(508c) A New Global Sensitivity Analysis Procedure Involving Quasi Linearization for Optimal Experimental Design

The general topic of experimental design includes qualitative design which consists of selecting inputs, outputs, and identifiable parameters, and quantitative design which consists of determining input shapes and sampling schedules based on optimization of a suitable criterion (Walter 1990). Local parametric sensitivities which are partial derivatives of the outputs with respect to the parameters play an important role in both qualitative and quantitative experimental design. Various criteria of experimental design have been developed based on local sensitivity analysis.

Local sensitivity analysis for a nonlinear model returns results that are dependent on the nominal values of the parameters. This situation can be a problem insofar as the true value of the parameters cannot be known prior to parameter estimation, however, experimental design is generally performed in order to collect data which is then used for parameter estimation. As a result, this dependence of the sensitivity value and in turn the experimental design criterion on the parameter values represents one of the main difficulties for experimental design of a nonlinear model.

Several approaches have been developed to decouple this dependence. The most widely used one is the local design (Box 1959) which assumes that the true parameter values are close to the nominal values. Using this assumption, it is possible to evaluate the sensitivity vectors at the nominal values of the parameters to design an experiment. This approach neglects parameter uncertainty, however, it is a relatively simple alternative to more advanced techniques. Another approach is the sequential design (Box 1962) which iterates between experimental design and parameter estimation. Using this methodology, the experiment is designed based on the sensitivity evaluated at the previously estimated parameter values and then the parameter values are updated using data from the newly designed experiment. However, iterating between experimental design and parameter estimation still requires that a large number of experiments need to be performed which may be prohibitive in practice. Another alternative for experimental design is robust design (Asprey 2002; Dette 2005). Instead of evaluating the sensitivities at only one point in the parameter space, this approach evaluates the values at different parameter values over the uncertain range. Techniques based upon this idea include the mini-max method (Pronzato 1988; Goodwin 2008) and the Bayesian method (Pronzato 1985; Chaloner 1995). One drawback to these techniques is that they can be computationally expensive.

Global sensitivity analysis has been developed as an alternative to local approaches. Global sensitivity analysis characterizes the effect of the parameters on the outputs by explicitly taking the information of the parameter uncertainty into account (Saltelli 2006). While there has been a significant number of contributions to global sensitivity analysis, most works focus on qualitative experimental dealing with identification of important parameters. It has been recognized that global sensitivity analysis outperforms local sensitivity for identification of the influential parameters, however, reports of quantitative experimental design by global sensitivity analysis, e.g. selection of sampling points and determination of input profiles, are rare.

The main obstacle for application of global sensitivity analysis to quantitative experimental design is that the computed global sensitivities fail to reduce to the local sensitivities in the case where the model under investigation is linear or the parameter uncertainty is negligible. Even though many experimental design criteria exists that are based on the local sensitivity analysis results, none of them can be easily extended to deal with global sensitivity analysis.

It is the goal of this work is to develop a global sensitivity measure which can be used for optimal experimental design. The global sensitivity is calculated by using a quasi linearization. It will be shown that this result will reduce to the local sensitivity value in the case of a linear model or if the range of the parameter uncertainty approaches zero. Due to this, the presented global sensitivity can be viewed as an extension of the local sensitivity applied to a model with uncertainties. Accordingly, experimental design criteria based on local sensitivity analysis will be extended in this work to be applicable to global sensitivity analysis. Since uncertainty in the model parameters is explicitly taken into account in the calculation of the global sensitivity, an experimental design based on global sensitivity analysis returns a better performance than one computed using a local approach. The technique is illustrated in two case studies where the optimal sampling points and an optimal input profile are determined for a reactor described by a nonlinear process model.

Reference:

1. Asprey SP, Macchietto S. Designing robust optimal dynamic experiments. Journal of Process Control 12 (4): 545-556 2002.

2. Box GEP, Lucas HL. Design of experiments in non-linear situations. Biometrika, 46 (1-2): 77-90 1959

3. Box GEP, Hunter WG. Useful method for model-building. Technometrics 4 (3): 301-318 1962

4. Chaloner K, Verdinelli I. Bayesian experimental design: A review. Statistical Science 10 (3) 273-304 1995

5. Dette H, Melas VB, Pepelyshev A, Strigul N. Robust and efficient design of experiments for the Monod model. Journal of Theoretical Biology 234 (4): 537-550 2005

6. Goodwin GC, Aguero JC, Welsh JS, Yuz JI, Adams GJ, Rojas CR. Robust identification of process models from plant data. Journal of Process Control 18 (9): 810-820 2008

7. Pronzato L, Walter E. Robust experiment design via stochastic-approximation. Mathematical Biosciences 75 (1): 103-120 1985

8. Pronzato L, Walter E. Robust experiment design via maximin optimization. Mathematical Biosciences 89 (2): 161-176 1988

9. Saltelli A, Ratto M, Tarantola S, Campolongo F. Sensitivity analysis practices: Strategies for model-based inference. Reliability Engineering & System Safety 91 (10-11) 1109-1125 2006

10. Walter E, Pronzato L. Qualitative and quantitative experiment design for phenomenological models - A survey. Automatica 26 (2): 195-213 1990.