MATHEMATICS OF COMPUTATION Volume 71, Number 237, Pages 275{296 S 0025-5718(00)01281-3 Article electronically published on October 16, 2000 CENTERED L2-DISCREPANCY OF RANDOM SAMPLING AND LATIN HYPERCUBE DESIGN, AND CONSTRUCTION OF UNIFORM DESIGNS KAI-TAI FANG, CHANG-XING MA, AND PETER WINKER Abstract. In this paper properties and construction of designs under a cen- tered version of the L2-discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Op- timization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance. 1. Introduction Many problems arising in industry, statistics, physics, and finance require mul- tivariate integration, the canonical form of which can be expressed as Z (1.1) I(f)= f(x)dx; Cs s s where C =[0; 1) and f(x)=f(x1; ··· ;xs). The sample mean method has been recommended to give an approximation to I(f)by 1 Xn (1.2) I^(f;P)= f(x ); n i i=1 s where P = fx1; ··· ; xng is a set of points on C .Ifx1; ··· ; xn are i.i.d. uniformly distributed on Cs, the set is called simple random sampling or simple random design (SRD) and is denoted by Rn;s.ItisknownthatI^(f;Rn;s)isunbiased and has an asymptotic variance O(n−1). This rate of convergence is too slow for the applications. Therefore, McKay, Beckman and Conover [MBC79] proposed the so-called Latin hypercube design (LHD), which also provides an unbiased estimate I^(f;P) with smaller asymptotic variance than that of SRD. LHD has been widely used in conducting computer experiments. A systematic study on LHD and various modified versions of LHD that can significantly reduce the asymptotic variance of LHD is given by [Owen92], [Owen94], [Owen95], [KO96], and [Tang93]. In this Received by the editor July 20, 1999 and, in revised form, February 25, 2000. 2000 Mathematics Subject Classification. Primary 68U07; Secondary 65D17, 62K99. Key words and phrases. Uniform design, Latin hypercube design, threshold accepting heuristic, quasi-Monte Carlo methods. This work was partially supported by a Hong Kong RGC-grant and SRCC of Hong Kong Baptist University. c 2000 American Mathematical Society 275 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 276 KAI-TAI FANG, CHANG-XING MA, AND PETER WINKER article, we only consider a special case of LHD, where x1; ··· ; xn are i:i:d: uniformly distributed on the lattice set 2ai−1 xi = 2n ;i=1;::: ;n; T = x =(x1;::: ;xn) : (a1;::: ;an)isapermutationoff1;::: ;ng We denote this LHD by Ln;s. There exist different measures to assess the performance of various designs on Cs. The Koksma-Hlawka inequality gives an upper bound for the approximation error (1.3) jI(f) − I^(f;P)|≤D(P)V (f); where D(P) is the discrepancy of P that will be defined in (1.4), and V (f)isa measure of the variation of f [Nie92]. In fact, we can find a number of other pairs fD(P);V(f)g satisfying the Koksma-Hlawka inequality, where D(P)isameasure of nonuniformity of P and V (f) is a measure of variation of f. An excellent study −1=2 on this topic is given by [Hic98]. For SRD, D(Rn;s)=O(n )asn !1.The determination of the order of convergence of D(P)forLHDP is still an open problem. It will be answered for the centered L2-discrepancy in this paper. s s Let P = fx1; ··· ; xng be a set of n points on C =[0; 1) .ThestarLp- discrepancy (Lp-discrepancy for simplicity) has been widely used in quasi-Monte Carlo methods (or number-theoretic methods) as well as in uniform design theory (cf. [Nie92] and [FWa94]). It is defined as Z p 1=p N(P; [0; x)) (1.4) Dp(P)= − Vol([0; x)) dx ; Cs n where [0; x) denotes the interval [0;x1) × ··· × [0;xs), N(P; [0; x)) the number of points of P falling in [0; x), and Vol(A)thevolumeofA.AmongtheLp- discrepancies, the D2(P)andD(P)=D1(P) (called discrepancy for short) are used most frequently. Hickernell [Hic98] pointed out some weakness of the Lp-discrepancy and pro- posed several modified Lp-discrepancies, among which the centered L2-discrepancy (CL2) seems most interesting. Ma and Fang [MF98] and Fang and Mukerjee [FM00] found some connections between CL2 and orthogonality, minimum aberration, and confounding for a certain class of