The Mean Square Discrepancy of Randomized Nets Fred J. Hickernell Hong Kong Baptist University One popular family of low discrepancy sets are the (t; m; s)-nets. Recently a randomization of these nets has been introduced that preserves their net property. In this article a formula for the mean square L2 -discrepancy of (0; m; s)-nets in base b is derived. This formula has a computational complexity of only O(s log(N )+ s2 ) for large N or s, where N = bm is the number of points. Moreover, the root mean square L2 -discrepancy of (0; m; s)-nets is shown to be O(N ?1 [log(N )](s?1)=2 ) as N tends to in nity, the same asymptotic order as the known lower bound for the L2 -discrepancy of an arbitrary set. Categories and Subject Descriptors: G.1.4 [Numerical Analysis]: Quadrature and Numerical Di erentiation|Error analysis ; Multiple quadrature; G.3 [Probability and Statistics]: Probabilistic algorithms (including Monte Carlo); I.6.8 [Simulation and Modeling]: Types of Simulation|Monte Carlo General Terms: Theory, Algorithms, Performance Additional Key Words and Phrases: multidimensional integration, number-theoretic nets and sequences, quadrature, quasi-Monte Carlo methods, quasi-random sets
1. INTRODUCTION Multidimensional integrals over the s-dimensional unit cube C s = [0; 1)s may be approximated by the sample mean of the integrand evaluated on a point set, P , with N points. (Here, in contrast to ordinary sets, P may have multiple copies of the same point [Niederreiter 1992, p. 14].) The quadrature error depends on how uniformly the points in P are distributed on the unit cube and on how much the integrand varies from a constant. For example, if D(P ) is the L1 -star discrepancy [Niederreiter 1992, De nition 2.1], and V (f ) is the variation of f on C s = [0; 1]s in the sense of Hardy and Krause, then the Koksma-Hlawka inequality [Niederreiter 1992, Theorem 2.11] is
Err(f )
Z Cs
f (x) dx ? N1
X
z2P
)
f (z D(P )V (f ):
(1.1)
This research was partially supported by a Hong Kong RGC grant 94-95/38 and HKBU FRG grant 96-96/II-01. Address: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, E-mail:
[email protected], URL: http://www.math.hkbu.edu.hk/~fred Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or direct commercial advantage and that copies show this notice on the rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior speci c permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or
[email protected]. ACM Transactions on Modeling and Computer Simululation, 1997
The Discrepancy of Randomized Nets
2
Error bounds of this form with other de nitions of D(P ) and V (f ) appear in the literature as well. A good set P for quadrature is one that has a small discrepancy, D(P ). Several deterministic sets have been found that have smaller discrepancies than simple random points. These sets are often called quasi-random points. Calculating the L1 -star discrepancy of a particular set is impractical unless both N and s are small because it requires O(N s ) operations. Much e ort has been directed towards nding quasi-random sequences that have asymptotically small L1 -star discrepancies as N tends to in nity. The (t; m; s)-nets [Niederreiter 1992, Chapter 4] are one example. By comparison the L2 -star discrepancy is easier to compute, requiring only O(N 2 ) operations using a naive algorithm, or O(N [log(N )]s ) operations using the algorithm of Heinrich [1996]. Owen [1995, 1996a, 1996b] has proposed a randomization of (t; m; s)-nets that preserves their net properties. His aim is to obtain probabilistic quadrature error estimates in a similar manner as one would for simple Monte Carlo quadrature. In this article a formula for the mean square L2 -discrepancy of randomized (0; m; s)nets is derived which requires only O(s log(N ) + s2 ) mathematical operations to evaluate as N and/or s tend to in nity. The L2 -discrepancy for these randomized nets is shown to decay like O(N ?1 [log(N )](s?1)=2 ) for large N | a result matching the asymptotic lower bound obtained by Roth [1954]. In the remainder of this section we de ne (t; m; s)-nets and describe Owen's randomization. Also, a generalized L2 -discrepancy that arises in quadrature error bounds recently derived by the author is de ned. In Section 2 the mean square L2 discrepancy is computed for simple random samples and randomized (0; m; s)-nets. The asymptotic behavior of the L2 -discrepancy is studied in Section 3. This article concludes with some discussion. Any point z = (z1 ; : : : ; zs ) 2 C s may be written in base b as z = (0:z11 z21 z31 : : : ; 0:z12 z22 z32 : : : ; : : : : : : ; 0:z1sz2s z3s : : : ); where the b-nary digits zij range from 0 to b ? 