Application of Global Sensitivity Indices for measuring the effectiveness of Quasi-Monte Carlo methods and parameter estimation. parameter.

Transcription

1 Applcato of Global estvty Idces for measurg the effectveess of Quas-Mote Carlo methods ad parameter estmato parameter estmato Kuchereko Emal:

2 Outle Advatages ad dsadvatages of Mote Carlo method Why obol sequeces are so effectve? Is ths true that for hgh-dmesoal problems Quas Mote Carlo methods offer o practcal advatage over Mote Carlo? Effectve dmeso versus omal dmeso ad ther lk wth global estvty Idces Classfcato of fuctos ad fuctoals based o global sestvty dces Applcato of parametrc GA for optmal expermetal desg 2

3 Comparso determstc ad Mote Carlo tegrato methods r r I[ f] = f( x) dx H Determstc tegrato method of p-order, k pots each drecto: = k p /p Error: = Ok ( ), =O(/ ) 2 Estmate:, p 2, 5 = d ε 5 ε ε = = = the total umber of partcles the uverse I[ f] s mpossble to evaluate! "Curse of Dmesoal ty " 3

4 Mote Carlo tegrato methods r I[ f] = E[ f( x)] r Mote Carlo : I[ f] = f( z) = r { z} s a sequece of radom pots H Error: ε = I[ f] I [ f] 2 2 σ ( f ) Expectato: E( ε ) = 2 /2 σ ( f ) ε = ( E( ε )) = /2 Covergece does ot depet o dmesoalty but t s slow 4

5 low covergece: ε = How to mprove MC? σ ( f ) /2 How to mprove MC? I Decrease σ ( f ) varace reducto II Use better ( more uformly dstrbuted ) sequeces Dscrepacy s a measure of devato from uformty: r r Qy ( ) H, Q( y) = [, y ) [, y ) [, y ), mq ( ) volume of Q D = r Q( y) sup m( Q), r Q( y) H 2 radom sequeces: D (l l ) / ~ / /2 /2 5

6 Quas radom sequeces (l ) D c( d) Low dscrepacy sequeces (LD) Covergece: ε = I[ f] I [ f] V( f) D V( f) Varato of f O(l ) εqm C = Assymptotcally ε ~ O(/ ) much hgher tha ε MC ~ O(/ ) QMC QMC 6

7 Projectos of Dfferet 2D sequeces to D Regular Grd Radom umbers LD 7

8 ormally dstrbuted obol equeces ormal probablty plots Hstograms 8

9 Dscrepacy I Low Dmesos Dscrepacy, =5 Radom Halto obol Dscrepacy, =2 Radom Halto obol e-5 e

10 Dscrepacy II Hgh Dmesos e-5 e-6 Dscrepacy, =5 Radom Halto obol e-7 e-8 e-9 e- e e-6 e-7 Dscrepacy, = Radom obol MC hgh-dmesos has smaller dscrepacy e-8 e-9 e

11 Are QMC effcet for hgh dmesoal problems O(l ) εqmc = Assymptotcally ε ~ O(/ ) QMC but ε creseas wth utl exp( ) QMC 2 5, 5 ot achevable for practcal applcatos = For hgh-dmesoal problems ( > 2), QMC offers o practcal advatage over Mote Carlo ( Bratley, Fox, ad ederreter (992))?!

12 Pros: Cos: Is MC more effcet for hgh-dmesoal problems tha QMC? MC hgh-dmesos has smaller dscrepacy ome studes show degradato of the covergece rate of QMC methods hgh-dmesos to O(/ ) Huge success of QMC methods face: QMC methods were prove to be much more effcet tha MC eve for problems wth thousads of varables Log-log plot of the root mea square error vrs umber of paths for QMC ad MC methods Asa call optos =252 Log(RME) QMC MC redle -QMC, /^82 redle - MC, /^5 Log(_path) 2

