Generalized Sensitivities and Optimal Experimental Design

We consider the problem of estimating a modeling parameter μ using a weighted least squares criterion Jd(y, μ) = ∑n i=1 1 3⁄4(ti) ( y(ti) − f(ti, μ) )2 for given data y by introducing an abstract framework involving generalized measurement procedures characterized by probability measures. We take an optimal design perspective, the general premise (illustrated via examples) being that in any data collected, the information content with respect to estimating μ may vary considerably from one time measurement to another, and in this regard some measurements may be much more informative than others. We propose mathematical tools which can be used to collect data in an almost optimal way, by specifying the duration and distribution of time sampling in the measurements to be taken, consequently improving the accuracy (i.e., reducing the uncertainty in estimates) of the parameters to be estimated. We recall the concepts of traditional and generalized sensitivity functions and use these to develop a strategy to determine the “optimal” final time T for an experiment; this is based on the time evolution of the sensitivity functions and of the condition number of the Fisher information matrix. We illustrate the role of the sensitivity functions as tools in optimal design of experiments, in particular in finding “best” sampling distributions. Numerical examples are presented throughout to motivate and illustrate the ideas. AMS subject classifications: 62G08, 62H99, 90C31, 65K10, 93B51, 62B10.