EARG presentation Oct 3, 2005 by Frank Hutter

1. Physical Experiments &Computer Experiments,PreliminariesChapters 1&2 “Design and Analysis of Experiments“by 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: x=xc • 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 • Can‘t be computed since y(x) unknown, but in Chapter 6 we‘ll 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: x=Xe (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 x=xm (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) = E{y(xc,Xe)} • Upper alpha quantile  = (xc): • P{y(xc,Xe) >= } = 

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 doesn‘t matter so much for that xc • Maximize (xc) subject to constraints on variance w.r.t. Xe