1. CHAPTER 17 O PTIMAL D ESIGN FOR E XPERIMENTAL I NPUTS Organization of chapter in ISSO* –Background Motivation Finite sample and asymptotic (continuous) designs Precision matrix and D-optimality –Linear models Connections to D-optimality Key equivalence theorem –Response surface methods –Nonlinear models *Note: Appendix to these slides is brief discussion of factorial design (not in ISSO) Slides for Introduction to Stochastic Search and Optimization (ISSO) by J. C. Spall

2. 17-2 Optimal Design in Simulation Two roles for experimental design in simulation –Building approximation to existing large-scale simulation via “metamodel” –Building simulation model itself Metamodels are “curve fits” that approximate simulation input/output –Usual form is low-order polynomial in the inputs; linear in parameters  –Linear –Linear design theory useful Building simulation model nonlinear –Typically need nonlinear design theory Some terminology distinctions: FactorsInputs –“Factors” (statistics term)  “Inputs” (modeling and simulation terms) LevelsValues –“Levels”  “Values” TreatmentsRuns –“Treatments”  “Runs”

3. 17-3 Unique Advantages of Design in Simulation Simulation experiments may be considered special case of general experiments Some unique benefits occur due to simulation structure Can control factors not generally controllable (e.g., arrival rates into network) Direct repeatability due to deterministic nature of random number generators –Variance reduction (CRNs, etc.) may be helpful Not necessary to randomize runs to avoid systematic variation due to inherent conditions –E.g., randomization in run order and input levels in biological experiment to reduce effects of change in ambient humidity in laboratory –In simulation, systematic effects can be eliminated since analyst controls nature

4. 17-4 Design of Computer Experiments in Statistics There exists significant activity among statisticians for experimental design based on computer experiments –T. J. Santner et al. (2003), The Design and Analysis of Computer Experiments, Springer-Verlag –J. Sacks et al (1989), “Design and Analysis of Computer Experiments (with discussion),” Statistical Science, 409–435 –Etc. Above statistical work differs from experimental design with Monte Carlo simulations –Above work assumes deterministic function evaluations via computer (e.g., solution to complicated ODE) One implication of deterministic function evaluations: no need to replicate experiments for given set of inputs Contrasts with Monte Carlo, where replication provides variance reduction

5. 17-5 General Optimal Design Formulation (Simulation or Non-Simulation) Assume model z = h( ,  x) + v, where x is an input we are trying to pick optimally Experimental design  consists of N specific input values x =  i and proportions (weights) to these input values w i : Finite-sample asymptotic (continuous)Finite-sample design allocates n  N available measurements exactly; asymptotic (continuous) design allocates based on n  

6. 17-6 D-Optimal Criterion Picking optimal design  requires criterion for optimization Most popular criterion is D-optimal measure Let M( ,  ) denote the “precision matrix” for an estimate of  based on a design  –M( ,  ) is inverse of covariance matrix for estimate and/or –M( ,  ) is Fisher information matrix for estimate D-optimal solution is

7. 17-7 Equivalence Theorem Consider linear model Prediction based on parameter estimate and “future” measurement vector h T is Kiefer-Wolfowitz equivalence theorem states: is the same  D-optimal solution for determining  to be used in forming is the same  that minimizes the maximum variance of predictor Useful in practical determination of optimal 

8. 17-8 Variance Function as it Depends on Input: Optimal Asymptotic Design for Example 17.6 in ISSO

9. 17-9 Orthogonal Designs With linear models, usually more than one solution is D-optimal Orthogonality is means of reducing number of solutions Orthogonality also introduces desirable secondary properties –Separates effects of input factors (avoids “aliasing”) –Makes estimates for elements of  uncorrelated Orthogonal designs are not generally D-optimal; D-optimal designs are not generally orthogonal some –However, some designs are both Classical factorial (“cubic”) designs are orthogonal (and often D-optimal)

10. 17-10 Example Orthogonal Designs, r = 2 Factors x k2 x k1 Cube (2 r design) x k2 x k1 Star (2r design)

11. 17-11 Example Orthogonal Designs, r = 3 Factors Star (2r design) xk1xk1 xk2xk2 xk3xk3 Cube (2 r design) xk2xk2 xk1xk1 xk3xk3

12. 17-12 Response Surface Methodology (RSM) Suppose want to determine inputs x that minimize the mean response z of some process (E(z)) –There are also other (nonoptimization) uses for RSM RSM can be used to build local models with the aim of finding the optimal x –Based on building a sequence of local models as one moves through factor (x) space Each response surface is typically a simple regression polynomial Experimental design can be used to determine input values for building response surfaces

13. 17-13 Steps of RSM for Optimizing x Step 0 (Initialization) Step 0 (Initialization) Initial guess at optimal value of x. Step 1 (Collect data) Step 1 (Collect data) Collect responses z from several x values in neighborhood of current estimate of best x value (can use experimental design). Step 2 (Fit model) Step 2 (Fit model) From the x, z pairs in step 1, fit regression model in region around current best estimate of optimal x. Step 3 (Identify steepest descent path) Step 3 (Identify steepest descent path) Based on response surface in step 2, estimate path of steepest descent in factor space. Step 4 (Follow steepest descent path) Step 4 (Follow steepest descent path) Perform series of experiments at x values along path of steepest descent until no additional improvement in z response is obtained. This x value represents new estimate of best vector of factor levels. Step 5 (Stop or return) Step 5 (Stop or return) Go to step 1 and repeat process until final best factor level is obtained.

