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Accessible Unlicensed Requires Authentication Published by De Gruyter January 10, 2021

On the dependence structure and quality of scrambled (t, m, s)-nets

Jaspar Wiart, Christiane Lemieux and Gracia Y. Dong

Abstract

In this paper we develop a framework to study the dependence structure of scrambled (t,m,s)-nets. It relies on values denoted by Cb(𝒌;Pn), which are related to how many distinct pairs of points from Pn lie in the same elementary 𝒌-interval in base b. These values quantify the equidistribution properties of Pn in a more informative way than the parameter t. They also play a key role in determining if a scrambled set P~n is negative lower orthant dependent (NLOD). Indeed, this property holds if and only if Cb(𝒌;Pn)1 for all 𝒌s, which in turn implies that a scrambled digital (t,m,s)-net in base b is NLOD if and only if t=0. Through numerical examples we demonstrate that these Cb(𝒌;Pn) values are a powerful tool to compare the quality of different (t,m,s)-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: 238959

Funding source: Austrian Science Fund

Award Identifier / Grant number: F5506-N26

Award Identifier / Grant number: F5509-N26

Funding statement: The authors wish to acknowledge the support of the Natural Science and Engineering Research Council (NSERC) of Canada for its financial support via grant #238959. The first author is also partially supported by the Austrian Science Fund (FWF): Projects F5506-N26 and F5509-N26, which are parts of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

A Appendix

Proof of Lemma 2.5.

When either x or y is 1, from Lemma 2.4 we know that

Vi=x(b-1)bi+1

and thus bVi-Vi-1=0 in this case. So for the remainder of the proof, we assume x,y[0,1). Let

x=k=1xkbkandy=k=1ykbk

be the base-b digital expansion of x and y chosen so that only finitely many digits are non-zero. Recall that ki=bimin(x,y)b-i for i0. When γb(x,y)1, then for i{1,,γb(x,y)} we have

hi=k=1ixkbk=k=1iykbk,

and k0=0. We also define rxi=x-hi, and ryi=y-hi for i0. Without loss of generality we assume xy. There are four cases.

Case 1: γb(x,y)<i-1. In this case

bVi-Vi-1=bxbi-xbi-1=0.

Case 2: γb(x,y)=i-1. In this case, bVi-Vi-1 becomes

xbi-1-xy+hi-1(x+y-hi-1-1bi-1)
=hi-1+rxi-1bi-1-(hi-1+rxi-1)(hi-1+ryi-1)+hi-1(hi-1+rxi-1+ryi-1-1bi-1)
=hi-1+rxi-1bi-1-rxi-1ryi-1-hi-1bi-1hi-1+rxi-1bi-1-rxi-1bi-1-hi-1bi-1=0

because ryi-11bi-1.

Case 3: γb(x,y)=i. We use the calculation in Case 2 and the identities rxi-1=xibi+rxi and ryi-1=xibi+ryi to simplify bVi-Vi-1:

b(xy-xbi+1-ki(x+y-ki-1bi))-(ki(x+y-ki-1bi)-hi-1(x+y-hi-1-1bi-1))
=(b+1)(xy-xbi-ki(x+y-ki-1bi))-(xy-xbi-1-hi-1(x+y-hi-1-1bi-1))
=(b+1)(rxiryi-rxibi)-(rxi-1ryi-1-rxi-1bi-1)
=brxiryi-rxibi-xi2b2i-xi(rxi+ryi)bi+xib2i-1.

Multiply by bi to get

bi+1rxiryi-rxi-xi2bi-xirxi-xiryi+xibi-1,

which will be shown to be non-negative. Note that by assumption x<y and since their base b expansions differ for the first time at the (i+1)st digit, we always have xi+1<yi+1.

Case 3a: xixi+1<yi+1. The assumption implies 0bi+1rxi-xi and xi+1bi+1ryi. We estimate

(bi+1rxi-xi)ryi-rxi-xi2bi-xirxi+xibi-1(bi+1rxi-xi)xi+1bi+1-rxi-xi2bi-xirxi+xibi-1
=xibi+1(b2-(b+1)xi-1)
xibi+1(b2-(b+1)(b-1)-1)=0.

