Non-lattice covering and quanitization of high dimensional sets
NNon-lattice covering and quanitization of high dimensional sets
Jack Noonan and Anatoly Zhigljavsky
Abstract
The main problem considered in this paper is construction and theoretical study of efficient n -pointcoverings of a d -dimensional cube [− , ] d . Targeted values of d are between 5 and 50; n can be in hundredsor thousands and the designs (collections of points) are nested. This paper is a continuation of our paper [4],where we have theoretically investigated several simple schemes and numerically studied many more. In thispaper, we extend the theoretical constructions of [4] for studying the designs which were found to be superiorto the ones theoretically investigated in [4]. We also extend our constructions for new construction schemeswhich provide even better coverings (in the class of nested designs) than the ones numerically found in [4]. Inview of a close connection of the problem of quantization to the problem of covering, we extend our theoreticalapproximations and practical recommendations to the problem of construction of efficient quantization designsin a cube [− , ] d . In the last section, we discuss the problems of covering and quantization in a d -dimensionalsimplex; practical significance of this problem has been communicated to the authors by Professor MichaelVrahatis, a co-editor of the present volume. The problem of the main importance in this paper is the following problem of covering a cube [− , ] d by n balls. Let Z , . . . , Z n be a collection of points in R d and B d ( Z j , r ) = { Z : (cid:107) Z − Z j (cid:107) ≤ r } be the Euclidean ballsof radius r centered at Z j ( j = , . . . , n ) . The dimension d , the number of balls n and their radius r could bearbitrary.We are interested in choosing the locations of the centers of the balls Z , . . . , Z n so that the union of the balls ∪ j B d ( Z j , r ) covers the largest possible proportion of the cube [− , ] d . More precisely, we are interested inchoosing a collection of points (called ‘design’) Z n = { Z , . . . , Z n } so that C d ( Z n , r ) : = vol ([− , ] d ∩ B d ( Z n , r ))/ d (1)is as large as possible (given n , r and the freedom we are able to use in choosing Z , . . . , Z n ). Here B d ( Z n , r ) isthe union of the balls J. NoonanSchool of Mathematics, Cardiff University, Cardiff, CF244AG, UKe-mail:
A. ZhigljavskySchool of Mathematics, Cardiff University, Cardiff, CF244AG, UKe-mail:
[email protected] a r X i v : . [ m a t h . S T ] J un Jack Noonan and Anatoly Zhigljavsky B d ( Z n , r ) = n (cid:216) j = B d ( Z j , r ) (2)and C d ( Z n , r ) is the proportion of the cube [− , ] d covered by B d ( Z n , r ) . If Z j ∈ Z n are random then we shallconsider E Z n C d ( Z n , r ) , the expected value of the proportion (1); for simplicity of notation, we will drop E Z n while referring to E Z n C d ( Z n , r ) .For a design Z n , its covering radius is defined by CR ( Z n ) = max X ∈C d min Z j ∈ Z n (cid:107) X − Z j (cid:107) . In computerexperiments, covering radius is called minimax-distance criterion, see [2] and [9]; in the theory of low-discrepancy sequences, covering radius is called dispersion, see [3, Ch. 6].The problem of optimal covering of a cube by n balls has very high importance for the theory of globaloptimization and many branches of numerical mathematics. In particular, the n -point designs Z n with smallestCR provide the following: (a) the n -point min-max optimal quadratures, see [10, Ch.3,Th.1.1], (b) min-max n -point global optimization methods in the set of all adaptive n -point optimization strategies, see [10, Ch.4,Th.2.1],and (c) worst-case n -point multi-objective global optimization methods in the set of all adaptive n -pointalgorithms, see [14]. In all three cases, the class of (objective) functions is the class of Liptshitz functions,where the Liptshitz constant may be unknown. The results (a) and (b) are the celebrated results of A.G.Sukharevobtained in the late nineteen-sixties, see e.g. [11], and (c) is a recent result of A. Žilinskas.If d is not small (say, d >
5) then computation of the covering radius CR ( Z n ) for any non-trivial design Z n is avery difficult computational problem. This explains why the problem of construction of optimal n -point designswith smallest covering radius is notoriously difficult, see for example recent surveys [12, 13]. If r = CR ( Z n ) ,then C d ( Z n , r ) defined in (1) is equal to 1, and the whole cube C d gets covered by the balls. However, we areonly interested in reaching the values like 0.95 or 0.99, when only a large part of the ball is covered.We will say that B d ( Z n , r ) makes a ( − γ ) -covering of [− , ] d if C d ( Z n , r ) = − γ ; (3)the corresponding value of r will be called ( − γ ) -covering radius and denoted r − γ or r − γ ( Z n ) . If γ = ( − γ ) -covering becomes the full covering and 1-covering radius r ( Z n ) becomes the covering radius CR ( Z n ) .The problem of construction of efficient designs with smallest possible ( − γ ) -covering radius (with some small γ >
