Enhancement of Nonlinear Estimation Capability of MLAB

The heart of the MLAB modeling language is the facility for parameter estimation in mathematical models by fitting to data. This computation is performed using the Marquardt-Levenburg algorithm and is applicable to nonlinear models subject to linear constraints. By the use of appropriate weighing choices, the algorithm is capable of a variety of extensions. These extensions include norms other than L2, iteratively reweighted least squares, and, for special cases only, maximum likelihood estimation. The robustness of the algorithm has been enhanced through the provision of automatically-calculated symbolic partial derivatives and by heuristic use of gradient search. Despite all these features, there are still many classes of estimation problem that are beyond the current capability of MLAB. These include more general maximum likelihood estimation, stochastic estimation, and estimation for partial differential equation-defined models. For each of these classes, we have developed new strategies and intend to incorporate them into MLAB. The present project deals with three particular aspects of nonlinear estimation that we have identified for enhancement: (1) generalization of the constraint facility to include nonlinear equality and inequality constraints using Lagrange multiplier techniques, (2) generalization of the facility for estimation of the confidence intervals and joint confidence regions for parameters in nonlinear models with active constraints, and (3) provision of alternative methods for function optimization that will increase the robustness of parameter estimation for certain classes of nonlinear models. In particular, simplex methods and simulated annealing have been identified as suitable candidates. For Phase I, we will identify a wide range of further extensions to be developed for Phase II.