Three-Toed Sloth

"Statistical Methods for Modeling Dynamic Systems" Workshop

...will be held July 9--13, 2007, at the Centre de Recherches Mathématiques, Université de Montréal, organized by David Campell, Giles Hooker and Jim Ramsay. They could hardly have come up with something I'd be more into if they'd been trying (which they weren't):

The term "dynamic system" typically implies a mathematical model expressed by a system of nonlinear differential or difference equations. Models of this nature have had a very long history in the physical sciences. More recently, these models have been employed for new areas such as clinical medicine, ecology, neurophysiology and the social sciences. There is, in addition, more and more attention given to assessing how well these models fit measured data in addition to displaying characteristics of the system being modeled at a qualitative level.

Statisticians have played a relatively limited role in these developments, in part because methods for fitting data with models of this nature that could spin off approaches to testing hypotheses and supplying confidence intervals for estimated quantities have not been easy to develop. Consequently, we have proposed this workshop as a means of bringing those working with dynamic models together with statisticians so as to stimulate further development, collaboration and application of statistical methodology in this important area.

Official website here (or, in French, here). Financial support is available for graduate students.

This is also a good occasion to plug some of the work of the organizers, which I've been meaning to do since Hooker came here to give a talk about a year ago:

James Ramsay, Giles Hooker, David Campbell and Jiguo Cao, "Parameter Estimation for Differential Equations: A Generalized Smoothing Approach", Journal of the Royal Statistical Society forthcoming (2007), with discussion [PDF preprint]

Abstract: We propose a new method for estimating parameters in non-linear differential equations. These models represent change in a system by linking the behavior of a derivative of a process to the behavior of the process itself. Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to statistical techniques such as inference and interval estimation. This paper describes a new method that uses noisy data to estimate the parameters defining a system of nonlinear differential equations. The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation. We derive interval estimates and show that these have good coverage properties on data simulated from chemical engineering and neurobiology. The method is demonstrated using real-world data from chemistry and from the progress of the auto-immune disease lupus.

There has been a lot of work in the physical and nonlinear dynamics communities on reconstructing the state space of smooth dynamical systems (a.k.a. "geometry from a time series"), which more or less assumes that the time series you're interested in is the solution to a set of nonlinear differential equations. (Much of my own work has been based on these ideas, as extended to certain kinds of discrete stochastic processes.) What it concentrates on are the "qualitative" properties of the system, like the geometric type of the attractor, or the Lyapunov exponents. (More exactly, "qualitative" here means "left alone by a smooth change of coordinates", or in the jargon "invariant under a diffeomorphism". This is how the numerical values of the Lyapunov exponents, i.e., quantities, get to be "qualitative".) The strength of these methods --- not needing to know the actual variables comprising the physical state, or the precise form of the dynamics --- is also their weakness; they can tell us that we're dealing with a limit cycle, but not (say) how strongly the calcium and potassium concentrations are coupled.

To answer questions of the latter sort, we need information about the form of the equations of motion and their parameters. Perhaps oddly, the nonlinear dynamics community has done less work on these questions. (Less, but not exactly none.) But this is precisely what Ramsay et al. are doing, by ingeniously using existing spline-smoothing techniques to learn the parameters of the equations of motion in a statistically reliable manner. This opens the door to testing hypotheses about those parameters (are calcium and potassium concentrations coupled? are they as strongly coupled as other experiments would suggest?, etc.), estimating errors, and all the other conveniences of statistical inference. Those of us who care about modeling dynamics should all be very interested in what these two approaches can be made to say to each other.

Enigmas of Chance; Physics