Goodness of fit in nonlinear dynamics: Misspecified rates or misspecified states?

Abstract

This paper introduces diagnostic tests for the nature of lack of fit in ordinary differential equation models (ODEs) proposed for data. We present a hierarchy of three possible sources of lack of fit: unaccounted-for stochastic variation, misspecification of functional forms in rate equations, and omission of dynamic variables in the description of the system. We represent lack of fit by allowing a parameter vector to vary over time, and propose generic testing procedures that do not rely on specific alternative models. Instead, different sources for lack of fit are characterized in terms of nonparametric relationships among latent variables. The tests are carried out through a combination of residual bootstrap and permutation methods. We demonstrate the effectiveness of these tests on simulated data and on real data from laboratory ecological experiments and electro-cardiogram data.

A tracking approach to parameter estimation in linear ordinary differential equations

“…In addition to the parameter estimate, we obtain also directly an estimate of the model discrepancy through the perturbation u that can give hints for analyzing and discussing the relevancy of a parametric model. More can be done with the estimated u about model analysis, and complementary analysis about model testing could be done following the results given in [20].…”

“…This forcing functionû θ is an outcome of the optimization process and can be relatively hard to analyze or understand, as it depends on the basis expansion used and it depends also on the data via the minimization of J n (β|θ, λ). Nevertheless, Hooker et al have proposed goodness-of-fit tests based on this so-called "empirical forcing function"û θ , asû θ are the residuals but at the derivative scale and not at the state scale [19,20]. Based on these remarks, we introduce the pertubed linear ODĖ…”

Linking mosquito surveillance to dengue fever through Bayesian mechanistic modeling

“…However, the fact that the mosquito mortality forcing terms (� ν , Eqs 13 and 16) were the only source of variability in the transmission process implies that the estimated mosquito mortality time series could have absorbed other sources of stochasticity or model misspecification. Hooker and Ellner [64] provide a framework for diagnosing such model misspecification in differential equation models using forcing functions similar to our implementation of the � ν . In that framework, Hooker and Ellner [64] estimate nonparametric forcing functions that modify a fitted differential equation model to provide a good fit to the data.…”

“…Hooker and Ellner [64] provide a framework for diagnosing such model misspecification in differential equation models using forcing functions similar to our implementation of the � ν . In that framework, Hooker and Ellner [64] estimate nonparametric forcing functions that modify a fitted differential equation model to provide a good fit to the data. These forcing functions serve as residuals on the time derivatives, and can be more readily interpreted as indicators of lack-of-fit than residuals on the state variables [16,64].…”

Control Theory and Experimental Design in Diffusion Processes

“…Recent statistical literature has given considerable attention to providing methods for performing inference in mechanistic, non-linear dynamic systems models, both those described by ordinary differential equations (Brunel, 2008;Ramsay, Hooker, Campbell, and Cao, 2007;Girolami, Calderhead, and Chin, 2010;Huang and Wu, 2006) and explicitly stochastic models (Ionides, Bretó, and King, 2006;Aït-Shahalia, 2008;Wilkinson, 2006) along with more general modeling concerns such as providing diagnostic methods for goodness of fit and model improvement (Hooker, 2009;Müller and Yang, 2010;Hooker and Ellner, 2013). However, little attention has been given to the problem of designing experiments on dynamic systems so as to yield maximal information about the parameters of interest.…”