Generalized sensitivities and optimal experimental design

Abstract We consider the problem of estimating amodeling parameter θ using a weighted least squares criterion 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 θ 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.

[1]  Harvey Thomas Banks,et al.  Inverse problems for a class of measure dependent dynamical systems , 2005 .

[2]  Claudio Cobelli,et al.  Generalized Sensitivity Functions in Physiological System Identification , 1999, Annals of Biomedical Engineering.

[3]  A. Gallant,et al.  Nonlinear Statistical Models , 1988 .

[4]  Marie Davidian,et al.  Nonlinear Models for Repeated Measurement Data , 1995 .

[5]  Mansour Eslami,et al.  Theory of Sensitivity in Dynamic Systems: An Introduction , 1994 .

[6]  Jose B. Cruz,et al.  System Sensitivity Analysis , 1982 .

[7]  Harvey Thomas Banks,et al.  Sensitivity functions and their uses in inverse problems , 2007 .

[8]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[9]  Weng Kee Wong,et al.  Applied Optimal Designs , 2006 .

[10]  Georgios E. Stavroulakis,et al.  Parameter sensitivity in nonlinear mechanics. Theory and finite element computations: by M. Kleiber, H. Antúney, T. D. Hien, P. Kowalczyck (Wiley, Chichester, 1997) , 1998 .

[11]  W. J. Studden,et al.  Theory Of Optimal Experiments , 1972 .

[12]  F. Pukelsheim.  Optimal Design of Experiments , 1993 .

[13]  S Dediu,et al.  Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space. , 2007, Mathematical biosciences and engineering : MBE.

[14]  A. Račkauskas,et al.  Approximation Theory in the Central Limit Theorem , 1989 .

[15]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[16]  Mark Kot,et al.  Elements of Mathematical Ecology: Frontmatter , 2001 .

[17]  H T Banks,et al.  Stochastic and deterministic models for agricultural production networks. , 2007, Mathematical biosciences and engineering : MBE.

[18]  Harvey Thomas Banks,et al.  Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls , 2007 .

[19]  Steven E. Rigdon,et al.  Model-Oriented Design of Experiments , 1997, Technometrics.

[20]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[21]  Harvey Thomas Banks,et al.  Modeling and Imaging Techniques with Potential for Application in Bioterrorism , 2003 .

[22]  Harvey Thomas Banks,et al.  An Inverse Problem Statistical Methodology Summary , 2007 .

[23]  M. Eslami,et al.  Introduction to System Sensitivity Theory , 1980, IEEE Transactions on Systems, Man, and Cybernetics.

[24]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[25]  Harvey Thomas Banks,et al.  Modeling and estimating uncertainty in parameter estimation , 2001 .

[26]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[27]  Andrej Pázman,et al.  Nonlinear Regression , 2019, Handbook of Regression Analysis With Applications in R.

[28]  Daniel Schneditz,et al.  Cardiovascular and Respiratory Systems: Modeling, Analysis, and Control , 2006 .

[29]  J. Davenport.  Editor , 1960 .

[30]  Yu. V. Prokhorov.  Convergence of Random Processes and Limit Theorems in Probability Theory , 1956 .

[31]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[32]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[33]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[34]  R. Haftka,et al.  Sensitivity Analysis of Discrete Structural Systems , 1986 .