Parameter estimation for dynamical systems

Parameter estimation is a vital component of model development. Making use of data, one aims to determine the parameters for which the model behaves in the same way as the system observations. In the setting of differential equation models, the available data often consists of time course measurements of the system. We begin by examining the parameter estimation problem in an idealized setting with complete knowledge of an entire single trajectory of data which is free from error. This addresses the question of uniqueness of the parameters, i.e. identifiability. We derive novel, interrelated conditions that are necessary and sufficient for identifiability for linear and linear-in-parameters dynamical systems. One result provides information about identifiability based solely on the geometric structure of an observed trajectory. Then, we look at identifiability from a discrete collection of data points along a trajectory. By considering data that are observed at equally spaced time intervals, we define a matrix whose Jordan structure determines the identifiability. We further extend the investigation to consider the case of uncertainty in the data. Our results establish regions in data space that give inverse problem solutions with particular properties, such as uniqueness or stability, and give bounds on the maximal allowable uncertainty in the data set that can be tolerated while maintaining these characteristics. Finally, the practical problem of parameter estimation from a collection of data for the system is addressed. In the setting of Bayesian parameter inference, we aim to improve the accuracy of the Metropolis-Hastings algorithm by introducing a new informative prior density called the Jacobian prior, which exploits knowledge of the fixed model structure. Two approaches are developed to systematically analyze the accuracy of the posterior density obtained using this prior