to the estimation of total sensitivity indices
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1 Applcato of the cotrol o varate ate techque to the estmato of total sestvty dces S KUCHERENKO B DELPUECH Imperal College Lodo (UK) skuchereko@mperalacuk B IOOSS Electrcté de Frace (Frace) S TARANTOLA Jot Research Cetre of the Europea Commsso, Ispra (Italy)
2 Outle ANOVA decomposto ad Sobol Sestvty Idces Improved drect formula for evaluato of Sobol Ma Effect Sestvty Idces wth small values Mote Carlo estmates Varace reducto techques Improved drect formula for evaluato of Sobol Total Effect Sestvty Idces usg cotrol varates Results
3 ANOVA decomposto ad Sestvty Idces Y = f ( x) Cosder a model x s a vector of put varables f(x) s square tegrable 2 x = ( x, x,, x ) H ANOVA decomposto s uque f varables are depedet x k k ( ) ( ) ( ) Y = f ( x) = f + f x + f x, x + + f x, x,, x, j j,2,, k 2 k = j> f ( x,,, x ) dx =, k, k s, f f dx dx =, s s s k l k k l l Varace decomposto: D = D + D + D , j 2,2,, j Sobol SI: k = j S + S + S + + S, 2,, = < j < j < l jl k
4 Evaluato of Sobol Sestvty Idces Cosder two subsets of the total put vector: x = ( y, z) Drect formulas for evaluato of Sobol Sestvty dces [ (, ') ', S = y f y z dydzdz f D tot 2 S y = [ f( y, z) f( y', z)] dydzdz ', 2 2D = (, ) D f y z dydz f Evaluato of sestvty dces s reduced to hgh-dmesoal tegrato usg MC/QMC methods
5 Evaluato of Sobol Ma Effect Sestvty Idces wth small values x = ( y, z ), x' = ( y', z') usg values f( y, z), f( y, z'), f( y', z) 2 Sy = f( y, z) f( y, z') dydzdz' f 2 D for small dces << S y f ( yz, ) f ( yz, ') dydzdz d d ' f loss of accuracy 2
6 Improved formula for evaluato of Sobol Ma Effect Sestvty Idces 2 Notce that f f ( y, z) dydz f ( y ', z ') dy ' dz ' = usg values f( y, z), f( y', z), f( y', z') 2 Sy = f( y, z) f( y', z) dydy' dz f 2 D S y = 2 [ f ( y ', z ')[ f ( y ', z ) f ( y, z )] dydy ' dzdz ' D - gves much more accurate results (Kuchereko, Mautz, 22) Addtoal advatage (Saltell 22): Requres N ( +2) model evaluto rather tha N(2 +) for orgal Sobol' formulas Improved formulas for small dces: S Kuchereko: 22, 27: Further mprovemets: Sobol ad Mshetskaya 27, A Owe 22
7 Improved formula for Sobol Ma Effect Sestvty Idces Test T 6 f( x) = x, S = S = = ( + )(2 + ) = 8, S = 5 7 Comparso betwee mproved ad orgal formulas for S Comparso betwee mproved ad orgal formulas for S E- -5 E-2 lo og2(error) - -5 log2(s) E-3 E-4 E-5-2 E log2(n) E log2(n) mproved Sobol orgal Sobol mproved Sobol orgal Sobol aalytcal value
8 Mote Carlo tegrato I[ f] = f( x) dx H Mote ecarlo: I[ f] = E[ f( x)] N CrudeMoteCarloEstmate: Carlo IN[ f] = f( z) N = { z } s a sequece of radom (MC) or quas-radom (QMC) pots H Error: ε = I[ f] I [ f] N MC 2 /2 σ ( f ) εn = ( E( ε )) = /2 N Covergece does ot depet o dmesoalty but t s slow
9 How to mprove MC? σ ( f ) Slow covergece: ε N = /2 N To mprove MC covergece: Decrease σ ( f ) by applyg varace reducto techques: atthetc varables; cotrol varates; stratfed samplg; mportace samplg
10 Mote Carlo Itegrato Varace reducto Cotrol Varate Method Defe a ew fucto: f ( x) = f( x) + C( g( x ) µ g ) gx f ( x ) The cotrol ( ) s sampled alog wth µ = g( xdx ), so that E[ f ( x)] = E[ f( x)] g Varace Decomposto Var f Var f C Cov f g CVarg 2 [ f ] = [ f ] + 2 ( f, g ) + [ g ] Cov( f, g) fx () gx ( ) Where s the covarece betwee ad The optmal cotrol parameter C s obtaed by mmzg the varace: Reducto of varace: Cov( f, g) C = Var( g) Cov( f, g) Var[ f ] = Var[ f ] Var[ f ] Var[ g]
