Nonlinear Least Squares or Trajectory Matching

Abstract

This chapter examines a direct means of fitting the parameters ordinary differential equation models to data: solve the ODE at each candidate set of parameter values and choose the ones which lead to best agreement with the data. Often, we need to search over both parameter values and initial conditions. This chapter introduces Gauss–Newton methods for minimizing a squared error criterion along with the sensitivity equations need to calculate derivatives with respect to parameters. We introduce the concepts of statistical inference—confidence intervals and hypothesis tests—as they apply to nonlinear regression and discuss complications when multiple quantities are measured. We also present Bayesian methods for parameter estimation and ODE-specific schemes designed to overcome difficult optimization surfaces.