length, and it is the uncertainties in (the distribution of) these finite time Lyapunov exponents which are of interest. Within this context a bootstrap algorithm for quantifying sampling uncertainties is shown to be inappropriate for multiplicative-ergodic statistics of deterministic chaos. This result remains unchanged in the presence of observational noise. As originally proposed, the algorithm is also inappropriate for general nonlinear stochastic processes, a modified version is presented which may prove of value in the case of stochastic dynamics. A new approach towards quantifying the minimum duration of observations required to estimate global Lyapunov exponents is suggested and is explored in a companio...

In this paper, we provide a H1{norm lower bound on the worst{case identi cation error of least{squares estimation when using FIR model structures. This bound increases as a logarithmic function of model complexity and is valid for a wide class of inputs characterized as being quasi{stationary with covariance function falling o suciently quickly.

A functional equation of the form a f x f bx has general solution f x logx x where logx is periodic with period logb. In scaling functions reflecting the mass distribution of self-similar fractal sets, the lacunarity of the set is reflected in the oscillation due to. Scaling measures of chaos may also vary with parameter in a universal manner. The universal functions logr which arise in these measures as a function of the magnitude of the parameter above its value at the point of accumulation of period doubling are identified. Exact results are derived for one dimensional maps, and illustrated both in that context and in a system of ordinary differential equations.

Modeling of uncertain systems with normalized coprime factor description is investigated where the experimental data is given by a finite set of frequency response measurement samples of the open loop plant that is linear, time-invariant, and possibly infinite-dimensional. The objective is not only to identify the nominal model but also to quantify the modeling error with sup-norm bounds in frequency domain. The uncertainty to be identified and quantified is chosen as the ν-metric, proposed by Vinnicombe [43], because of its compatibility with H∞-based robust control. An algorithm is developed to model the normalized coprime factors of the given plant using techniques of discrete Fourier analysis (DFA) and balanced stochastic truncation (BST), and is shown to be robust in presence of the worst-case noise. Upper bounds are derived for the associated modeling error based on the minimum a priori information of the underlying model set and of the noise level in the measurement data. A simul...

In this contribution it is shown that log cos(ßx=(2C)) is the optimally robust norm for prediction error methods with respect to amplitude bounded stochastic disturbances. This norm minimizes the maximum asymptotic covariance matrix of the parameter estimates for the family of innovations of the system which are amplitude bounded by the constant C. Furthermore, the stochastic worst-case performance of the estimate corresponding to the norm log cos(ßx=(2C)) is better than the worst-case performance of the least-squares estimate even if the constant C is chosen larger than the actual amplitude bound on the innovations. In addition to its favorable properties in a stochastic setting this norm also generates estimates which are unfalsified in a deterministic framework.