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Smooth functional tempering for nonlinear differential equation models

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Abstract

Differential equations are used in modeling diverse system behaviors in a wide variety of sciences. Methods for estimating the differential equation parameters traditionally depend on the inclusion of initial system states and numerically solving the equations. This paper presents Smooth Functional Tempering, a new population Markov Chain Monte Carlo approach for posterior estimation of parameters. The proposed method borrows insights from parallel tempering and model based smoothing to define a sequence of approximations to the posterior. The tempered approximations depend on relaxations of the solution to the differential equation model, reducing the need for estimating the initial system states and obtaining a numerical differential equation solution. Rather than tempering via approximations to the posterior that are more heavily rooted in the prior, this new method tempers towards data features. Using our proposed approach, we observed faster convergence and robustness to both initial values and prior distributions that do not reflect the features of the data. Two variations of the method are proposed and their performance is examined through simulation studies and a real application to the chemical reaction dynamics of producing nylon.

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Authors and Affiliations

  1. Department of Statistics and Actuarial Science, Simon Fraser University, 13450 102nd avenue, Surrey, BC, Canada, V3T 0A3

    David Campbell

  2. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Ouest, Montreal, QC, Canada, H3A 2K6

    Russell J. Steele

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Campbell, D., Steele, R.J. Smooth functional tempering for nonlinear differential equation models. Stat Comput 22, 429–443 (2012). https://doi.org/10.1007/s11222-011-9234-3

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