0) will be referred to as the problem of weak covering.Let us give two strong arguments why the problem of weak covering could be even more practically importantthan the problem of full covering.• Numerical checking of weak covering (with an approximate value of γ ) is straightforward while numericalchecking of the full covering is practically impossible, if d is large enough.• For a given design Z n , C d ( Z n , r ) defined in (1) and considered as a function of r , is a cumulative distributionfunction (c.d.f.) of the random variable (r.v.) (cid:37) ( U , Z n ) = min Z i ∈ Z n (cid:107) U − Z i (cid:107) , where U is a random vectoruniformly distributed on [− , ] d , see (29) below. The covering radius CR ( Z n ) is the upper bound of this r.v.while in view of (3), r − γ ( Z n ) is the ( − γ ) -quantile. Many practically important characteristics of designssuch as quantization error considered in Section 7 are expressed in terms of the whole c.d.f. C d ( Z n , r ) andtheir dependence on the upper bound CR ( Z n ) is marginal. As shown in Section 7.5, numerical studies indicatethat comparison of designs on the base of their weak coverage properties is very similar to quantization errorcomparisons, but this may not be true for comparisons with respect to CR ( Z n ) . This phenomenon is similar tothe well-known fact in the theory of space covering by lattices (see an excellent book [1] and surveys [12, 13]),where best lattice coverings of space are often poor quantizers and vice-versa. Moreover, Figures 1-2 belowshow that CR ( Z n ) may give a totally inadequate impression about the c.d.f. C d ( Z n , r ) and could be muchlarger than r − γ ( Z n ) with very small γ > C d (· , r ) , black line, and also indicatethe location of the r =CR and r . by vertical red and green line respectively. In Figure 1, we take d = on-lattice covering and quanitization of high dimensional sets 3 n =
512 and use a 2 d − design of maximum resolution concentrated at the points (± / , . . . , ± / ) ∈ R d as design Z n ; this design is a particular case of Design 4 of Section 8 and can be defined for any d >
2. InFigure 2, we keep d =
10 but take the full factorial 2 d design with m = d points, again concentrated at thepoints (± / , . . . , ± / ) ; denote this design Z (cid:48) m .For both designs, it is very easy to analytically compute their covering radii (for any d > ( Z n ) = √ d + / ( Z (cid:48) m ) = √ d /
2; for d =
10 this gives CR ( Z n ) (cid:39) . ( Z (cid:48) m ) (cid:39) . . The values of r . are: r . ( Z n ) (cid:39) . r . ( Z (cid:48) m ) (cid:39) . r . . We will return tothis example in Section 2.1.Fig. 1: C d ( Z n , r ) with r . and r : d = Z n is a 2 d − -factorial design with n = d − Fig. 2: C d ( Z (cid:48) m , r ) with r . and r : d = Z m is a 2 d -factorial designOf course, for any Z n = { Z , . . . , Z n } we can reach C d ( Z n , r ) = r . Likewise, for anygiven r we can reach C d ( Z n , r ) = n → ∞ . However, we are not interested in very large values of n and try to get the coverage of the most part of the cube C d with the radius r as small as possible. We will keepin mind the following typical values of d and n which we will use for illustrating our results: d = , , , n = k with k = , . . . ,
11 (we have chosen n as a power of 2 since this a favorable number for Sobol’s sequence(Design 3) as well as Design 4 defined in Section 8).The structure of the rest of the paper is as follows. In Section 2 we discuss the concept of weak covering in moredetail and introduce three generic designs which we will concentrate our attention on. In Sections 3, 4 and 5 wederive approximations for the expected volume of intersection the cube [− , ] d with n balls centred at the pointsof these designs. In Section 6, we provide numerical results showing that the developed approximations are veryaccurate. In Section 7, we derive approximations for the mean squared quantization error for chosen families ofdesigns and numerically demonstrate that the developed approximations are very accurate. In Section 8, we nu-merically compare covering and quantization properties of different designs including scaled Sobol’s sequenceand a family of very efficient designs defined only for very specific values of n . In Section 9 we try to answerthe question raised by Michael Vrahatis by numerically investigating the importance of the effect of scalingpoints away from the boundary (we call it δ -effect) for covering and quantization in a d -dimensional simplex.In Appendix, Section 10, we formulate a simple but important lemma about the distribution and moments of acertain random variable.Our main theoretical contributions in this paper are:• derivation of accurate approximations (16) and (22) for the probability P U ,δ,α, r defined in (9); For simplicity of notation, vectors in R d are represented as rows. Jack Noonan and Anatoly Zhigljavsky • derivation of accurate approximations (18), (24) and (27) for the expected volume of intersection of the cube [− , ] d with n balls centred at the points of the selected designs;• derivation of accurate approximations (32), (34) and (35) for the mean squared quantization error for theselected designs.We have performed a large-scale numerical study and provided a number of figures and tables. The followingare the key messages containing in these figures and tables.• Figures 1–2: weak covering could be much more practically useful than the full covering;• Figures 3–14: developed approximations for the probability P U ,δ,α, r defined in (9) are very accurate;• Figures 15–28: (a) developed approximations for C d ( Z n , r ) are very accurate, (b) there is a very strong δ -effect for all three types of designs, and (c) this δ -effect gets stronger as d increases;• Tables 1 and 2 and Figures 29-30: smaller values of α are beneficial in Design 1 but Design 2 (where α = n gets close to 2 d ;• Figures 31–44: developed approximations for the quantization error are very accurate and there is a verystrong δ -effect for all three types of designs used for quantization;• Tables 3–4 and Figures 45-46: (a) Designs 2a and especially 2b provide very high quality coverage for suitable n , (b) properly δ -tuned deterministic non-nested Design 4 provides superior covering, (c) coverage propertiesof δ -tuned low-discrepancy sequences are much better than of the original low-discrepancy sequences, and(d) coverage properties of unadjusted low-discrepancy