1. Let Zs+ denote the space of s-dimensional non-negative integer vectors. For any k = (k1 ; : : : ; ks ) 2 Zs+ , let (k) = k1 + + ks . There are bk1 bks = b(k) di erent ways to choose the rst k b-nary digits of a point z : z11 ; : : : ; zk1 1 ; z12; : : : ; zk2 2 ; : : : ; z1s ; : : : ; zks s : (1.2) (If kj = 0, then no digit zij is being speci ed.) A (t; m; s)-net contains at least one point with every possible choice of the rst k digits, provided that (k) is small enough. De nition 1. Let t and m be non-negative integers with 0 t m, and let s be a positive integer. A (t; m; s)-net in base b is a set, P , containing N = bm points in C s . For any possible choice of the rst k b-nary digits (1.2) there exist bm?(k) points in P with these digits provided that (k) m ? t. A smaller value of t tends to imply a net with better uniformity (smaller discrepancy). Any (t1 ; m; s)-net is also a (t2 ; m; s)-net for t1 < t2 , and any set is an (m; m; s)-net. For a fuller discussion of (t; m; s)-nets see Niederreiter [1992, Chapter 4].
The Discrepancy of Randomized Nets
3
Owen randomizes the digits of the points in a given (t; m; s)-net P0 to obtain a new (t; m; s)-net P . For every digit index i = 1; 2; : : : , every coordinate index j = 1; : : : ; s and every y 2 P0 one obtains a random digit zij that contributes to a random point z 2 P . The randomized net, P satis es the following assumptions: Assumption 1. a. For any z 2 P each digit zij is uniformly distributed on the set f0; : : : ; b ? 1g. b. For any two points z; z 0 2 P the random vectors (z1 ; z10 ); : : : ; (zs ; zs0 ) are mutually independent. c. For any two points y; y0 2 P0 let z; z 0 2 P be the corresponding points in the randomized net. Suppose that yj and yj0 share the same rst kj digits, but that their kj + 1st digits are di erent. Then i. zij = zij0 for i = 1; : : : ; kj , ii. the random vector (zkj +1;j ; zk0 j +1;j ) is uniformly distributed on the set f(n; n0) : n 6= n0 ; n; n0 = 0; : : : ; b ? 1g, and iii. zij ; zi0 j are mutually independent for i; i0 = kj + 2; : : : . iv. zkj +2;j ; zkj +3;j ; : : : zk0 j +2;j ; zk0 j +3;j ; : : : are mutually independent. Assumptions 1a and 1b imply that the marginal probability distribution of any z = (z1 ; : : : ; zs ) 2 P is uniform on C s and that the z1; : : : ; zs are mutually independent. Assumption 1c maintains the correlation between di erent points in P that is necessary for retaining its net properties and thus a low discrepancy. For a simple random sample Assumptions 1a and 1b are also satis ed, but instead of Assumption 1c one has z1j ; z2j ; : : : ; z10 j ; z20 j ; : : : mutually independent for z 6= z 0. Let S = f1; : : :; sg be the set of coordinate indices. For any u S let juj denote the number of points in u. Let C u = [0; 1)u denote the juj-dimensional unit cube involving the coordinates in u, and likewise, let Zu+ be the juj-dimensional nonnegative integer vectors. This notation allows us to distinguish spaces of the same dimension in di erent coordinate directions. The Lp -star discrepancy is sometimes de ned as the Lp -norm of the di erence between the empirical distribution associated with the sample P and the uniform distribution on the unit cube:
jP \ [0; x)j
? Vol([0; x)) ; (1.3) D (P ) 0
p;S
N
p kp denotes the Lp -norm.
where j j denotes the number of points in a set, and k Although this de nition is appropriate for p = 1, it does not admit an error bound like (1.1) for other values of p. A more suitable de nition [Hickernell 1997] is
Dp (P )
8 < X
:
9
p =1=p
Vol([0 u ))
; p
Vol([0 u ))
1
))
jPu \ [0; xu )j ?