Case 3b: (xi+1<xi<yi+1). The assumption implies rxixibi+1 and (bi+1ryi-xi-1)0. We estimate

xibi-1-xi2bi-xiryi+(bi+1ryi-xi-1)rxixibi-1-xi2bi-xiryi+(bi+1ryi-xi-1)xibi+1
=xibi+1(b2-(b+1)xi-1))
xibi+1(b2-(b+1)(b-1)-1))=0.

Case 3c: (xi+1<yi+1xi). The assumption implies 0(bi+1ryi-xi-1). We estimate

bi+1rxiryi-rxi-xi2bi-xirxi-xiryi+xibi-1=xibi-1-xi2bi-xiryi+(bi+1ryi-xi-1)rxi
xibi-1-xi2bi-xiryi
xi(1bi-1-b-1bi-1bi)=0.

Case 4: γb(x,y)>i. In this case we need to show that

(A.1)bhi+1(x+y-hi+1-1bi+1)-(b+1)ki(x+y-ki-1bi)+hi-1(x+y-hi-1-1bi-1)

is greater than or equal to zero. Using the identities

hi+1=hi-1+xibi+xi+1bi+1,ki=hi-1+xibi,x=hi-1+xibi+xi+1bi+1+rxi+1,y=hi-1+xibi+xi+1bi+1+ryi+1,

write

hi+1(x+y-hi+1-1bi+1)=(hi-1+xibi+xi+1bi+1)(hi-1+xibi+xi+1bi+1+rxi+1+ryi+1-1bi+1)
=hi-12+xi2b2i+xi+12b2i+2+2hi-1xibi+2hi-1xi+1bi+1+2xixi+1b2i+1+hi-1(rxi+1+ryi+1)
+xi(rxi+1+ryi+1)bi+xi+1(rxi+1+ryi+1)bi+1-hi-1bi+1-xib2i+1-xi+1b2i+2,

and

ki(x+y-ki-1bi)=(hi-1+xibi)(hi-1+xibi+2xi+1bi+1+rxi+1+ryi+1-1bi)
=hi-12+xi2b2i+2hi-1xibi+2hi-1xi+1bi+1+2xixi+1b2i+1
+hi-1(rxi+1+ryi+1)+xi(rxi+1+ryi+1)bi-hi-1bi-xib2i,

and

hi-1(x+y-hi-1-1bi-1)=hi-1(hi-1+2xibi+2xi+1bi+1+rxi+1+ryi+1-1bi-1)
=hi-12+2hi-1xibi+2hi-1xi+1bi+1+hi-1(rxi+1+ryi+1)-hi-1bi-1.

Now substituting into (A.1) and simplifying, we get

bVi+1-Vi=0(hi-12+2hi-1xibi+2hi-1xi+1bi+1+hi-1(rxi+1+ryi+1))
-(xi2b2i+2xixi+1b2i+1+xi(rxi+1+ryi+1)bi)+b(xi+12b2i+2+xi+1(rxi+1+ryi+1)bi+1)
-hi-1(bbi+1-b+1bi+1bi-1)-xi(bb2i+1-b+1b2i)-bxi+1b2i+2
=xi+12b2i+1+xib2i-1-xi2b2i-2xixi+1b2i+1-xi+1b2i+1+(xi+1-xi)(rxi+1+ryi+1)bi.

By multiplying the above by b2i+1 we see that to finish the proof we need to show that

xi+12+b2xi-bxi2-2xixi+1-xi+1+bi+1(xi+1-xi)(rxi+1+ryi+1)

is non-negative.

Case 4a: xi<xi+1. We have

xi+12+b2xi-bxi2-2xixi+1-xi+1+bi+1(xi+1-xi)(rxi+1+ryi+1)
xi+12+b2xi-bxi2-2xixi+1-xi+1
xi+12+bxi(xi+1+1)-bxixi+1-2xixi+1-xi+1
=xi+12+bxi-2xixi+1-xi+1
xi+12+(xi+1+1)xi-2xixi+1-xi+1
xi+12-xixi+1-xi+1+xi
xi+1(xi+1-xi-1)
0.