13 estvty Idces (I) Cosder a model x s a vector of put varables Y s the model output Y = f ( x) x = ( x, x,, x ) 2 x k s s k k ( ) ( ) ( ) Y = f ( x) = f + f x + f x, x + + f x, x,, x, AOVA decomposto (HDMR): j j,2,, k 2 k = j> f ( x,,, x ) dx =, k, k s Varace decomposto: K σ = σ + σ + σ , j,2,, j obol I: k = + = + j + jl +, 2,, < j < j < l 3 k

14 Effectve dmeso Let u be a cardalty of a set of varables u he effectve dmeso of f( x) the superposto sese s the smallest teger d such that < u< d u ( ε), ε << It meas that f( x) s almost a sum of d -dmesoal fuctos he fucto f( x) has effectve dmeso the trucato sese d f u {,2,, d } Example: d d ( ε), ε << u ( ) f x = = x d =, d = does ot deped o the order whch the put varables are sampled, - depeds o the order by reoderg varables d ca be reduced 4

15 x = ( x,, x ) For may problems oly low order terms the AOVA decomposto are mportat d << Cosder a approxmato error δ ( f, h) Lk betwee a approxmato error ad effectve dmeso superposto sese Approxmato errors heorem : d = d : δ( f, h) d hx ( ) = f + f ( x,, x) s s= < < 2 δ ( f, h) = [ f ( x) h( x) ] dx σ d δ ( f, h) = ( x,, x ) ε s= < < s s s s s s s x= ( y, z) : y = ( x,, x ), z = ( z,, x ) et of varables ca be regarded as ot mportat tot f If = ad << z d d+ z z z Cosder a approxmato error ( ) f ( x) f y, z δ ( z ) heorem 2: Lk betwee a approxmato error ad effectve dmeso trucato sese d = d : δ ( z [ ] 2 ) = f( x) f( y, z) dx D tot Eδ ( z ) = 2 Eδ( z ) 2ε z 5

16 Classfcato of fuctos ype A Varables are ot equally mportat y y z >> d << z ype B,C Varables are equally mportat d j ype B Domat low order dces = d << ype C Domat hgher order dces = d << 6

17 estvty dces for type A fuctos y y z >> d << z 7

18 Itegrato error vs ype A (a) f(x) = j= (-) Π j= x j, = 36, (b) f(x) = Π s = 4x -2 /(+a ), = K k ε = ( I I ) K k = 2 /2-5 y y z >> d << z log2(ε ) - QMC (-94) MC (-52) ε ~ α, < α < (a) = 94, = {,2} {3,4,36} log 2 () -2 (b) log2(ε ) -6 - QMC (-69) MC (-49) = = 64 {,2} {3,4,} log 2 () 8

19 estvty dces for type B fuctos Domat low order dces = d << 9

20 Itegrato error vs ype B Domat low order dces = d << - (a) log2(ε ) -4-8 QMC (-66) MC (-5) x f ( x) = 5 = 36 = log 2 () -7 (b) log2(ε ) - QMC (-66) MC (-53) f ( x) = ( + = 36 = ) x / log 2 () 2

21 estvty dces for type C fuctos Domat hgher order dces = << d 2

22 he tegrato error vs ype C Domat hgher order dces: = << d (a) log2(ε ) QMC (-46) MC (-45) 4x 2 + a f( x) =, a = + a = = 2 = 4x 2 log 2 () - (b) log2ε) -3-5 QMC (-44) MC (-44) f( x) = (/2) = 2 / = x log 2 () 22

23 Applcato of GA to path depedet tegrals I = F[ x( t)] d x, C W xt ( ) cotuous t, x() = x I = E( F[ ξ( t)]), ξ( t) radom Weer processes (a Browa moto) Mote Carlo approach: to costruct may radom paths ξ( t), evaluate ad average results Practcal applcatos: hare prce follows geometrcal Browa moto: /2 d =µ dt +σdξ, dξ= z( dt), z ~ (,) 2 t () = exp[( µ σ ) t+σξ())], t ξ() t Weer path 2 23

