Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods

Abstract

Mechanistic models based on systems of nonlinear differential equations can help provide a quantitative understanding of complex physical or biological phenomena. The use of such models to describe nonlinear interactions in molecular biology has a long history; however, it is only recently that advances in computing have allowed these models to be set within a statistical framework, further increasing their usefulness and binding modelling and experimental approaches more tightly together. A probabilistic approach to modelling allows us to quantify uncertainty in both the model parameters and the model predictions, as well as in the model hypotheses themselves. In this paper, the Bayesian approach to statistical inference is adopted and we examine the significant challenges that arise when performing inference over nonlinear ordinary differential equation models describing cell signalling pathways and enzymatic circadian control; in particular, we address the difficulties arising owing to strong nonlinear correlation structures, high dimensionality and non-identifiability of parameters. We demonstrate how recently introduced differential geometric Markov chain Monte Carlo methodology alleviates many of these issues by making proposals based on local sensitivity information, which ultimately allows us to perform effective statistical analysis. Along the way, we highlight the deep link between the sensitivity analysis of such dynamic system models and the underlying Riemannian geometry of the induced posterior probability distributions.

Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation

“…Moreover, as noted by Girolami & Calderhead [22] the negative Hessian of the prior is added to the Fisher information in order to form the metric tensor used during MCMC sampling. This has the added benefit of regularizing the Fisher information when it is near-singular [9], although we have not observed such problems in the simulations presented here.…”

“…These algorithms rely on the gradient and Fisher information matrix of the likelihood function to automatically tune the proposal mechanism such that large moves on the parameter space are possible and therefore improve convergence and mixing of the chains. In the study of Calderhead & Girolami [9], this approach has successfully been applied for the MRE approximation of chemical reaction networks. For the LNA, the Fisher information and the gradient of the likelihood function can easily be obtained [2].…”

Inference in complex systems

“…Systems of nonlinear differential equations have been of fundamental importance in deterministic mathematical modelling. Calderhead & Girolami [5] here advance recent efforts to put this approach into a statistical framework, allowing principled parameter estimation and hypothesis evaluation for differential equation models, as well as measures of uncertainty in model predictions. Their implementation uses Markov chain Monte Carlo (MCMC) with Riemannian geometry to model the local covariance structure of the parameter space, and is illustrated with application to cell signalling pathways and enzymatic circadian control.…”

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

“…A model of circadian control in the Arabidopsis thaliana plant comprised a system of six nonlinear differential equations, with twenty two parameters to be inferred. Another model for cell signalling consisted of a system of six nonlinear differential equations with eight parameters, with inference complicated by the fact that observations of the model are not recorded directly [54]. The resulting inference was performed using RWM, MALA and geometric methods, with the results highlighting the benefits of taking the latter approach.…”

“…Calderhead and Girolami [54] dealt with two models for biological phenomena based on nonlinear dynamical systems. A model of circadian control in the Arabidopsis thaliana plant comprised a system of six nonlinear differential equations, with twenty two parameters to be inferred.…”