Likelihood based observability analysis and confidence intervals for predictions of dynamic models

Jens Timmer,Andreas Raue,Clemens Kreutz

doi:10.1186/1752-0509-6-120

Abstract

BackgroundPredicting a system’s behavior based on a mathematical model is a primary task in Systems Biology. If the model parameters are estimated from experimental data, the parameter uncertainty has to be translated into confidence intervals for model predictions. For dynamic models of biochemical networks, the nonlinearity in combination with the large number of parameters hampers the calculation of prediction confidence intervals and renders classical approaches as hardly feasible.ResultsIn this article reliable confidence intervals are calculated based on the prediction profile likelihood. Such prediction confidence intervals of the dynamic states can be utilized for a data-based observability analysis. The method is also applicable if there are non-identifiable parameters yielding to some insufficiently specified model predictions that can be interpreted as non-observability. Moreover, a validation profile likelihood is introduced that should be applied when noisy validation experiments are to be interpreted.ConclusionsThe presented methodology allows the propagation of uncertainty from experimental to model predictions. Although presented in the context of ordinary differential equations, the concept is general and also applicable to other types of models. Matlab code which can be used as a template to implement the method is provided at http://www.fdmold.uni-freiburg.de/∼ckreutz/PPL.

Highlights

Predicting a system’s behavior based on a mathematical model is a primary task in Systems Biology
For model predictions, there are still demands for methodology that is applicable for mathematical models like ordinary differential equations (ODEs) used to describe the dynamics of a system in a variety of scientific fields e.g. in molecular biology [7,8], and in medical research, chemistry, engineering, and physics
Existing approaches for prediction confidence intervals like Markov Chain Monte Carlo (MCMC) [26] or bootstrap procedures are based on forward evaluations of the model for many parameter values

Summary

Introduction

Predicting a system’s behavior based on a mathematical model is a primary task in Systems Biology. The task of establishing a realistic mathematical model which is able to reliably predict a systems behavior is to comprehensively use the existing knowledge, e.g. in terms of experimental data, to adjust the models’ structures and parameters. The major steps of this mathematical modeling process comprise model discrimination, i.e. identification of an appropriate model structure, model calibration, i.e. estimation of unknown model parameters, as well as prediction and model validation. For all these topics it is essential to have appropriate methods assessing the certainty or ambiguity of any result for given experimental information. For model predictions, there are still demands for methodology that is applicable for mathematical models like ordinary differential equations (ODEs) used to describe the dynamics of a system in a variety of scientific fields e.g. in molecular biology [7,8], and in medical research, chemistry, engineering, and physics

Methods

Results

Discussion

Conclusion