N ;uS
(P ) max
jPu \ [0; xu )j ? D1 N ;uS
jP \ [0; x)j =
N ? Vol([0; x
;x
;x
1
;
(1 p < 1);
The Discrepancy of Randomized Nets
4
where Pu denotes the projection of the sample P into the cube C u . This means (P ) is only one term in the de nition of Dp (P ) for p < 1. An error that Dp;S bound of the form (1.1) based on this de nition was derived by Zaremba [1968] for p = 2 and Hickernell [1997] for all p. The Lp -star discrepancy is in general dicult to compute, except in the case p = 2 where it reduces to a double sum: !
s XY
s 3 ? zj2 1 X Y 0 ) : + 2 ? max( z ; z j j 2 2 N z;z 2P j=1 z2P j =1 Hickernell [1997] has derived a family of quadrature error bounds of form (1.1) and Lp -discrepancies, Dp (P ), that include the star discrepancy as a special case. The L2 -discrepancy is the simplest to compute. Let be an arbitrary positive constant and be an arbitrary function on [0; 1) whose rst derivative is essentially R1 bounded and that satis es 0 (xj ) dxj = 0: Furthermore, let
[D2 (P )]2 =
s
4 3
? N2
M =
0
Z
1 d 2
0
dx
dx;
M = 1 + 2 M:
The generalized L2 -discrepancy de ned in [Hickernell 1997] is [D2 (P )]2 = M s ? N2
s XY
z2P j =1
[M + 2 (zj )] + N12
s X Y
z;z 2P j =1
fM
0
+ 2 (zj ) + (zj0 ) + 21 B2 (zj ) + 12 B2 (zj0 ) + 16 ? 21 jzj ? zj0 j
; (1.4)
where B2 denotes the quadratic Bernoulli polynomial [Abramowitz and Stegun 1964, Chapter 23]. The star discrepancy is obtained by choosing 2 (1.5) (x) = 16 ? x2 ; = 1; M = 13 ; M = 34 : Another choice of and yields a discrepancy derived by Hickernell [1996] that is similar to the gure of merit, P , used in the study of lattice rules [Sloan and Joe 1994, Eq. (4.8)]. Several di erent choices of and are discussed by Hickernell [1997]. The L2 -discrepancy de ned in (1.4) can also be written as X 2juj [D2;u (P )]2 (1.6) [D2 (P )]2 = ;uS
where
[D2;u (P )]2 = M juj ? N2
XY
z2P j 2u
[M + (zj )] + N12
X Y
z;z 2P j 2u 0
M
+(zj ) + (zj0 ) + 21 B2 (zj ) + 12 B2 (zj0 ) + 61 ? 21 jzj ? zj0 j : (1.7)
Choosing and according to (1.5) and setting u = S in the above equation yields
The Discrepancy of Randomized Nets
5
a formula for D2;S of (1.3) originally derived by Warnock [1972]: !
s XY
s 1 ? zj2 1 X Y 0 ); : + 1 ? max( z ; z j j 2 3 N z2P j=1 2 N z;z 2P j=1 Not only does the L2 -discrepancy appear in worst-case quadrature error bounds, such as (1.1), it also arises in average-case quadrature error analysis. Let Ef denote the expected value over some space of integrands. Then Ef [Err(f )]2 = D2 (P ); where the choice of the space of integrands determines the speci c form of the discrepancy [Sacks and Ylvisacker 1970; Ritter 1995]. The case of the star discrepancy has been studied by Wozniakowski [1991] and Moroko and Ca isch [1994]. s 2 [D2;S (P )] = 1 ? 2
0
2. COMPUTING THE MEAN SQUARE L2 -DISCREPANCY Let E denote the expected value over a space of random sets P . In this section we compute E f[D2(P )]2 g for simple random samples and for randomized (0; m; s)nets. Both kinds of samples satisfy Assumptions 1a and 1b. This allows a simple treatment of terms in (1.4) involving only zj or only zj0 . The dicult term is jzj ? zj0 j since its expected value depends on the correlation of z and z 0 . Lemma 1. If P is a random set satisfying Assumptions 1a and 1b, then
E f[D2 (P )]2 g 2 s = M+
? 2
8 juj