sequences is very low, if dimension d is not small;• Tables 5 and 6, Figures 47 and 48: very similar conclusions to the above but made with respect to thequantization error;• Figures 51–62: the δ -effect for covering and quantization schemes in a simplex is definitely present (thiseffect is more apparent in quantization) but it is much weaker than in a cube. In this section, we consider the problem of weak covering defined and discussed in Section 1. The maincharacteristic of interest will be C d ( Z n , r ) , the proportion of the cube covered by the union of balls B d ( Z n , r ) ;it is defined in (1). We start the section with short discussion on comparison of designs based on their coveringproperties. Two different designs will be differentiated in terms of covering performance as follows. Fix d and let Z n and Z (cid:48) n be two n -point designs. For ( − γ ) -covering with γ ≥
0, if C d ( Z n , r ) = C d ( Z (cid:48) n , r (cid:48) ) = − γ and r < r (cid:48) ,then the design Z n provides a more efficient ( − γ ) -covering and is therefore preferable. Moreover, the naturalscaling for the radius is r n = n / d r and therefore we can compare an n -point design Z n with an m -point design Z (cid:48) m as follows: if C d ( Z n , r ) = C d ( Z (cid:48) m , r (cid:48) ) = − γ and n / d r < m / d r (cid:48) , then we say that the design Z n provides amore efficient ( − γ ) -covering than the design Z (cid:48) m .As an example, consider the designs used for plotting Figures 1 and 2 in Section 1: Z n with n = d − and Z (cid:48) m with m = d . For the full covering, we have for any d : n / d r ( Z n ) = − / d √ d + > √ d = r ( Z (cid:48) m ) m / d so that the design Z (cid:48) m is better than Z n for the full covering for any d and the difference between normalizedcovering radii is quite significant. For example, for d =
10 we have n / d r ( Z n ) (cid:39) . r ( Z (cid:48) m ) m / d (cid:39) . on-lattice covering and quanitization of high dimensional sets 5 For 0.999-covering, however, the situation is reverse, at least for d =
10, where we have: n / d r . ( Z n ) (cid:39) . < . (cid:39) r ( Z (cid:48) m ) m / d and therefore the design Z n is better for 0.999-covering than the design Z (cid:48) m for d = In the designs Z n , which are of most interest to us, the points Z j ∈ Z n are i.i.d. random vectors in R d witha specified distribution. Let us show that for these designs, we can reduce computation of C d ( Z n , r ) to theprobability of covering [− , ] d by one ball.Let Z , . . . , Z n be i.i.d. random vectors in R d and B d ( Z n , r ) be as defined in (2). Then, for given U = ( u , . . . , u d ) ∈ R d , P { U ∈ B d ( Z n , r )} = − n (cid:214) j = P (cid:8) U (cid:60) B d ( Z j , r ) (cid:9) = − n (cid:214) j = (cid:0) − P (cid:8) U ∈ B d ( Z j , r ) (cid:9)(cid:1) = − (cid:18) − P Z {(cid:107) U − Z (cid:107) ≤ r } (cid:19) n . (4) C d ( Z n , r ) , defined in (1), is simply C d ( Z n , r ) = E U P { U ∈ B d ( Z n , r )} , (5)where the expectation is taken with respect to the uniformly distributed U ∈ [− , ] d . For numerical convenience,we shall simplify the expression (4) by using the approximation ( − t ) n (cid:39) e − nt , (6)where t = P Z {(cid:107) U − Z (cid:107) ≤ r } . This approximation is very accurate for small values of t and moderate values of nt , which is always the case of our interest. Combining (4), (5) and (6), we obtain the approximation C d ( Z n , r ) (cid:39) − E U exp (− n · P Z {(cid:107) U − Z (cid:107) ≤ r }) . (7)In the next section we will formulate three schemes that will be of theoretical interest in this paper. For eachscheme and hence different distribution of Z , we shall derive accurate approximations for P Z {(cid:107) U − Z (cid:107) ≤ r } and therefore, using (7), for C d ( Z n , r ) . The three designs that will be the focus of theoretical investigation in this paper are:
Design 1. Z , . . . , Z n ∈ Z n are i.i.d. random vectors on [− δ, δ ] d with independent components distributedaccording to the following Beta δ ( α, α ) distribution with density: Jack Noonan and Anatoly Zhigljavsky p α,δ ( t ) = ( δ ) − α Beta ( α, α ) [ δ − t ] α − , − δ < t < δ , for some α > ≤ δ ≤ . (8) Design 2a. Z , . . . , Z n ∈ Z n are i.i.d. random vectors obtained by sampling with replacement from the verticesof the cube [− δ, δ ] d . Design 2b. Z , . . . , Z n ∈ Z n are random vectors obtained by sampling without replacement from the vertices ofthe cube [− δ, δ ] d . All three designs above are nested so that Z n ⊂ Z n + for all eligible n . Designs 1 and 2a are defined for all n = , , . . . whereas Design 2b is defined for n = , , . . . , d . The appealing property of any design whosepoints Z i are i.i.d. is the possibility of using (4); this is the case of Designs 1 and 2a. For Design 2b, we willneed to make some adjustments, see Section 5.In the case of α = δ ( α, α ) becomes uniform on [− δ, δ ] d . This case hasbeen comprehensively studied in [4] with a number of approximations for C d ( Z n , r ) being developed. Theapproximations developed in Section 3 are generalizations of the approximations of [4]. Numerical resultsof [4] indicated that Beta-distribution with α < α →
