Research

Differential Equation Models

Differential Equation (DE) models are theoretically built mechanistic models for complex systems.  A single nonlinear (DE) model can describe a wide variety of behaviours including oscillations, steady states and exponential growth and decay, with few but readily interpretable parameters. Consequently, DE models are routinely used in describing chemical reaction dynamics, predator-prey interactions, heat transfer, economic growth, epidemiological outbreaks, climate and weather prediction, gene regulatory pathways, turbulence,... The wide range of complex dynamics that a single model can produce complicates all areas of estimation and motivates most of my research and has led me to work on Uncertainty Quantification, Parameter Estimation, Computational Bayesian Methods, and Poorly Identified Models.  I’ve also done some work on Time-Frequency Methods.

Uncertainty Quantification

Differential Equation models typically require numerical methods to solve for their implicitly defined states.  In complex models, especially those involving chaotic or stiff dynamics, small errors introduced by the numerical solver accumulate leading to exponential divergence from the true system solution.  As such classic numerical error tolerance bounds are inadequate to describe the functional uncertainty in the produced solution.  I’m interested in how to define probability measures on the function space of  possible differential equation solutions and how these functional distributions can be incorporated into the inference problem.

• McDonald, S., Campbell, D. (2021) "A Review of Uncertainty Quantification for Density Estimation” Statistics Surveys (open access)
• Chkrebtii, O., and Campbell, D. (2019) “ Adaptive step-size selection for state-space based probabilistic differential equation solvers” Statistics and Computing
• Chkrebtii, O. , Campbell, D., Girolami, M. and Calderhead, B. (2015), “Bayesian Uncertainty Quantification for Differential Equations” Bayesian Analysis, with discussion
• Golchi, S., Bingham, D., Chipman, H., Campbell, D. (2015) Monotone Function Estimation for Computer Experiments, Journal of Uncertainty Quantification

Parameter Estimation for Differential Equation Models

I'm most interested in DEs without an analytic solution. Unfortunately the flexibility of this class of model implies that a likelihood centred on the solution to the DE is full of local maxima, ridges, ripples, flat sections and other difficult topologies. These give rise to a host of statistical challenges that I have been researching from Bayesian and frequentist perspectives. Here's a talk I gave in Banff giving an overview on statistical methods for differential equation models

• Campbell, D. and Chkrebtii, O. (2013), “Maximum profile likelihood estimation of differential equation parameters through model based smoothing state estimatesMathematical Biosciences, doi:10.1016/j.mbs.2013.03.011
• Campbell, D., Hooker, G., McAuley, K. (2012), "Parameter estimation in differential equation models with constrained states" Journal of Chemometrics
• Ramsay, J. O., Hooker, G., Campbell, D., and Cao, J. (2007), "Parameter estimation for differential equations: a generalized smoothing approach (with discussion)Journal of the Royal Statistical Society Series B, 69, 741-796.

Computational Bayesian Methods

I am working on Bayesian sampling methods aimed at overcoming the unique challenges of dynamic system models. Often posterior densities for differential equation models have multiple modes and deep posterior valleys, sometimes thousands of units deep on the log scale. I am working on methods that can bypass these prohibitive topologies and produce a reasonable sample from the target posterior.