Parameter inference and model selection for differential equation models

Abstract

Firstly, we consider the problem of estimating parameters of stochastic differential equations with discrete-time observations that are either completely or partially observed. The transition density between two observations is generally unknown. We propose an importance sampling approach with an auxiliary parameter when the transition density is unknown. We embed the auxiliary importance sampler in a penalized maximum likelihood framework which produces more accurate and computationally efficient parameter estimates. Simulation studies in three different models illustrate promising improvements of the new penalized simulated maximum likelihood method. The new procedure is designed for the challenging case when some state variables are unobserved and moreover, observed states are sparse over time, which commonly arises in ecological studies. We apply this new approach to two epidemics of chronic wasting disease in mule deer. Next, we consider the problem of selecting deterministic or stochastic models for a biological, ecological, or environmental dynamical process. In most cases, one prefers either deterministic or stochastic models as candidate models based on experience or subjective judgment. Due to the complex or intractable likelihood in most dynamical models, likelihood-based approaches for model selection are not suitable. We use approximate Bayesian computation for parameter estimation and model selection to gain further understanding of the dynamics of two epidemics of chronic wasting disease in mule deer. The main novel contribution of this work is that under a hierarchical model framework we compare three types of dynamical models: ordinary differential equation, continuous time Markov chain, and stochastic differential equation models. To our knowledge model selection between these types of models has not appeared previously. The practice of incorporating dynamical models into data models is becoming more common, the proposed approach may be useful in a variety of applications. Lastly, we consider estimation of parameters in nonlinear ordinary differential equation models with measurement error where closed-form solutions are not available. We propose a new numerical algorithm, the data driven adaptive mesh method, which is a combination of the Euler and 4th order Runge-Kutta methods with different step sizes based on the observation time points. Our results show that both the accuracy in parameter estimation and computational cost of the new algorithm improve over the most widely used numerical algorithm, the 4th Runge-Kutta method. Moreover, the generalized profiling procedure proposed by Ramsay et al. (2007) doesn't have good performance for sparse data in time as compared to the new approach. We illustrate our approach with both simulation studies and ecological data on intestinal microbiota.