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CHAPTER 17 OPTIMAL DESIGN FOR EXPERIMENTAL INPUTS

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Title: CHAPTER 17 OPTIMAL DESIGN FOR EXPERIMENTAL INPUTS


1
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
  • Note Appendix to these slides is brief
    discussion of factorial design (not in ISSO)

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 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

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
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

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 wi
  • Finite-sample design allocates n ? N available
    measurements exactly asymptotic (continuous)
    design allocates based on n ? ?

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
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 ?

8
Variance Function as it Depends on Input Optimal
Asymptotic Design for Example 17.6 in ISSO
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
  • However, some designs are both
  • Classical factorial (cubic) designs are
    orthogonal (and often D-optimal)

10
Example Orthogonal Designs, r 2 Factors
x
x
k
2
k
2
x
x
k
1
k
1
r
Cube (2
design)
Star (2r
design)

11
Example Orthogonal Designs, r 3 Factors
xk3
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
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.

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
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 Fn(?, 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
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
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).

18
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

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
  • 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(?)

20
Appendix to Slides for Chapter 17 Factorial
Design (not in ISSO)
  • Classical experimental design deals with linear
    models
  • Factorial design is most popular classical method
  • All r inputs (factors) changed at one time
  • Factorial design provides two key advantages over
    one-at-a-time changes
  • Greater efficiency in extracting information
    from a given number of experiments
  • Ability to determine if there are interaction
    effects
  • Standard method is 2r factorial 2 comes about
    by looking at each input at two levels low (?)
    and high ()
  • E.g., if r 3, then have 23 8 input
    combinations
  • (? ? ?), ( ? ?), (? ?), (? ? ),
  • ( ?), ( ? ), (? ), ( )

21
Appendix to Slides (contd) Factorial Design
with 3 Inputs
  • Consider r 3 linear model
  • zk ?0 ?1xk1 ?2xk2 ?3xk3 ?4xk1xk2
    ?5xk1xk3 ?6xk2xk3 ?7xk1xk2xk3
    noise,
  • where ? ?0,? ?1,,? ?7T represents vector of
    (unknown) parameters and xki represents i?th term
    in input vector xk
  • 23 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, 23 factorial design must be augmented in
    some way if wish to add quadratic (e.g., )
    or other higher-order polynomial terms to model

22
Appendix to Slides (contd) Illustration of
Interaction with 2 Inputs
  • Example responses for r 2 no interaction and
    interaction between input variables
  • Left plot (no interaction) shows that change in
    zk with change in xk2 does not depend on xk1
    right plot (interaction) shows change in zk does
    depend on xk1

No interaction
Interaction
zk
Xk1 high
( ?)
(? )
Xk1 low
(? ?)
( )
xk2
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