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Physical Experiments

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Title: Physical Experiments


1
Physical Experiments Computer
Experiments, Preliminaries Chapters 12 Design
and Analysis of Experiments by Thomas J.
Santner, Brian J. Williams and William I. Notz
  • EARG presentation Oct 3, 2005 by Frank Hutter

2
Preface
  • What they mean by Computer Experiments
  • Code that serves as a proxy for physical process
  • Can modify inputs and observe how process output
    is affedcted
  • Math
  • Should be understandable with a Masters level
    training of Statistics

3
Overview of the book
  • Chapter 1 intro, application domains
  • Chapter 2
  • Research goals for various types of inputs
    (random, controlled, model parameters)
  • Lots of definitions, Gaussian processes
  • Chapters 3-4 Predicting Output from Computer
    Experiments
  • Chapter 5 Basic Experimental Design (similar to
    last term)
  • Chapter 6 Active Learning (sequential
    experimental design)
  • Chapter 7 Sensitivity analysis
  • Appendix C code

4
Physical experiments a few concepts we saw last
term (2f)
  • Randomization
  • In order to prevent unrecognized nuisance
    variables from systematically affecting response
  • Blocking
  • Deals with recognized nuisance variables
  • Group experimental units into homogeneous groups
  • Replication
  • Reduce unavoidable measurement variation
  • Computer experiments
  • Deterministic outputs
  • None of the traditional principles ... are of
    use

5
Types of input variables (15f)
  • Control variables xc
  • Can be set by experimenter / engineer
  • Engineering variables, manufactoring variables
  • Environmental variables Xe
  • Depend on the environment/user
  • Random variables with known or unknown
    distribution
  • When known for a particular problem xe
  • Noise variables
  • Model variables xm
  • Parameters of the computer model that need to be
    set to get the best approximation of the physical
    process
  • Model parameters, tuning parameters

6
Examples of Computer Models (6ff)
  • ASET (Available Safe Egress Time)
  • 5 inputs, 2 outputs
  • Design of Prosthesis Devices
  • 3 environment variables, 2 control variables
  • 2 competing outputs
  • Formation of Pockets in Sheet Metal
  • 6 control variables, 1 output
  • Other examples
  • Optimally shaping helicopter blade 31 control
    variables
  • Public policy making greenhouse gases 30 input
    variables, some of them modifiable (control
    variables)

7
ASET (Available Safe Egress Time) (4f)
  • Evolution of fires in enclosed areas
  • Inputs
  • Room ceiling height and room area
  • Height of burning object
  • Heat loss fraction for the room (depends on
    insulation)
  • Material-specific heat release rate
  • Maximum time for simulation (!)
  • Outputs
  • Temperature of the hot smoke layer
  • Distance of hot smoke layer from fire source

8
Design of Prosthesis Devices (6f)
  • 2 control variables
  • b, the length of the bullet tip
  • d, the midstem parameter
  • 3 environment variables
  • ?, the joint angle
  • E, the elastic modulus of the surrounding
    cancellous bone
  • Implant-bone interface friction
  • 2 conflicting outputs
  • Femoral stress shielding
  • Implant toggling
  • (flexible prostheses minimize stress, but toggle
    more ? loosen)

9
Formation of Pockets in Steel (8ff)
  • 6 control variables
  • Length l
  • Width w
  • Fillet radius f
  • Clearance c
  • Punch plan view radius p
  • Lock bead distance d
  • Output
  • Failure depth (depth at which the metal tears)

10
Research goals for homogeneous-input codes (17f)
  • Homogeneous-input only one of the three possible
    variable types present
  • All control variables xxc
  • Predict y(x) well for all x in some domain X
  • Global perspective
  • Integrated squared error sX y(x) - y(x)2 w(x)
    dx
  • Cant be computed since y(x) unknown, but in
    Chapter 6 well replace y(x)-y(x)2 by a
    computable posterior mean squared value
  • Local perspective
  • Level set Find x such that y(x) t0
  • t0 maximum value

11
All environmental variables xXe (18)
  • How does the variability in Xe transmit through
    the computer code ?
  • Find the distribution of y(Xe)
  • When the problem is to find the mean
  • Latin hypercube designs for choosing the training
    sites

12
All model variables xxm (18f)
  • Mathematical modelling contains unknown
    parameters (unknown rates or physical constants)
  • Calibration (parameter fitting)
  • Choose the model variables xm so that the
    computer output best matches the output from the
    physical experiment

13
Research goals for mixed inputs (19ff)
  • Focus on case with control and environmental
    variables x(xc,Xe) where Xe has a known
    distribution
  • Example hip prosthesis
  • y(xc,Xe) is a random variable whose distribution
    is induced by Xe
  • Mean ?(xc) Ey(xc,Xe)
  • Upper alpha quantile ?? ??(xc)
  • Py(xc,Xe) gt ?? ?

14
Research goals for mixed inputs simple adaption
of previous goals (20ff)
  • Predict y well over its domain
  • Minimize sX ?(x) - ?(x)2 w(x) dx
  • (again, there is a Bayesian analog with
    computable mean)
  • Maximize the mean output maxxc ?(xc)

15
When the distribution of Xe is unknown
  • Various flavours of robustness
  • G-robust minimax
  • Want to minimize your maximal loss
  • Pessimistic
  • ?(.)-robust
  • Minimize weighted loss (weighted by prior density
    on distribution over Xe)
  • M-robust
  • Suppose for a given xc, y(xc,xe) is fairly flat
  • Then value of Xe doesnt matter so much for that
    xc
  • Maximize ?(xc) subject to constraints on variance
    w.r.t. Xe
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