Bayesian ODE-Penalized B-Spline Model

Bayesian ODE-penalized B-spline model

Jonathan Jaeger1, Philippe Lambert1,2

1 Institut de Statistique, Biostatistique et Sciences Actuarielles.
Université Catholique de Louvain, Louvain-la-Neuve, Belgium.

2 Institut des Sciences Humaines et Sociales.
Université de Liège, Liège, Belgium

Ordinary Differential Equations (ODEs) are widely used to model physical, chemical and biological processes. For example, the study of artherogenesiscanpartly be doneby understanding and analyzing the perfusion ratio between ischemic and non-ischemic blood flow.The evolution of this perfusion ratio is simply modeled by an ODE.

Currently, the most commonly used estimation procedures rely on nonlinear least squares (Biegler et al., 1986). It uses minimization techniques for the estimation of the ODE parameters and a numerical solver for the approximation of the solution. These approaches are computationally intensive and often poorly suited for statistical inference. Lunn et al. (2002) propose a Bayesian framework for these NLS procedures. The possible use of prior information about the ODE parameters is a definite advantage, but the numerical approximation to the solution leads to a posterior distribution with no-closed form for the ODE parameters.

Alternative estimation method of the state functions and of the ODE parameters was recently proposed in Ramsay et al. (2007). This may be viewed as a generalization of the P-spline theory (Eilers and Marx, 1996) that involves some basis function expansion of each state function and a penalty term expressed using the set of differential equations. It converges quickly, gives less biased and more accurate estimates. It allows also estimating unobserved state functions. This approach has two major weaknesses: the choice of the ODE-adhesion and the approximation for the variance of the parameters estimates.

We propose a full Bayesian ODE-penalized B-spline approach to jointly estimate parameters and state functions in affine systems of differential equations (Jaeger and Lambert, 2011). The two major drawbacks of the frequentist ODE-penalized smoothing approach are overcome: the selection of the ODE-adhesion parameter is now automatic and uncertainty measures about parameters can simply be obtained using MCMC. The basic Bayesian ODE-penalized B-spline approach can also be easily extended to a hierarchical setting.

In this talk, we first give motivating examples takenfroma pharmacokinetic study and anischemic/non-ischemic perfusion ratio study. We then present the Bayesian ODE-penalized B-spline model and extend it to a hierarchical context.We illustrate the usefulness of this methodology on the two motivating examples and we explain how simply posterior probability and predictions can be computed.

Reference

Biegler, L.T., Damiano, J.J., and Blau, G.E. (1986). Nonlinear parameter estimation: A case study comparison. AIChE Journal, 32, 29-45.

Eilers, P.H.C., and Marx, B.D. (1996).Flexible smoothing with B-splines and penalties. Statistical Science, 11, 89-121.

Jaeger, J., and Lambert, P. (2011).Bayesian Generalized Profiling Estimation in Hierarchical Linear Dynamic Systems.Discussion Paper 1101, Université Catholique de Louvain.

Lunn, D.J., Best, N., Thomas, A., Wakefield J., and Spiegelhalter, D. (2002).Bayesian Analysis of Population PK/PD Models: General Concepts and Software.Journal of Pharmacokinetics and Pharmacodynamics, 29, 271-307.

Ramsay, J.O., Hooker, G., Campbell, D., and Cao, J. (2007).Parameter estimation for differential equations: a generalized smoothing approach.Journal of the Royal Statistical Society, Series B, 69, 741-796.