designs. The centered Lp-discrepancy is a modi- fication of the Lp-discrepancy by the requirement that it becomes invariant under reflections of P about any plane xj =0:5. It is defined by Z X N(P ;J ) p P p u xu − (1.5) (Dp( )) = Vol(Jxu ) du; u n u=6 ; C where u is a nonempty subset of the set of coordinate indices S = f1; ··· ;sg, juj denotes the cardinality of u, Cu is the juj-dimensional unit cube involving the coordinates in u, Jx is an s-dimensional interval uniquely determined by x, Pu is P u u As the projection of to C ,andJxu is the projection of Jx on C .Let denote s s s the set of 2 vertices of the cube C and α =(a1;::: ;as) 2A be the closest one to x. Define s Jx = fy 2 C j min(aj;xj ) ≤ yj < max(aj ;xj); for 1 ≤ j ≤ sg: License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE CL2 OF RANDOM SAMPLING AND LATIN HYPERCUBE DESIGN 277 For CL2 Hickernell [Hic98] derived an analytical expression 13 s 2 Xn Ys 1 1 CL (P)2 = − (1 + jx − 0:5|− jx − 0:5j2) 2 12 n 2 kj 2 kj k=1 j=1 1 Xn Xn Ys 1 1 1 (1.6) + 1+ jx − 0:5j + jx − 0:5|− jx − x j ; n2 2 ki 2 ji 2 ki ji k=1 j=1 i=1 where xk =(xk1; ··· ;xks) 2P. From the definition (1.5) the centered Lp- discrepancy takes into account not only the uniformity of P over Cs, but also uniformity of all the projections of P over Cu. In Sections 2 and 3 we shall derive the expectation and variance for square CL2 of simple random designs Rn;s and Latin hypercube designs Ln;s,andgive comparisons of these statistics. It will be shown that the LHD has much lower expected CL2-value and variance than SRD. Our results are consistent with the results of comparing variance of I^(f;P) between SRD and LHD [Owen92]. Note that the LHD Ln;s can be defined in terms of U-type designs. Definition 1.1. AU-typedesignUn;qs is an n × s matrix U =(uij ) of which each column has q entries 1;::: ;q appearing equally often. The induced matrix of U, − denoted by XU =(xij ), is defined by xij =(uij 0:5)=q.Whenq = n,weusethe notation Un;s instead of Un;ns .LetUn;qs and Un;s be the set of all Un;qs and the set of all Un;s, respectively. Any induced matrix XU defined in Definition 1.1 corresponds to a set of n s P P points on C , denoted by U . Each row of XU corresponds to a point of U on s P L C .TheCL2(U) is defined as CL2( U ). The LHD n;s is a design XU ,where 2U 2U U n;s. The design XU ; where U n;qs , can be considered as an extension of LHD and is denoted by Ln;qs . In Sections 2 and 3 we also derive the expectation 2 and variance of CL2(Ln;qs ) . The U-type design is the basis of the uniform design. The latter is one of \space filling” designs (Cheng and Li (1995) [CL95], and Koehler and Owen (1996) [KO96]). The uniform design allocates experimental points uniformly scattered on the domain in the sense of low-discrepancy [FWa94]. Any discrepancy mentioned before can be used as a measure of nonuniformity. In the past, most uniform designs are obtained in terms of the discrepancy and the L2-discrepancy. Fang and Winker [FW98] found that both discrepancy and L2-discrepancy are not suitable mea- sures of nonuniformity for searching the UD, since the discrepancy is not sensitive enough for identifying different designs while the L2-discrepancy ignores differences j N(P;[0;x)) − j2 n Vol([0; x)) on any low-dimensional subspace. Therefore, they recom- mend the use of the three modified L2-discrepancies proposed by [Hic98]. In this paper we concentrate on the centered L2-discrepancy for construction of uniform designs. s ∗ Let Pn be the class of sets of n points on C .AsetP 2Pn is called a uniform design if it has the smallest CL2-value over Pn, i.e., ∗ (1.7) CL2(P )= min CL2(P): P2Pn In Section 4 we propose a heuristic optimization algorithm for the construction of uniform designs based on U-type designs under CL2. The results obtained in Sections 2 and 3 provide information which can be used to reduce the computing time of searching uniform designs and low-discrepancy designs. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 278 KAI-TAI FANG, CHANG-XING MA, AND PETER WINKER The paper is organized as follows.