TD C. Space filling designs in R

Transcription

1 TD C Sace filling designs in R 8/05/0

2 Tyical engineering ractice : One-At-a-Time (OAT design X P P3 P X Main remarks : OAT brings some information, but otentially wrong Eloration is oor : on monotonicity? Discontinuity? Interaction? Summer school CEA-EDF-IRIA 0 - TD C 04/07/

3 Illustration of the curse of dimensionality log(ratio(k oints in a 3D sace Bad sace covering = =3 vol. shere( r 0.5 / log(ratio Surf. circle / Surf. square ~ 3/4 Vol. shere / Vol. cube ~ / =0 Ratio ~ k hyercube volume >> (included and tangent hyershere volume For large dimensions, all the oints will be in the corner of the hyercube Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 3

4 Model eloration goal GOAL : elore as best as ossible the behaviour of the code Put some oints in the whole inut sace in order to «maimize» the amount of information on the model outut Contrary to an uncertainty roagation ste, it deends on Regular mesh with n levels =n simulations To minimize, needs to have some techniques ensuring good «coverage» of the inut sace Simle random samling (Monte Carlo does not ensure this E: =, n =3 =9 = 0, n=3 = E: = = 0 Monte Carlo Otimized design Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 4

5 Eloration in hysical eerimentation Design of eeriments develos strategies to define eeriments in order to obtain the required information as efficiently as ossible Designs for real eeriments Estimate arameters of linear regression with a minimal number of oints Eamles : Full factorial design 3 Fractional factorial design 3- Designs for numerical eeriments Characteristics Deterministic eeriments (no error, Large number of inut variables, Large range of inut variation domain, Multile outut variables, arameter Strong interactions between inuts, High non linearity in the model arameter arameter 3 sace filling designs (uniform coverage in the inut sace Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 5

6 Objectives When the objectives is to discover what haens inside the model and when no model comutations have been realized, we want to resect the two following constraints: To sread the oints over the inut sace in order to cature non linearities of the model outut, To ensure that this inut sace coverage is robust with resect to dimension reduction. Therefore, we look some design whech insures the «best coverage» of the inut sace How to define this «best»? How to choose the right number of oints? How to measure the reresentativity? Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 6

7 The design of numerical eeriments: Sace filling Sarsity of the sace of the inut variables in high dimension The learning design choice is made in order to have an otimal coverage of the inut domain The sace filling designs are good candidates. Simle Random Samle (SRS Sace Filling Design (SFD Eamle: Sobol sequence Two ossible criteria:. Distance criteria between the oints: minima, maimin,. Uniformity criteria of the design (discreancy measures Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 7

8 Distance criteria between the oints [ Johnson et al., 990 ] Minima design D MI : Minimize the maimal distance between the oints min ma d(, D D where d(, D All oints in [0,] are not too far from a design oint => One of the best design but too eensive to find D MI ma d(, D min d(, ( 0 D MI (0 Maimin design D MA : Maimize the minimal distance between the oints ma D ( ( ( min d(, min d(, ( ( ( (, D, D MA ( [ From: Owen, 996 ] Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 8

9 Eercise a Build a design based on the maimin criterion Characteristics : = variables U[0,] ; = 9 oints The idea is to generate a high number od random designs, then to select the best, by using the mindist( function of the DiceDesign ackage b Visualize this random maimin design with resect to a ure random design c Build a full factorial design, by using the factdesign( function of the DiceDesign ackage : variables with 3 levels => 9 oints Visualize this design with resect the two others Comare the maimin criteria of these 3 designs (random/maimin/factorial d Identify the roblems related to the full factorial designs Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 9

10 Sace filling measure of a design: the discreancy Measure of the maimal deviation between the distribution of the samle s oints to an uniform distribution Measure of deviation from the uniformity Geometrical interretation: Comarison between the volume of intervals and the number oints within these intervals Q( t [0,[, Q( t [0, t [ [0, t [ [0, t disc( D su Q( t [0,[ Q( t i t i [ Lower the discreancy is, the more the oints of the design D fill the all sace Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 0

11 Summer school CEA-EDF-IRIA 0 - TD C 04/07/ Discreancy comutation in ractice Modified L -discreancy (intervals with minimal boundary 0 Centered L -discreancy (intervals with boundary one verte of the unit cube Symetric L -discreancy (intervals with boundary one «even» verte of the unit cube Different definitions, deending on the chosen norm & considered intervals Classical choice (easy comutations: L discreancy j i k j k i k j k i k i k i k i k D, ( ( ( ( ( ( 3 ( disc

12 Relation with the integration roblem I [0,[ f ( d M ontecarlo : I with MC ( i i... i f ( ( i a sequence of random ointsin [0,[ MC MC Var( I I ; Var I O General roerty: V( f disc( D With a low discreancy sequence D (quasi Monte Carlo sequence : Well-known choice: Sobol sequence O ln Summer school CEA-EDF-IRIA 0 - TD C 04/07/

13 Sobol sequence vs. Random samle vs. regular grid [ From: Kucherenko, 00 ] Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 3

14 Eercise a Build a sequence of Sobol by using the sobol( function of the randtoolbo ackage Characteristics : = variables U[0,] ; = 9 oints Visualize this design and comare its maimin criterion with those of eercise b Comute a discreancy criterion of Sobol sequence and designs of eercise by using the discreancycriteria( function of the DiceDesign ackage c Build a sequence of Sobol with = 8 variables U[0,] ; = 00 oints Visualize all the scatterlots with the airs( function What kind of anomalies do you detect? PS: this roblem is much more regnant with Faure and Halton sequences Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 4

15 The design of numerical eeriments: LHS A lot of models are additive. If not, first order effects often dominate Proerty of uniform rojections on the margins It can be obtained via a Latin Hyercube Samle Divide each dimension in intervals Take one oint in each stratum Eamle : =, =0, X ~ U[0,], X ~ (0, Eamle : =, =4 Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 5

16 Algorithm of LHS Stein method Samle with oints of inuts ran = matri(runif(*,nrow=,ncol= # tirage de valeurs selon loi U[0,] for (i in : { id = samle(: # vecteur de ermutations des entiers {,,,} P = (id-ran[,i] / # vecteur de robabilites [,i] <- quantile_selon_la_loi (P } Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 6

17 [,] Otimization of LHS Joining the two roerties (sace filling and LHS One ossibility: generate a large number (for e: 000 of different LHS Then, choose the LHS which otimizes the criterion Eamle: = = 6 Maimin LHS Low wra-around For comarison: discreancy LHS [,] Sobol sequence Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 7

18 Eercise 3 a Build a random LHS and a maimin LHS by using the ackage lhs Characteristics : = variables U[0,] ; = 0 oints Visualize these two designs and comare them with a ure random design b BOUS : sensitivity analysis Look at the ackage sensitivity which allows to erform some global sensitivity analysis, and esecially to obtain variance-based sensitivity indices Run the eamle at the end of hel age of sobol00( (on Sobol g-function Relace the two initial indeendent Monte Carlo samles needed by sobol00( by some indeendent Sobol sequences ; then look at the results in terms of sensitivity estimates and errors Eact st order indices : S=0.76; S=0.79; S3=0.04 ; S4=0.007 ; S5=0 ; S6=0 ; S7=0 ; S8=0 Eact total indices : ST=0.786; ST=0.4; ST3=0.034; ST4=0.00; ST5=0; ST6=0; ST7=0 ; ST8=0 Summer school CEA-EDF-IRIA 0 - TD C 04/07/ 8