Model-Based Optimal Experiment Design for Nonlinear Parameter Estimation Using Exact Confidence Regions

Abstract

Optimal experiment design is usually performed as a search over a finitely-parameterized shape that (over-) approximates the confidence region of parameters of a model. In general, there exists no such shape to exactly enclose the confidence region of a nonlinear parameter estimation problem. Due to this fact, the design-of-experiment techniques are not well established for this problem and approximate designs are conducted. In this contribution, assuming Gaussian (normally distributed) noise, we propose and study (a) two schemes to over-approximate the confidence region of parameters using an ellipsoid and an orthotope and (b) a framework for optimal experiment design. We formulate the over-approximation of the confidence region as an optimization problem. The optimal experiment design is then proposed as a bi-level optimization problem. In line with the existing optimal experiment design methodology for a linear parameter estimation problem, we also propose several design criteria that optimize some measure of the over-approximated confidence region for the nonlinear case. The proposed bi-level optimization problem is solved (i) as a nonlinear programming problem using the necessary conditions for optimality or (ii) as a nested problem with globally optimized inner-level problem. We illustrate the proposed schemes on a benchmark test case.