Alistair Mees

Convergence of an annealing algorithm

The annealing algorithm is a stochastic optimization method which has attracted attention because of its success with certain difficult problems, including NP-hard combinatorial problems such as the travelling salesman, Steiner trees and others. There is an appealing physical analogy for its operation, but a more formal model seems desirable. In this paper we present such a model and prove that the algorithm converges with probability arbitrarily close to 1. We also show that there are cases where convergence takes exponentially long—that is, it is no better than a deterministic method. We study how the convergence rate is affected by the form of the problem. Finally we describe a version of the algorithm that terminates in polynomial time and allows a good deal of ‘practical’ confidence in the solution.

Dynamics of brain electrical activity

SummaryIn addition to providing important theoretical insights into chaotic deterministic systems, dynamical systems theory has provided techniques for analyzing experimental data. These methods have been applied to a variety of physical and chemical systems. More recently, biological applications have become important. In this paper, we report applications of one of these techniques, estimation of a signals correlation dimension, to the characterization of human electroencephalographic (EEG) signals and event-related brain potentials (ERPs). These calculations demonstrate that the magnitude of the technical difficulties encountered when attempting to estimate dimensions from noisy biological signals are substantial. However, these results also suggest that this procedure can provide a partial characterization of changes in cerebral electrical activity associated with changes in cognitive behavior that complements classical analytic procedures.

On selecting models for nonlinear time series

Abstract Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series.

Dynamics from multivariate time series

Abstract Multivariate time series data are common in experimental and industrial systems. If the generating system has nonlinear dynamics, we may be able to construct a model that reproduces the dynamics and can be used for control and other purposes. In principle, multivariate time series are not necessary for recovering dynamics: according to the embedding theorem, only one time series should be needed. However, for real data, there may be large gains in using all of the measurements. In this paper we examine the issues of how to use multiple data streams most effectively for modeling and prediction. For example, perhaps the data are redundant in that only a subset of the data streams is useful. And how should we embed the data, if indeed embedding is required at all? We show how these questions can be answered, and describe some numerical experiments which show that using multivariate time series can significantly improve predictability. We also demonstrate a somewhat surprising synchronization between different reconstructions.

Embedding as a modeling problem

Abstract Standard approaches to time-delay embedding will often fail to provide an embedding that is useful for many common applications. This happens in particular when there are multiple timescales in the dynamics. We present a modified procedure, non-uniform embedding, which overcomes such problems in many cases. For more complex nonlinear dynamics we introduce variable embedding, where, in a suitable sense, the embedding changes with the state of the system. We show how to implement these procedures by combining embedding and modeling into a single procedure with a single optimization goal.

Dangers of geometric filtering

Abstract We argue that there are unrecognised dangers in the popular and powerful geometric filtering method. Carelessly applied, geometric filtering can produce apparent structure in data which is pure noise, or can cause severe distortions in clean deterministic data. The explanations for these effects are straightforward and the dangers are easily avoided by taking simple precautions in the filtering process.

Optimal simultaneous stabilization of linear single-input systems via linear state feedback control

Simultaneous stabilization of a collection of linear time-invariant systems is considered. The aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses. Hurwitzs necessary and sufficient conditions are used as the required set of constraints for simultaneous stability. For transient behaviour, we introduce the usual quadratic objective function as performance index for each system. Their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the Hurwitz stability constraints. A computational technique is proposed for solving this problem. The results obtained give good transient behaviour. For illustration, two numerical examples are solved.

DYNAMICAL SYSTEMS AND TESSELATIONS: DETECTING DETERMINISM IN DATA

Data measurements from a dynamical system may be used to build triangulations and tesselations which can — at least when the system has relatively low-dimensional attractors or invariant manifolds — give topological, geometric and dynamical information about the system. The data may consist of a time series, possibly reconstructed by embedding, or of several such series; transients can be put to good use. The topological information which can be found includes dimension and genus of a manifold containing the state space. Geometric information includes information about folds, branches and other chaos generators. Dynamical information is obtained by using the tesselation to construct a map with stated smoothness properties and having the same dynamics as the data; the resulting dynamical model may be tested in the way that any scientific theory may be tested, by making falsifiable predictions.

Deterministic prediction and chaos in squid axon response

Abstract We make deterministic predictive models of apparently complex squid axon response to periodic stimuli. The result provides evidence that the response is chaotic (and therefore partially predictable) and implies the possibility of identifying deterministic chaos in other kinds of noisy data even when explicit models are not available.

Parsimonious dynamical reconstruction

Many nonlinear deterministic models of time series have large numbers of parameters and tend to overfit in the presence of noise. This paper shows how to generate radial basis function models with small numbers of parameters for a given quality of fit. It also addresses questions of how to select subsets from candidate sets of centers for radial basis function models, and what kinds of basis functions to use, as well as how large a model is appropriate.