Sequence Construction For Integral Estimation
Transcription
Sequence Construction For Integral Estimation
Sequence Construction For Integral Estimation S. Varet1, S. Lefebvre1, A. Roblin1, G. Durand1, S. Cohen2 1ONERA/DOTA 2IMT/LSP - Chemin de la Hunière - 91761 Palaiseau Cedex - 118 route de Narbonne - 31062 Toulouse Cedex 9 Context Dimension Reduction Data input x1,…,xp Integral estimation Usual methods : - Factorial design + F-test : Gaussian assumption Code IRS ≡f Uncertainty on the input data (x(i))i=1,..., N Low discrepancy sequence (x(i))i=1,..., N Uniform distribution - Functional ANOVA : Envelope describing the set of possible values of the signature Quasi-Monte Carlo method Monte Carlo method The IRS is not Gaussian Convergence speed in O( Independence assumption and 1 ) N Convergence speed in O( (log N ) p ) N requires a lot of function evaluation Too time consuming The input variables are not independent Estimation of P ( a < f ( x1 , . . . ,x p ) < b ) Better rate of convergence Too time consuming Combination of the 2 steps Sobol indices estimation with a 2 levels Definition of the effective discrepancy: Propose a point x Gives more importance to important projections IS u ≈ IS frac u D i ( x ( 1 ) ,..., x ( i ) ) = ∑ IS frac u D L 2 ( x u(1 ) ,..., x u( i ) ) (i) = ( x 1i ,..., x i p ) (1) (i ) Compute the effective discrepancy Di of ( x ,..., x ) If Di-1 > kiDi then accept the point u Function dependent Construction algorithm While i <N: Weighted sum of the projections discrepancies with the Sobol indices as weights fractional factorial design Else accept the point with probability pi Points dependent i = i + 1 The effective discrepancy The effective discrepancy of several sequences for f(x): f(x) : [-1; 1]20 → 20 134x25x26 f ( x1 ,..., x20 ) = ∑ λi xi + 31x1 x3 + 34 x2 x4 + 0,00001x18 x19 + 0,000015x14 x15 i =1 i 1 2 3 4 5 6 7 8 9 10 λi 30 4.10-5 32 3.10-5 2.10-5 1 1 1 5.10-5 10-5 12 13 14 15 16 17 i 11 λi 6.10-5 1,5.10-5 2,5.10-5 3,5.10-5 4,5.10-5 5,5.10-5 6,5.10-5 18 19 20 7.10-5 7,5.10-5 8.10-5 Important factors: x1, x3, x2, x4 Moderately important factors: x6, x7, x8 Important interactions: x1x3, x2x4 The effective discrepancy is correlated to the integration error Algorithm: Test 2 Algorithm: Test 1 f(x) : [-1; 1]30 → Estimation of E(f(X)) = 1,3 30 g(x) : [0; 1]6 → f ( x1 ,..., x30 ) = ∑ λi xi + 129 x29 x30 − 132 x27 x28 + 133x29 x26 − 134 x25 x26 + 135 x26 x27 + 1,3 i =1 i λi 16-20 21 22 23 24 25 26 27 28 29 30 ( −1) i +1 i10 −3 ( − 1) i +1 (i − 9 )10 −2 (− 1) i i 1-10 131 124 -150 -160 125 126 127 128 140 130 11-15 Algorithm parameters : ki = 1+1/(2i) and pi = 50/i Estimation of E(g(X)) = 0,259699 g ( x1,..., x6 ) = 1I f(x) < -40 = 1I -6,5.10-5 −2 x −68x +2 x −68x +4.10−5 x +3.10−4 x +136x x 1 IS frac 2 3 4 5 6 2 4 < −40 of f Algorithm parameters : ki = 1+1/(2i) pi = 50/i References [1] J.-J. Droesbeke, J. Fine et G. Saporta - Plans d’expériences. Applications à l’entreprise, Technip, 2002 [2] B. Efron et C. Stein - The jackknife estimate of variance, The Annals of Statistics 9 (1981), no. 3, p. 586–596 [3] E. Hlawka - Funktionen von beschränkter variation in der theorie der gleichverteilung, Annali di Matematica Pura ed Applicata 54 (1961), p. 325–333. [4] W. J. Morokoff et R. E. Caflisch - Quasi-random sequences and their discrepancies, SIAM Journal on Scientific Computing 15 (1994), no. 6, p. 1251–1279. Conclusions - A new way to estimate the Sobol indices - A new criterion to quantify the adequacy of the sequence to the function of interest - A new algorithm to select the points adapted to the function of interest