11 Evaluato of Sobol Total Sestvty Idces Recall Sobol-Jase formula for oe put varable: x= ( x, x ) j ~ j tot S j = f x f x 2 j x~ j dydzdz 2D ' 2 [ ( ) (, )] ' Ideally we eed to fd the cotrol fucto for ' [ f( x) f( xj, x~ j)] But t s ot possble a geeral case
12 Evaluato of Sobol Total Sestvty Idces usg cotrol varates approach ANOVA decomposto: ( ) ( ) ( ) f ( x) = f + f x + f x, x + + f x, x,, x j j,2,, 2 = j > g( x) = f + f( x) = Use the frst order terms as cotrol varates Theorem: tot ' 2 ' S = [ f ( x) f ( x ) [ f ( x ') f ( x )]] dxdx S 2 2D + j j j j j j j It s preseted for the case of a sgle varable x j The same approach ca be easly geeralsed for the case of a set of varables The effcecy of ths method ca be creased further by addg hgher order terms to cotrol varate gx ( ) = f + f ( x ) + f j ( x, x j ) = j> Ca result sgfcat mprovemet the effcecy of MC estmates, provded the frst order terms f j( x j) ad correspodg S j are kow They ca be foud for A) explctly tl gve aalytcal l fucto; B) fuctos approxmated by metamodels
13 Improved formula for Sobol Total Sestvty Idces, Test: G-fucto 4x 2+ a f = g( x), g( x) =, + a = a = {,, 45, 9, 99, 99, 99, 99} Comparso of covergeces umercally computed total SI to the aalytcal values Stadard blue, Reduced varace- red
14 Improved formula for Sobol Total Sestvty Idces, Test: G-fucto 4x 2+ a f = g( x), g( x) =, + a = a = {,, 45, 9, 99, 99, 99, 99} Root Mea Square Error vs Number of sampled pots
15 Improved formula for Sobol Total Sestvty Idces, Test: G-fucto 4x 2+ a f = g( x), g( x) =, + a = a = {,, 45, 9, 99, 99, 99, 99} Comparso of Hstograms: Dramatc varace reducto! [ f( x) f( x')] ' [ f ( x) f j( x j) [ f ( x ') f j( x j)]]
16 Improved formula for Sobol Total SI, Test: Modfed G-fucto f = = g( x), 4x a gx ( ) =, + a a = {9, 9, 4} Comparso of covergeces umercally computed total SI to the aalytcal values Stadard blue, Reduced varace- red
17 f = = g( x), 4x a gx ( ) =, + a a = {9, 9, 4} Improved formula for Sobol Total Sestvty Idces, Test: Modfed G-fucto Root Mea Square Error vs Number of sampled pots
18 f = = g( x), 4x a gx ( ) =, + a a = {9, 9, 4} Improved formula for Sobol Total Sestvty Idces, Test: Modfed G-fucto Comparso of Hstograms: Dramatc varace reducto! [ f( x) f( x')] ' [ f ( x) f j( x j) [ f ( x ') f j( x j)]]
19 Improved formula for Sobol Total Sestvty Idces, Test: Ishgam fucto 2 4 ( ) ( ) ( ) fxxx 2 3 x x2 x3 x π x (,, ) = s + 7s + s, π, =,2,3 Varable Varable 2 Root Mea Square Error vs N orgal Sobol-Jase formula (dark blue le); mproved formulas: based o metamodel (red le), aalytcal results for f j( xj) ad S j usg MC (gree le), aalytcal results for f j( x j) ad S j usg QMC (lght blue le)
20 Thak you for your atteto! Ackowledgmets Facal support: EPSRC Grat Publcatos: Sobol, IM, Taratola S, Gatell D, Kuchereko S, Mautz W Estmatg the Approxmato Error whe fxg Uessetal Factors Global Sestvty Aalyss, Relablty Egeerg g& System Safety, 92(7): , 27 Kuchereko S, Fel B, Shah N, Mautz W The detfcato of model effectve dmesos usg global sestvty aalyss Relablty Egeerg & System Safety 96 (2) Sobol dces for models wth depedet varables: S Kuchereko, S Taratola, P Ao Estmato of global sestvty dces for models wth depedet varables, Computer Physcs Commucatos, V 83 (22)
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