24 Approxmatos of path depedet tegrals usg stadard ad Browa brdge dscretzatos DE: dξ= z dt, z ~ (,) tadard algorthm: Browa brdge algorthm: ξ( t ) = ξ( t ) + tz, t = /, + + ξ( ) = ξ + z, ξ( /2) = ( ξ( ) + ξ) + z2, 2 2 ξ( /4) = ( ξ( /2) + ξ) + /2 z3, 2 2 ξ(3 / 4) = ( ξ( / 2) + ξ( )) + / 2 z4, 2 2 M ξ(( ) / ) = ( ξ(( 2) / ) + ξ( )) + 2 / z

25 Global sestvty aalyss of F Fuctoal: umercal approxmato: a, a j F[ ξ ( t)] = ξ ( t) dt 2 2 = + j j = < j F az a zz F( z,, z ) deped o ad o the type of the approxmato Expectato: AOVA decomposto: Varace: estvty dces: I = E( F ) = a 2 = + + j j < j F I a ( z ) a z z 2 2 = + j < j DF ( ) 2 a a 2 = 2 a / D( F ), 2 j = j a / D( F ) 25

26 Frst order sestvty dces versus dex umber, =32 Browa brdge tadard approx Log() e-5 e-6 e Idex 26

27 GA of two algorthms at dfferet tadard approxmato: the effectve dmeso d ¾ Browa Brdge approxmato: both effectve dmesos are close to 2 27

28 What s the optmal way to arrage pots two dmesos? Regular Grd obol equece Low dmesoal projectos of low dscrepacy sequeces are better dstrbuted tha hgher dmesoal projectos 28

29 Covergece curves at =64 K k ε = ( I I ), K k = 2 /2 Browa brdge, obol seq tadard approx, obol seq Browa brdge, Radom tadard approx, Radom Log(Error) MC ~ O(/ ) QMC ~ O(/ ) e-5 e Log2() Browa Brdge dscretzatos - the effectve dmeso reducto techque 29

30 Orgal vrs Improved formulae for evaluato of obol estvty Idces y = f ( y, z) f ( y, z ') dydzdz ' f f ( y, z) dydz f for small dces << f ( y, z) f ( y, z ') dydzdz ' f loss of accuracy otce that f = f( y, z) dydz f( y', z') dy ' dz' y 2 [ ] f( y, z) f( y, z') f( y', z') dydzdz' y = f ( y, z) f f( y, z) + f dydz much more accurate 2 Requres ( +2) model evaluto orgal obol' formulas (2 +) he same model evaluatos ca be used for computg secod order dces 3

31 Improved formula for obol estvty Idces f( x) = x, = = = = 8, = ( + )(2+ ) Comparso betwee mproved ad orgal formulas for Comparso betwee mproved ad orgal formulas for E- -5 E-2 log2(error) - -5 log2() E-3 E-4 E-5-2 E log2() E log2() mproved obol orgal obol mproved obol orgal obol aalytcal value 3

32 Optmal expermetal desg (OED) for parameter estmato Fd values of expermetally mapulable varables (cotrols) ad the tme samplg strategy for a set of exp expermets whch provdes maxmum formato for the subsequet parameter estmato problem subject to: ystem dyamcs (ODEs, DAEs) Other algebrac costrats Upper ad lower bouds: u L u u U o-lear programmg problem (LP) wth partal dfferetal-algebrac (PDAEs) costrats 32

33 Case study: fed-batch reactor Bomass: ubstrate: dy dt Reacto rate: ( rm u ) y p = y dy r y m dt ( u ) 2 = + u y 2 2 p2 5y2 r m = 5 + y 2 Parameters to be estmated: p, p 2 5 < p < 98, 5 < p 2 < 98 Cotrol varables: u, u 2 Dluto factor: 5 < u < 5 Feed substrate cocetrato: 5 < u 2 < 5 33

