Physical Experiments

1
Physical Experiments Computer
Experiments,PreliminariesChapters 12 Design
and Analysis of Experimentsby Thomas J.
Santner, Brian J. Williams and William I. Notz

• EARG presentation Oct 3, 2005by 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 inputssimple 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)

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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