ABSTRACT
In inverse or parameter estimation problems, as discussed in the Introduction, an important but practical question is how successful the mathematical model is in describing the physical or biological phenomena represented by the experimental data. In general, it is very unlikely that the residual sum of squares (RSS) in the least-squares formulation is zero. Indeed, due to measurement noise as well as modeling error, there may not be a “true” set of parameters so that the mathematical model will provide an exact fit to the experimental data. Even if one begins with a deterministic model and has no initial interest
in uncertainty or stochasticity, as soon as one employs experimental data in the investigation, one is led to uncertainty that should not be ignored. In fact, all measurement procedures contain error or uncertainty in the data collection process and hence statistical questions arise regarding that sampling error. To correctly formulate, implement and analyze the corresponding inverse problems, one requires a framework entailing a statistical model as well as a mathematical model. In this chapter we discuss mathematical, statistical and computational as-
pects of inverse or parameter estimation problems for deterministic dynamical systems based on the Ordinary Least Squares (OLS), Weighted Least Squares (WLS) and Generalized Least Squares (GLS) methods with appropriate corresponding data noise assumptions of constant variance and non-constant variance (e.g., relative error). Among the topics included are the interplay between the mathematical model, the statistical model, and observation or data assumptions, and some techniques (residual plots) for analyzing the uncertainties associated with inverse problems employing experimental data. We also outline a standard theory underlying the construction of confidence intervals for parameter estimators. This asymptotic theory for confidence intervals can be found in Seber and Wild [35]. Finally, we also compare this asymptotic error approach to the popular “bootstrapping” approach.