0. Theoreticalapproximations developed below for C d ( Z n , r ) for Design 2a are, however, more precise than the limiting casesof approximations obtained for C d ( Z n , r ) in case of Design 1. For numerical comparison, in Section 6 we shallalso consider several other designs. C d ( Z n , r ) for Design 1 As a result of (7), our main quantity of interest in this section will be the probability P U ,δ,α, r : = P Z {(cid:107) U − Z (cid:107) ≤ r } = P Z (cid:8) (cid:107) U − Z (cid:107) ≤ r (cid:9) = P d (cid:213) j = ( u j − z j ) ≤ r (9)in the case when Z has the Beta-distribution with density (8). We shall develop a simple approximation basedon the Central Limit Theorem (CLT) and then subsequently refine it using the general expansion in the CLT forsums of independent non-identical r.v. P U ,δ,α, r Let η u ,δ,α = ( z − u ) , where z has density (8). In view of Lemma 10, the r.v. η u ,δ,α is concentrated on the interval [( max ( , δ − | u |)) , ( δ + | u |) ] and its first three central moments are: µ ( ) u = E η u ,δ,α = u + δ α + , (10) µ ( ) u = var ( η u ,δ,α ) = δ α + (cid:20) u + δ α ( α + ) ( α + ) (cid:21) , (11) µ ( ) u = E (cid:104) η u ,δ,α − µ ( ) u (cid:105) = α δ ( α + ) ( α + ) (cid:20) u + δ ( α − ) ( α + ) ( α + ) (cid:21) . (12)For a given U = ( u , . . . , u d ) ∈ R d , consider the r.v. on-lattice covering and quanitization of high dimensional sets 7 (cid:107) U − Z (cid:107) = d (cid:213) i = η u i ,δ,α = d (cid:213) j = ( u j − z j ) , where we assume that Z = ( z , . . . , z d ) is a random vector with i.i.d. components z i with density (8). From (10),its mean is µ = µ d ,δ,α, U : = E (cid:107) U − Z (cid:107) = (cid:107) U (cid:107) + d δ α + . Using independence of z , . . . , z d and (11), we obtain σ d ,δ,α, U : = var ((cid:107) U − Z (cid:107) ) = δ α + (cid:20) (cid:107) U (cid:107) + d δ α ( α + ) ( α + ) (cid:21) , and from independence of z , . . . , z d and (12) we get µ ( ) d ,δ,α, U : = E (cid:2) (cid:107) U − Z (cid:107) − µ (cid:3) = d (cid:213) j = µ ( ) u j = α δ ( α + ) ( α + ) (cid:20) (cid:107) U (cid:107) + d δ ( α − ) ( α + ) ( α + ) (cid:21) . If d is large enough then the conditions of the CLT for (cid:107) U − Z (cid:107) are approximately met and the distribution of (cid:107) U − Z (cid:107) is approximately normal with mean µ d ,δ,α, U and variance σ d ,δ,α, U . That is, we can approximate theprobability P U ,δ,α, r = P Z {(cid:107) U − Z (cid:107) ≤ r } by P U ,δ,α, r (cid:27) Φ (cid:18) r − µ d ,δ,α, U σ d ,δ,α, U (cid:19) , (13)where Φ (·) is the c.d.f. of the standard normal distribution: Φ ( t ) = ∫ t −∞ ϕ ( v ) d v with ϕ ( v ) = √ π e − v / . The approximation (13) has acceptable accuracy if the probability P U ,δ,α, r is not very small; for example, it fallsinside a 2 σ -confidence interval generated by the standard normal distribution. In the next section, we improveapproximations (13) by using an Edgeworth-type expansion in the CLT for sums of independent non-identicallydistributed r.v. P U ,δ,α, r General expansion in the central limit theorem for sums of independent non-identical r.v. has been derived byV.Petrov, see Theorem 7 in Chapter 6 in [6], see also Proposition 1.5.7 in [8]. The first three terms of thisexpansion have been specialized by V.Petrov in Section 5.6 in [7]. By using only the first term in this expansion,we obtain the following approximation for the distribution function of (cid:107) U − Z (cid:107) : P (cid:18) (cid:107) U − Z (cid:107) − µ d ,δ,α, U σ d ,δ,α, U ≤ x (cid:19) (cid:27) Φ ( x ) + µ ( ) d ,δ,α, U σ d ,δ,α, U ( − x ) ϕ ( x ) , (14)leading to the following improved form of (13): Jack Noonan and Anatoly Zhigljavsky P U ,δ,α, r (cid:27) Φ ( t ) + αδ (cid:104) (cid:107) U (cid:107) + d δ ( α − ) ( α + )( α + ) (cid:105) ( α + )( α + ) / (cid:104) (cid:107) U (cid:107) + d δ α ( α + )( α + ) (cid:105) / ( − t ) ϕ ( t ) , (15)where t : = r − µ d ,δ,α, U σ d ,δ,α, U = √ α + ( r − (cid:107) U (cid:107) − d δ α + ) δ (cid:113) (cid:107) U (cid:107) + d δ α ( α + )( α + ) . For α =
1, we obtain P U ,δ,α, r (cid:27) Φ ( t ) + δ (cid:2) (cid:107) U (cid:107) + d δ / (cid:3) √ (cid:2) (cid:107) U (cid:107) + d δ / (cid:3) / ( − t ) ϕ ( t ) with t = √ ( r − (cid:107) U (cid:107) − d δ / ) δ (cid:112) (cid:107) U (cid:107) + d δ / , which coincides with formula (16) of [4].A very attractive feature of the approximations (13) and (15) is their dependence on U through (cid:107) U (cid:107) only. Wecould have specialized for our case the next terms in Petrov’s approximation but these terms no longer depend on (cid:107) U (cid:107) only and hence the next terms are much more complicated. Moreover, adding one or two extra terms fromPetrov’s expansion to the approximation (15) does not fix the problem entirely for all U , δ , α and r . Instead, wepropose a slight adjustment to the r.h.s of (15) to improve this approximation, especially for small dimensions.Specifically, we suggest the approximation P U ,δ,α, r (cid:27) Φ ( t ) + c d ,α αδ (cid:104) (cid:107) U (cid:107) + d δ ( α − ) ( α + )( α + ) (cid:105) ( α + )( α + ) / (cid:104) (cid:107) U (cid:107) + d δ α ( α + )( α + ) (cid:105) / ( − t ) ϕ ( t ) , (16)where c d ,α = + /( α d ) .Below, there are figures of two types. In Figures 3–4, we plot P U ,δ,α, r over a wide range of r ensuring thatvalues of P U ,δ,α, r lie in the whole range [ , ] . In Figures 5–8, we plot P U ,δ,α, r over a much smaller range of r with P U ,δ,α, r lying roughly in the range [ , . ] . For the purpose of using formula (4), we need to assess theaccuracy of all approximations for smaller values of P U ,δ,α, r and hence the second type of plots are more useful.In these figures, the solid black line depicts P U ,δ,α, r obtained via Monte Carlo methods where for simplicity wehave set U = ( / , / , . . . , / ) and δ = /
2. Approximations (13) and (16) are depicted with a dotted blueand dash green line respectively. From numerous simulations and these figures, we can conclude the following.Whilst the basic normal approximation (13) seems adequate in the whole range of values of r , for particularlysmall probabilities, that we are most interested in, approximation (16) is much superior and appears to be veryaccurate for all values of α . on-lattice covering and quanitization of high dimensional sets 9 Fig. 3: P U ,δ,α, r and approximations: d = α = .
