A robust version of classical D-optimal design applied ... - Hydrologie.org

Integrated Design of Hydrological Networks (Proceedings of the Budapest Symposium, July 1986). IAHS Publ. no. 158,1986.

A robust version of classical D - o p t i m a l design applied t o dissolved-oxygen sag-curve calibration Une solide interprétation de la planification classique D appliquée à la calibration de la concave de 1'oxygène dissous

optimale courbe

E.A. CASMAN Interstate Commission on the Potomac River B a s i n , 6110 Executive Boulevard, R o c k v i l l e , Maryland, 2 0 8 5 2 - 3 9 0 3 , U.S.A. ABSTRACT This paper presents a modification to classical optimal-design theory for parameter precision intended to make this theory more relevant to water-quality modeling. A discussion of the classical techniques and their shortcomings precedes the development of the robust method. T h e appropriateness of this design technique for fitting river models is examined in the context of an example from the water-quality modeling literature: fitting a simple one-dimensional steady-state model of dissolved-oxygen kinetics in rivers. NOTATION

Ejj k, ko t a2 f X p p() n Uj. a k

Covariance matrix of the parameter estimates BOD decay coefficient (day ) Reaeration rate (day ) Time of travel (days) Error variance of the observations Fitted function Matrix of first partial derivatives of f w i t h respect to its parameters Number of unknown parameters in f Probability distribution function N u m b e r of samples in experimental design Expected value of k Variance of k

INTRODUCTION TO D-OPTIMAL DESIGN THEORY Optimal-sampling design methods are mathematical tools for identifying the ideal settings of the independent variables in an experiment before the data are collected. The purpose of these methods is to improve the information content of experiments while minimizing the sampling effort. Optimal designs are typically narrow in scope, emphasizing a single goal for an experiment. One such goal is the improvement of parameter-estimate precision. A major class of 105

106 E.A.Casman parameter-precision design criteria focuses on reducing the variance of the parameter estimates. This information resides in the covariance matrix of the parameters estimates, 2k.(k.,t_), w n i c n i s commonly approximated by the following expression^: £k(k,t) = a 2 (X'X) -1

(1)

where the circumflex signifies an estimated quantity, underlining indicates vectors, o is the error variance of the observations, and X is the matrix of first partial derivatives of the fitted function, f, with respect to its parameters, k. The vector, _t, representing the independent variable settings of the experiment, is included as the argument of £]