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CHAPTER 17 OPTIMAL DESIGN FOR EXPERIMENTAL

INPUTS

Slides for Introduction to Stochastic Search and

Optimization (ISSO) by J. C. Spall

- 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

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 design theory useful
- Building simulation model
- Typically need nonlinear design theory
- Some terminology distinctions
- Factors (statistics term) ? Inputs (modeling

and simulation terms) - Levels ? Values
- Treatments ? Runs

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

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, 409435 - 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

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 wi - Finite-sample design allocates n ? N available

measurements exactly asymptotic (continuous)

design allocates based on n ? ?

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

Equivalence Theorem

- Consider linear model
- Prediction based on parameter estimate and

future measurement vector hT is - Kiefer-Wolfowitz equivalence theorem states
- 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 ?

Variance Function as it Depends on Input Optimal

Asymptotic Design for Example 17.6 in ISSO

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 - However, some designs are both
- Classical factorial (cubic) designs are

orthogonal (and often D-optimal)

Example Orthogonal Designs, r 2 Factors

Example Orthogonal Designs, r 3 Factors

xk3

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

Steps of RSM for Optimizing x

- Step 0 (Initialization) Initial guess at optimal

value of x. - 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) 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) Based on

response surface in step 2, estimate path of

steepest descent in factor space. - 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) Go to step 1 and repeat

process until final best factor level is

obtained.

Conceptual Illustration of RSM for Two Variables

in x Shows More Refined Experimental Design Near

Solution

Adapted from Montgomery (2001), Design and

Analysis of Experiments, Fig. 11-3

Nonlinear Design

- Assume model
- z h(?,?x) v ,
- where ? enters nonlinearly
- D-optimality remains dominant measure
- Maximization of determinant of Fisher information

matrix (from Chapter 13 of ISSO Fn(?, x) is

Fisher information matrix based on n data points) - 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

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 ?.

Sequential Approach for Parameter Estimation and

Optimal Design

- Step 0 (Initialization) Make initial guess at

?, Allocate n0 measurements to initial

design. Set k 0 and n 0. - Step 1 (D-optimal maximization) Given Xn , choose

the nk inputs in X to maximize - Step 2 (Update ? estimate) Collect nk

measurements based on inputs from step 1. Use

measurements to update from to - 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 nk (updated Xn now

includes inputs from step 1).

Comments on Sequential Design

- Note two optimization problems being solved one

for ?, one for ? - Determine next nk input values (step 1)

conditioned on current value of ? - Each step analogous to nonlinear design with

fixed (nominal) value of ? - Full sequential mode (nk 1) updates ? based

on each new input?ouput pair (xk , zk) - Can use stochastic approximation to update ?
- where

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
- While log transform makes no difference with

fixed ?, it does affect integral-based solution. - To simplify integral, may be useful to choose

discrete prior p(?)