14. 17-14 Conceptual Illustration of RSM for Two Variables in x; Shows More Refined Experimental Design Near Solution Adapted from: Montgomery (2005), Design and Analysis of Experiments, Fig. 11-3

15. 17-15 Nonlinear Design Assume model z = h( ,  x) + v, where  enters nonlinearly and x is r-dimensional input vector D-optimality remains dominant measure –Maximization of determinant of Fisher information matrix (from Chapter 13 of ISSO: F n ( , X) is Fisher information matrix based on n inputs in n  ×  r matrix X) Fundamental distinction from linear case is that D- optimal criterion depends on  Leads to conundrum: Choosing X to best estimate , yet need to know  to determine X

16. 17-16 Strategies for Coping with Dependence on  Assume nominal value of  and develop an optimal design based on this fixed value Sequential design strategy based on an iterated design and model fitting process. Bayesian strategy where a prior distribution is assigned to , reflecting uncertainty in the knowledge of the true value of 

17. 17-17 Sequential Approach for Parameter Estimation and Optimal Design Step 0 (Initialization) Step 0 (Initialization) Make initial guess at , Allocate n 0 measurements to initial design. Set k = 0 and n = 0. Step 1 (D-optimal maximization) Step 1 (D-optimal maximization) Given X n, choose the n k inputs in X = to maximize Step 2 (Update  estimate) Step 2 (Update  estimate) Collect n k measurements based on inputs from step 1. Use measurements to update from to Step 3 (Stop or return) Step 3 (Stop or return) Stop if the value of  in step 2 is satisfactory. Else return to step 1 with the new k set to the former k + 1 and the new n set to the former n + n k (updated X n now includes inputs from step 1).

18. 17-18 Comments on Sequential Design Note two optimization problems being solved: one for , one for  Determine next n k input values (step 1) conditioned on current value of  –Each step analogous to nonlinear design with fixed (nominal) value of  “Full sequential” mode (n k = 1) updates  based on each new input  ouput pair (x k, z k ) Can use stochastic approximation to update  : where

19. 17-19 Bayesian Design Strategy Assume prior distribution (density) for , p(  ), reflecting uncertainty in the knowledge of the true value of . There exist multiple versions of D-optimal criterion One possible D-optimal criterion: Above criterion related to Shannon information doesWhile log transform makes no difference with fixed , it does affect integral-based solution To simplify integral, may be useful to choose discrete prior p(  )

20. 17-20 Appendix to Slides for Chapter 17: Factorial Design (not in ISSO; see ref. [1] below) Classical experimental design deals with linear models Factorial designFactorial design is most popular classical method – All r inputs (“factors”) changed at one time (note: ref. [1] uses notation m instead of r) Factorial design provides two key advantages over one-at- a-time changes: 1. Greater efficiency in extracting information from given number of experiments 2. Ability to determine if there are interaction effects Standard method is 2 r factorial; “2” comes about by looking at each input at two levels: low (  ) and high (+) – E.g., if r = 3, then have 2 3 = 8 input combinations: (    ), (+   ), (  +  ), (   +), (+ +  ), (+  +), (  + +), (+ + +) [1] Spall, J. C. (2010), “Factorial Design for Choosing Input Values in Experimentation: Generating Informative Data for System Identification,” IEEE Control Systems Magazine, vol. 30(5), pp. 38−53.

21. 17-21 Appendix to Slides (cont’d): Factorial Design with 3 Inputs Consider r = 3 linear model z k =  0 +  1 x k1 +  2 x k2 +  3 x k3 +  4 x k1 x k2 +  5 x k1 x k3 +  6 x k2 x k3 +  7 x k1 x k2 x k3 + noise, where  = [  0,   1,…,   7 ] T represents vector of (unknown) parameters and x ki represents i  th term in input vector x k all2 3 factorial design allows for efficient estimation of all parameters in  In contrast, one-at-a-time provides no information for estimating  4 to  7 However, 2 3 factorial design must be augmented in some way if wish to add quadratic (e.g., ) or other higher- order polynomial terms to model

22. 17-22 Appendix to Slides (cont’d): Illustration of Interaction with 2 Inputs no interaction interactionExample responses for r = 2: no interaction and interaction between input variables does not doesLeft plot (no interaction) shows that change in z k with change in x k2 does not depend on x k1 ; right plot (interaction) shows change in z k does depend on x k1 No interaction zkzk (  +) xk2xk2 ( )( ) (+ +) (+  ) X k1 = low X k1 = high Interaction zkzk (  +) xk2xk2 ( )( ) (+ +) (+  ) X k1 = low X k1 = high

23. 17-23 Appendix to Slides (cont’d): Efficiency of Factorial Design for Main Effects Factorial design estimates “main effects” (non-interaction) with greater efficiency than one-at-a-time changes Plot below based on same accuracy in estimation for the two methods 8 246810121416 1 2 3 4 5 6 7 Ratio of number of runs needed: one-at-a-time / factorial Input dimension r