5. Fig. 4: P U ,δ,α, r and approximations: d = α = . P U ,δ,α, r and approximations: d = α = .
5. Fig. 6: P U ,δ,α, r and approximations: d = α = P U ,δ,α, r and approximations: d = α = .
5. Fig. 8: P U ,δ,α, r and approximations: d = α = C d ( Z n , r ) for Design 1 Consider now C d ( Z n , r ) for Design 1, as expressed via P U ,δ,α, r in (7). As U is uniform on [− , ] d , E (cid:107) U (cid:107) = d / ((cid:107) U (cid:107) ) = d / . Moreover, if d is large enough then (cid:107) U (cid:107) = (cid:205) dj = u j is approximately normal.We will combine the expressions (7) with approximations (13) and (16) as well as with the normal approximationfor the distribution of (cid:107) U (cid:107) , to arrive at two final approximations for C d ( Z n , r ) that differ in complexity. If theoriginal normal approximation (13) of P U ,δ,α, r is used then we obtain: C d ( Z n , r ) (cid:39) − ∫ ∞−∞ ψ ,α ( s ) ϕ ( s ) ds (17)with ψ ,α ( s ) = exp {− n Φ ( c s )} , c s = ( α + ) / (cid:16) r − s (cid:48) − d δ α + (cid:17) δ √ s (cid:48) + κ , s (cid:48) = s (cid:114) d + d / , κ = d δ α ( α + )( α + ) . If the approximation (16) is used, we obtain: C d ( Z n , r ) (cid:39) − ∫ ∞−∞ ψ ,α ( s ) ϕ ( s ) ds , (18)with ψ ,α ( s ) = exp − n (cid:169)(cid:173)(cid:173)(cid:171) Φ ( c s ) + c d ,α αδ (cid:104) s (cid:48) + d δ ( α − ) ( α + )( α + ) (cid:105) ( α + )( α + ) / [ s (cid:48) + κ ] / ( − c s ) ϕ ( c s ) (cid:170)(cid:174)(cid:174)(cid:172) . For α =
1, we get ψ , ( s ) = exp − n (cid:169)(cid:173)(cid:173)(cid:171) Φ ( c s ) + c d ,α δ (cid:104) s (cid:48) + d δ (cid:105) √ (cid:104) s (cid:48) + d δ (cid:105) / ( − c s ) ϕ ( c s ) (cid:170)(cid:174)(cid:174)(cid:172) (19)and the approximation (18) coincides with the approximation (26) in [4]. The accuracy of approximations (17)and (18) will be assessed in Section 6.1. C d ( Z n , r ) for Design 2a Our main quantity of interest in this section will be the probability P U ,δ, , r defined in (9) in the case wherecomponents z i of the vector Z = ( z , . . . , z d ) ∈ R d are i.i.d.r.v with Pr ( z i = δ ) = Pr ( z i = − δ ) = /
2; this is alimiting case of P U ,δ,α, r as α → P U ,δ, , r Using the same approach that led to approximation (13) in Section 3.1, the initial normal approximation for P U ,δ, , r is: on-lattice covering and quanitization of high dimensional sets 11 P U ,δ, , r (cid:27) Φ (cid:18) r − µ d ,δ, U σ d ,δ, U (cid:19) , (20)where, from Lemma 10, we have µ d ,δ, U = (cid:107) U (cid:107) + d δ and σ d ,δ, U = δ (cid:107) U (cid:107) . P U ,δ, , r From (38), we have µ ( ) d ,δ,α, U = α = (cid:107) U − Z (cid:107) : P (cid:18) (cid:107) U − Z (cid:107) − µ d ,δ, U σ d ,δ, U ≤ x (cid:19) (cid:27) Φ ( x ) − ( x − x ) κ ( ) d ,δ, , U σ d ,δ, , U ϕ ( x ) , (21)where κ ( ) d ,δ, , U is the sum of d fourth cumulants of the centred r.v. ( z − u ) , where z is concentrated at two points ± δ with Pr ( z = ± δ ) = /
2. From (38), κ ( ) d ,δ, , U : = d (cid:213) j = ( µ ( ) u j − [ µ ( ) u j ] ) = − δ d (cid:213) i = u i . Unlike (14), the rhs of (21) does not depends solely on (cid:107) U (cid:107) . However, the quantities (cid:107) U (cid:107) and (cid:205) di = u i arestrongly correlated; one can show that for all d corr (cid:32) (cid:107) U (cid:107) , d (cid:213) i = u i (cid:33) = √ (cid:27) . . This suggests (by rounding the correlation above to 1) the following approximation: d (cid:213) i = u i (cid:27) √ d (cid:169)(cid:173)(cid:173)(cid:171) (cid:107) U (cid:107) − d / (cid:113) d (cid:170)(cid:174)(cid:174)(cid:172) + d . With this approximation, the rhs of (21) depends only on (cid:107) U (cid:107) . As a result, the following refined form of (20)is: P U ,δ, , r (cid:27) Φ ( t ) + ( t − t ) ((cid:107) U (cid:107) − d / )/√ + d / (cid:107) U (cid:107) ϕ ( t ) , where t : = r − µ d ,δ, , U σ d ,δ, , U = ( r − (cid:107) U (cid:107) − d δ ) δ (cid:107) U (cid:107) . Similarly to approximation (16), we propose a slight adjustment to the r.h.s of the approximation above: P U ,δ, , r (cid:27) Φ ( t ) + (cid:18) + d (cid:19) ( t − t ) ((cid:107) U (cid:107) − d / )/√ + d / (cid:107) U (cid:107) ϕ ( t ) . (22) In the same style as at the end of Section 3.2, below there are figures of two types. In Figures 9–10, we plot P U ,δ, , r over a wide range of r ensuring that values of P U ,δ, , r lie in the range [ , ] . In Figures 11–14, weplot P U ,δ, , r over a much smaller range of r with P U ,δ, , r lying in the range [ , . ] . In these figures, the solidblack line depicts P U ,δ,α, r obtained via Monte Carlo methods where we have set δ = / U is a pointsampled uniformly on [− , ] d ; for reproducibility, in the caption of each figure we state the random seed usedin R. Approximations (20) and (22) are depicted with a dotted blue and dash green line respectively. Fromthese figures, we can conclude the same outcome as in Section 3.2. Whilst the approximation (20) is rathergood overall, for small probabilities the approximation (22) is much superior and is very accurate. Note thatsince random vectors Z j are taking values on a finite set, which is the set of points (± δ, . . . , ± δ ) , the probability P U ,δ, , r considered as a function of r , is a piece-wise constant function.Fig. 9: P U ,δ, , r and approximations: d = seed =
10. Fig. 10: P U ,δ, , r and approximations: d = seed = P U ,δ, , r and approximations: d = seed =
10. Fig. 12: P U ,δ, , r and approximations: d = seed = on-lattice covering and quanitization of high dimensional sets 13 Fig. 13: P U ,δ, , r and approximations: d = seed =
10. Fig. 14: P U ,δ, , r and approximations: d = seed = C d ( Z n , r ) Consider now C d ( Z n , r ) for Design 2a, as expressed via P U ,δ,α, r in (7). Using the normal approximation for (cid:107) U (cid:107) as made in the beginning of Section 3.3, we will combine the expressions (7) with approximations (20)and (22) to arrive at two approximations for C d ( Z n , r ) that differ in complexity.If the original normal approximation (20) of P U ,δ, , r is used then we obtain: C d ( Z n , r ) (cid:39) − ∫ ∞−∞ ψ , n ( s ) ϕ ( s ) ds , (23)with ψ , n ( s ) = exp {− n Φ ( c s )} , c s = (cid:0) r − s (cid:48) − d δ (cid:1) δ √ s (cid:48) , s (cid:48) = s (cid:114) d + d / . If the approximation (22) is used, we obtain: C d ( Z n , r ) (cid:39) − ∫ ∞−∞ ψ , n ( s ) ϕ ( s ) ds , (24)with ψ , n ( s ) = exp (cid:40) − n (cid:32) Φ ( c s ) + (cid:18) + d (cid:19) ( c s − c s ) ( s (cid:48) − d / )/√ + d / ( s (cid:48) ) ϕ ( c s ) (cid:33)(cid:41) . (25)and c s = (cid:0) r − s (cid:48) − d δ (cid:1) δ √ s (cid:48) , s (cid:48) = s (cid:114) d + d / . The accuracy of approximations (23) and (24) will be assessed in Section 6.1. C d ( Z n , r ) for Design 2b Designs whose points Z i have been sampled from a finite discrete set without replacement have dependence,for example Design 2b, and therefore formula (4) cannot be used.In this section, we suggest a way of modifying the approximations developed in Section 4 for Design 2a. Thiswill amount to approximating sampling without replacement by a suitable sampling with replacement. Let S be a discrete set with k distinct elements, where k is reasonably large. In case of Design 2b, the set S consists of k = d vertices of the cube [− δ, δ ] d . Let Z n = { Z , . . . , Z n } denote an n − point design whose points Z i have been sampled without replacement from S ; n < k . Also, let Z (cid:48) m = { Z (cid:48) , . . . , Z (cid:48) m } denote an associated m − point design whose points Z (cid:48) i are sampled with replacement from the same discrete set S ; Z (cid:48) , . . . , Z (cid:48) m arei.i.d. random vectors with values in S . Our aim in this section is to establish an approximate correspondencebetween n and m .When sampling m times with replacement, denote by X i the number of times the i th element of S appears.Then the vector ( X , X , . . . , X k ) has the multinomial distribution with number of trials m and event proba-bilities ( / k , / k , . . . , / k ) with each individual X i having the Binomial distribution Binomial ( m , / k ) . Sincecorr ( X i , X j ) = − / k when i (cid:44) j , for large k the correlation between random variables X , X , . . . , X k is verysmall and will be neglected. Introduce the random variables: Y i = (cid:40) , if X i = , if X i > . Then the random variable N = (cid:205) ki = Y i represents the number of elements of S not selected. Given the weakcorrelation between X i , we approximately have N ∼ Binomial ( k , P ( X = )) . Using the fact P ( X = ) = ( − / k ) m , the expected number of unselected elements when sampling with replacement is approximately E N (cid:27) k ( − / k ) m . Since, when sampling without replacement from S we have chosen N = k − n elements,to choose the value of m we equate E N to k − n . By solving the equation k − n = k (cid:18) − k (cid:19) m [ (cid:27) E N ] for m we obtain m = log ( k − n ) − log ( k ) log ( k − ) − log ( k ) . (26) C d ( Z n , r ) for Design 2b. Consider now C d ( Z n , r ) for Design 2b. By applying the approximation developed in the previous section, thequantity C d ( Z n , r ) can be approximated by C d ( Z m , r ) for Design 2a with m given in (26): Approximation of C d ( Z n , r ) for Design 2b. We approximate it by C d ( Z m , r ) where m is given in (26) and C d ( Z m , r ) is approximated by (24) with n substituted by m from (26) . Specifying this, we obtain: on-lattice covering and quanitization of high dimensional sets 15 C d ( Z n , r ) (cid:39) − ∫ ∞−∞ ψ , m ( s ) ϕ ( s ) ds , (27)where m = m n , d = log ( d − n ) − d log ( ) log ( d − ) − d log ( ) (28)and the function ψ , · (·) is defined in (25). The accuracy of the approximation (27) will be assessed in Section 6.1. C d ( Z n , r ) and studying their dependence on δ In this section, we present the results of a large-scale numerical study assessing the accuracy of approximations(17), (18), (23), (24) and (27). In Figures 15–28, by using a solid black line we depict C d ( Z n , r ) obtained byMonte Carlo methods, where the value of r has been chosen such that the maximum coverage across δ isapproximately 0 .
9. In Figures 15–20, dealing with Design 1, approximations (17) and (18) are depicted witha dotted blue and dashed green lines respectively. In Figures 21–24 (Design 2a) approximations (23) and (24)are illustrated with a dotted blue and dashed green lines respectively. In Figures 25–28 (Design 2b) the dashedgreen line depicts approximation (27). From these figures, we can draw the following conclusions.• Approximations (18) and (24) are very accurate across all values of δ and α . This is particularly evident for d = , d , like d = δ close to one (for such values of δ the covering is very poor)and n close to 2 d this approximation begins to worsen, see Figures 26 and 28.• A sensible choice of δ can dramatically increase the coverage proportion C d ( Z n , r ) . This effect, which wecall ‘ δ -effect’, is evident in all figures and is very important. It gets much stronger as d increases.Fig. 15: Design 1: C d ( Z n , r ) and approximations; d = , α = . , n = C d ( Z n , r ) and approximations; d = , α = . , n = Fig. 17: Design 1: C d ( Z n , r ) and approximations; d = , α = . , n = C d ( Z n , r ) and approximations; d = , α = . , n = C d ( Z n , r ) and approximations; d = , α = . , n = C d ( Z n , r ) and approximations; d = , α = . , n = C d ( Z n , r ) and approximations; d = , α = , n = C d ( Z n , r ) and approximations; d = , α = , n = α In Table 1, for Design 2a and Design 1 with α = . , , . r required toachieve the 0.9-coverage on average. For these schemes, the value inside the brackets shows the average value on-lattice covering and quanitization of high dimensional sets 17 Fig. 23: Design 2a: C d ( Z n , r ) and approximations; d = , α = , n = C d ( Z n , r ) and approximations; d = , α = , n = C d ( Z n , r ) and approxima-tion (27); d = , n = C d ( Z n , r ) and approximation(27); d = , n = C d ( Z n , r ) and approxima-tion (27); d = , n = C d ( Z n , r ) and approximation(27); d = , n = δ required to obtain this 0.9-coverage. Design 2b is not used as d is too small (for this design, we must have n < d and in these cases Design 2b provides better coverings than the other designs considered).From Tables 1 and 2 we can make the following conclusions:• For small n ( n < d or n (cid:39) d ), Design 2a provides a more efficient covering than other three other schemesand hence smaller values of α are better. d = n = n = n = n = ( α = ) α = . α = α = . Table 1: Values of r and δ (in brackets) to achieve 0.9 coverage for d = d = n = n = n = n = ( α = ) α = . α = α = . Table 2: Values of r and δ (in brackets) to achieve 0.9 coverage for d = n > d , Design 2a begins to become impractical since a large proportion of points duplicate. This isreflected in Table 1 by comparing n =
100 and n =
500 for Design 2a; there is only a small reduction in r despite a large increase in n . Moreover, for values of n >> d , Design 2a provides a very inefficient covering.• For n >> d , from looking at Design 1 with α = . n = α ∈ ( , ) rather than α > α = C d ( Z n , r ) across δ for different choices of α .In Figures 29–30, the red line, green line, blue line and cyan line depict approximation (24) ( α =
0) andapproximation (18) with α = . α = α = . α , at least for these values of n and d .Fig. 29: d = n = r = .