34 OED tradtoal approach Fsher Iformato Matrx ( FIM ) based crtera: A-optmalty FIM = = y p A crtero = D crtero = ( t ) W ( t ) m max y p [ trace( FIM )] [ det( FIM )] θ 2 D-optmalty E-optmalty E crtero = [ ( FIM )] max λ m Modfed-E crtero = λ m max λ m ( FIM ) ( ) FIM θ Ma drawback: based o local I o-realstc lear ad local assumptos 34

35 Parametrc GA r r r olear dyamc model: Y = f ( p, u, t) r p - ucerta parameters, r u - cotrol varables, t - tme r r r olve: max r F( ( u, t)) u r OED u *for parameter estmato Fd ( u, t), ( u, t) deped o parameters! Optmal expermetal desg: detfcato of a set of expermets wth codtos that delver measuremet data that are the most sestve to the ukow parameters 35

36 Applcato of Parametrc GA for parameter optmzato r r = (, ) (, ) FIM = Q u t W Q u t r r = (, ) (, ) GIM = Q u t W Q u t ( ) Qt y r y r y r ( ut, ) ( ut, ) L ( ut, ) p p2 p p = M M O M y r s y r s y r s ( ut, ) ( ut, ) L ( ut, ) p p2 pp ( ) Qt r r r, ut,2 ut, ut r r r ut ut ut M M O M r r r ut ut ut (, ) (, ) (, ) L p (, ) (, ) (, ) L p 2, 2,2 2, = (, ) (, ) L (, ) s, s,2 s, p Ma advatage: based o global I allows to cosder a rage of values for the parameters to be estmated objectve fucto: max det Applcato of Global Optmzato method r u ( GIM ) 36

37 Case study: fed-batch reactor Bomass: ubstrate: dy dt Reacto rate: ( rm u ) y p = y dy r y m dt ( u ) 2 = + u y 2 2 p2 5y2 r m = 5 + y 2 Parameters to be estmated: p, p 2 5 < p < 98, 5 < p 2 < 98 Cotrol varables: u, u 2 Dluto factor: 5 < u < 5 Feed substrate cocetrato: 5 < u 2 < 5 37

38 Cocetrato (g/l) Problem costrats: Optmal Expermetal Desg Expermet durato: h umber of measuremet tmes: Results: y y2 Cotrols vared every 2 hours u (h - ) Optmal put profle for u ad u 2 : u2 (g/l) me (h) me (h) 38

39 teps to fd p: ettg of the Parameter Estmato Problem ake expermetal or geerated pseudo-expermetal pots Maxmum lkelhood optmzato y% p: set of parameters to be estmated : model predcto J ml E V M 2 ( p) = ( 2πσ jk ) = j= k = y k ( p) 2 exp 2 y jk ( p) σ ~ y jk jk 2 2 σ k : measuremets varace : expermetal measures y~k subject to: ystem dyamcs (ODEs, DAEs) Other algebrac costrats Upper ad lower bouds: p L p p U o-lear programmg problem (LP) wth partal dfferetal-algebrac (PDAEs) costrats 39

40 Results of parameter estmato p = 5 ± 5, p 2 = 5 ± Desty Desty p p = 37 ± 2, p 2 = 72 ± p2 2 Desty 5 Desty p p2 4

41 ummary Global estvty Aalyss ca be successfully used for measurg the effectveess of Quas-Mote Carlo methods Quas Mote Carlo methods based o obol sequeces outperform Mote Carlo eve hgh dmesos By reducg the effectve dmeso the effcecy of Quas Mote Carlo methods ca be further mproved Applcato of global I to OED results the reducto of the requred expermetal work ad the creased accuracy of parameter estmato 4

42 hak to AMO orgazg commttee for vtg me to the coferece! Prof obol Ackowledgmets Wolfgag Mautz, María Rodríguez Ferádez, lay hah, Costas Pateldes EPRC Grat 42