Case 4b: xi+1xi. Since rxi+1,ryi+11bi+1, we have

xi+12+b2xi-bxi2-2xixi+1-xi+1+bi+1(xi+1-xi)(rxi+1+ryi+1)
xi+12+b2xi-bxi2-2xixi+1+xi+1-2xi
=(xi+1-xi)2+b2xi-(b+1)xi2-2xi+xi+1
(xi+1-xi)2+xi(b2-(b+1)(b-1)-2)+xi+1
=(xi+1-xi)2+xi+1-xi
=(xi+1-xi)(xi+1-xi+1)
=(xi-xi+1)(xi-xi+1-1)0.

References

[1] J. Dick, F. Y. Kuo and I. H. Sloan, High-dimensional integration: The quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288. Search in Google Scholar

[2] J. Dick and F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010. Search in Google Scholar

[3] H. Faure, Discrépance de suites associées à un système de numération (en dimension s), Acta Arith. 41 (1982), no. 4, 337–351. Search in Google Scholar

[4] H. Faure and C. Lemieux, Implementation of irreducible Sobol’ sequences in prime power bases, Math. Comput. Simulation 161 (2019), 13–22. Search in Google Scholar

[5] M. Gerber, On integration methods based on scrambled nets of arbitrary size, J. Complexity 31 (2015), no. 6, 798–816. Search in Google Scholar

[6] M. Gnewuch, M. Wnuk and N. Hebbinghaus, On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds, Discrepancy Theory, Radon Ser. Comput. Appl. Math. 26, De Gruyter, Berlin (2020), 43–68. Search in Google Scholar

[7] F. J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996), 274–296. Search in Google Scholar

[8] H. S. Hong and F. J. Hickernell, Algorithm 823: implementing scrambled digital sequences, ACM Trans. Math. Software 29 (2003), no. 2, 95–109. Search in Google Scholar

[9] C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling, Springer Ser. Statist., Springer, New York, 2009. Search in Google Scholar

[10] C. Lemieux, Negative dependence, scrambled nets, and variance bounds, Math. Oper. Res. 43 (2018), no. 1, 228–251. Search in Google Scholar

[11] J. Matoušek, On the L2-discrepancy for anchored boxes, J. Complexity 14 (1998), no. 4, 527–556. Search in Google Scholar

[12] R. B. Nelsen, An Introduction to Copulas, 2nd ed., Springer Ser. Statist., Springer, New York, 2006. Search in Google Scholar

[13] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conf. Ser. in Appl. Math. 63, Society for Industrial and Applied Mathematics, Philadelphia, 1992. Search in Google Scholar

[14] A. B. Owen, Randomly permuted (t,m,s)-nets and (t,s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Las Vegas 1994), Lect. Notes Stat. 106, Springer, New York (1995), 299–317. Search in Google Scholar

[15] A. B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), no. 5, 1884–1910. Search in Google Scholar

[16] A. B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Statist. 25 (1997), no. 4, 1541–1562. Search in Google Scholar

[17] A. B. Owen, Scrambling Sobol’ and Niederreiter–Xing points, J. Complexity 14 (1998), no. 4, 466–489. Search in Google Scholar

[18] A. B. Owen, Variance and discrepancy with alternative scramblings, ACM Trans. Model. Comput. Simul. 13 (2003), 363–378. Search in Google Scholar

[19] I. M. Sobol’, On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comp. Math. Math. Phys. 7 (1967), 86–112. Search in Google Scholar

[20] I. M. Sobol’ and D. I. Asotsky, One more experiment on estimating high-dimensional integrals by quasi-Monte Carlo methods, Math. Comput. Simul. 62 (2003), 255–263. Search in Google Scholar

[21] M. Wnuk and M. Gnewuch, Note on pairwise negative dependence of randomly shifted and jittered rank-1 lattices, Oper. Res. Lett. 48 (2020), no. 4, 410–414. Search in Google Scholar

Received: 2020-07-28
Revised: 2020-11-08
Accepted: 2020-12-18
Published Online: 2021-01-10
Published in Print: 2021-03-01

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