228 Fig. 30: d = n = r = . In this section, we will study the following characteristic of a design Z n . on-lattice covering and quanitization of high dimensional sets 19 Quantization error.
Let U = ( u , . . . , u d ) be uniform random vector on [− , ] d . The mean squared quantizationerror for a design Z n = { Z , . . . , Z n } ⊂ R d is defined by θ ( Z n ) = E U (cid:37) ( U , Z n ) , where (cid:37) ( U , Z n ) = min Z i ∈ Z n (cid:107) U − Z i (cid:107) . (29)If the design Z n is randomized then we consider the expected value E Z n θ ( Z n ) of θ ( Z n ) as the main characteristicwithout stressing this.The mean squared quantization error θ ( Z n ) is related to our main quantity C d ( Z n , r ) defined in (1): indeed, C d ( Z n , r ) , as a function of r ≥
0, is the c.d.f. of the r.v. (cid:37) ( U , Z n ) while θ ( Z n ) is the second moment of thedistribution with this c.d.f.: θ ( Z n ) = ∫ r ≥ r dC d ( Z n , r ) . (30)This relation will allow us to use the approximations derived above for C d ( Z n , r ) in order to construct approxi-mations for the quantization error θ ( Z n ) . Using approximation (18) for the quantity C d ( Z n , r ) , we obtain ddr ( C d ( Z n , r )) (cid:27) f α,δ ( r ) : = n · r δ ∫ ∞−∞ ϕ ( s ) ϕ ( c s ) ψ ,α ( s ) ×× √ α + √ s (cid:48) + k + c d ,α α (cid:16) s (cid:48) + d δ ( α − ) ( α + )( α + ) (cid:17) ( α + ) ( s (cid:48) + k ) (cid:40) δ ( c s − c s ) − √ α + ( r − d δ α + − s (cid:48) )√ s (cid:48) + k (cid:41) ds . (31)By then using relation (30) we obtain the following approximation for the mean squared quantization error withDesign 1: θ ( Z n ) (cid:27) ∫ ∞ r f α,δ ( r ) dr . (32)By taking α = f ,δ ( r ) : = n · r δ ∫ ∞−∞ ϕ ( s ) ϕ ( c s ) ψ , ( s ) √ √ s (cid:48) + k + c d , (cid:16) s (cid:48) + d δ (cid:17) ( s (cid:48) + k ) (cid:40) δ ( c s − c s ) − √ ( r − d δ − s (cid:48) )√ s (cid:48) + k (cid:41) ds . with ψ , defined in (19). The resulting approximation θ ( Z n ) (cid:27) ∫ ∞ r f ,δ ( r ) dr . coincides with [4, formula 31]. Using approximation (24) for the quantity C d ( Z n , r ) , we have: ddr ( C d ( Z n , r )) (cid:27) f ,δ ; n ( r ) : = n · r δ ∫ ∞−∞ ϕ ( s ) ϕ ( c s ) ψ , n ( s )√ s (cid:48) (cid:34) + (cid:18) + d (cid:19) ( ( s (cid:48) − d / )/√ + d / )( c s − c s − ) ( s (cid:48) ) (cid:35) ds , (33)where ψ , n (·) is defined in (25). From (30) we then obtain the following approximation for the mean squaredquantization error with Design 2a: θ ( Z n ) (cid:27) ∫ ∞ r f ,δ ; n ( r ) dr . (34) Similarly to (34), for Design 2b, we use the approximation θ ( Z n ) (cid:27) ∫ ∞ r f ,δ ; m ( r ) dr . (35)where f ,δ, m ( r ) is defined by (33) and m = m n , d is defined in (28). δ -effect In this section, we assess the accuracy of approximations (32), (34) and (35). Using a black line we depict E Z n θ ( Z n ) obtained via Monte Carlo simulations. Depending on the value of α , in Figures 31–36 approximation(32) or (34) is shown using a red line. In Figures 41–44, approximation (35) is depicted with a red line. From thefigures below we can see that all approximations are generally very accurate. Approximation (34) is much moreaccurate than approximation (32) across all choices of δ and n and this can be explained by the additional termtaken in the general expansion; see Section 4.2. This high accuracy is also seen with approximation (35). Theaccuracy of approximation (32) seems to worsen for large δ , n and d not too large like d =
20, see Figures 33–34.For d =
50, all approximations are extremely accurate for all choices of δ and n . Figures 31–36 very clearlydemonstrate the δ -effect implying that a sensible choice of δ is crucial for good quantization. on-lattice covering and quanitization of high dimensional sets 21 Fig. 31: E θ ( Z n ) and approximation (32): d = α = n =
500 Fig. 32: E θ ( Z n ) and approximation (32): d = α = . n = E θ ( Z n ) and approximation (32): d = α = . n = E θ ( Z n ) and approximation (32): d = α = n = E θ ( Z n ) and approximation (32): d = α = . n = E θ ( Z n ) and approximation (32): d = α = n = Fig. 37: E θ ( Z n ) and approximation (34): d = α = n =
100 Fig. 38: E θ ( Z n ) and approximation (34): d = α = n = E θ ( Z n ) and approximation (34): d = α = n =
500 Fig. 40: E θ ( Z n ) and approximation (34): d = α = n = E θ ( Z n ) and approximation (35): d = n =
100 Fig. 42: E θ ( Z n ) and approximation (35): d = n = on-lattice covering and quanitization of high dimensional sets 23 Fig. 43: E θ ( Z n ) and approximation (35): d = n =
500 Fig. 44: E θ ( Z n ) and approximation (35): d = n = Let us extend the range of designs considered above by additing the following two designs.
