Sequence Construction For Integral Estimation

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