Design 3. Z , . . . , Z n are taken from a low-discrepancy Sobol’s sequence on the cube [− δ, δ ] d . Design 4. Z , . . . , Z n are taken from the minimum-aberration d − k fractional factorial design on the vertices ofthe cube [− δ, δ ] d . Unlike Designs 1, 2a, 2b and 3, Design 4 is non-adaptive and defined only for a particular n of the form n = d − k with some k ≥
0. We have included this design into the list of all designs as "the golden standard". In view ofthe numerical study in [4] and theoretical arguments in [5], Design 4 with k = δ provides the bestquantization we were able to find; moreover, we have conjectured in [5] that Design 4 with k = δ provides minimal normalized mean squared quantization error for all designs with n ≤ d . We repeat, Design4 is defined for one particular value of n only. In Tables 3–4, we present results of Monte Carlo simulations where we have computed the smallest values of r required to achieve the 0.9-coverage on average (on average, for Designs 1, 2a, 2b). The value inside the bracketsshows the value of δ required to obtain the 0.9-coverage.From Tables 3–4 we draw the following conclusions:• Designs 2a and especially 2b provide very high quality coverage (on average) whilst being online procedures(that is, nested designs);• Design 2b has significant benefits over Design 2a for values of n close to 2 d ;• properly δ -tuned deterministic non-nested Design 4 provides superior covering;• coverage properties of δ -tuned low-discrepancy sequences are much better than of the original low-discrepancy sequences;• coverage of an unadjusted low-discrepancy sequence is poor. d = n = n = n = n = α = . α = . δ = Table 3: Values of r and δ (in brackets) to achieve 0.9 coverage for d = d = n = n = n = n = α = . α = . δ = Table 4: Values of r and δ (in brackets) to achieve 0.9 coverage for d = n and δ , we plot C d ( Z n , r ) as a function of r for the following designs: Design 1with α = δ = α =
1, Design 2a and Design 2b, we have used approximations (19), (25) and (27) respectivelyto depict C d ( Z n , r ) whereas for Design 3, we have used Monte Carlo simulations. For the first three designs,depending of the choice of n , the value of δ has been fixed based on the optimal value for quantization; theseare the values inside the brackets in Tables 5–6.From Figure 45, we see that Design 2b is superior and uniformly dominates all other designs for this choiceof d and n (at least when the level of coverage is greater than 1/2). In Figure 46, since n << d , the values of C d ( Z n , r ) for Designs 2a and 2b practically coincide and the green line hides under the blue. In both figures wesee that Design 3 with an unadjusted δ provides a very inefficient covering.Fig. 45: C d ( Z n , r ) as a function of r for severaldesigns: d = n =
512 Fig. 46: C d ( Z n , r ) as a function of r for severaldesigns: d = n = on-lattice covering and quanitization of high dimensional sets 25 As follows from results of [3, Ch.6], for efficient covering schemes the order of convergence of the coveringradius to 0 as n → ∞ is n − / d . Therefore, for the mean squared distance (which is the quantization error) weshould expect the order n − / d as n → ∞ . Therefore, for sake of comparison of quantization errors θ n across n we renormalize this error from E θ n to n / d E θ n .In Figure 47–6, we present the minimum value of n / d E θ n for a selection of designs. In these tables, the valuewithin the brackets corresponds to the value of δ where the minimum of n / d E θ n was obtained. d = n = n = n = n = α = . α = α = . δ = Table 5: Minimum value of n / d E θ n and δ (in brackets) across selected designs; d = d = n = n = n = n = α = . α = α = . δ = Table 6: Minimum value of n / d E θ n and δ (in brackets) across selected designs; d = (cid:37) ( X , Z n ) for Design 2a with δ = . δ = . d =
10 and n = n = δ = . δ = .
82 for Design 3). Here we see a very clear stochastic dominance ofthe Design 2b over Design 4. All findings are consistent with Tables 5 and 6. In Figures 47 and 48, values ofthe parameter δ for all designs are chosen as numerically optimal, in accordance with Table 5.We make the following conclusions from analyzing results of this section:• Designs 2a and 2b provide very good quantization per point. As expected, Design 2b is superior over Design2a when n is close to 2 d ; see Table 5.• Properly δ -tuned non-nested Design 4 is provides the best quantization per point of all designs considered.• Properly δ -tuned Design 3 is comparable in performance to Design 1 but it is not as efficient as Designs 2a,2b and 4. Fig. 47: d = , n = δ = . δ = .
8. Fig. 48: d = , n = δ = . δ = . d -simplex Consider the standard orthogonal d -simplex S d : = (cid:40) ( u , u , . . . , u d ) ∈ R d (cid:12)(cid:12) d (cid:213) i = u i ≤ u i ≥ i (cid:41) with vol (S d ) = / d !. For a design Z n = { Z , . . . , Z n } , consider the following two characteristics:(a) the proportion of the simplex S d covered by B d ( Z n , r ) : C d ( Z n , r ) : = d ! vol (S d ∩ B d ( Z n , r )) , (36)(b) θ ( Z n ) = E U min i = ,..., n (cid:107) U − Z i (cid:107) , the mean squared quantization error for Z n , where U = ( u , . . . , u d ) is arandom vector uniformly distributed in S d .In this section, we investigate whether the δ -effect seen in Sections 6, 7.5 and 8 for the cube is present for thesimplex S d . We will consider two possible ways of scaling points in S d . Define the two δ -simplices S ( δ ) d , and S ( δ ) d , as follows: S ( δ ) d , : = δ · S d , S ( δ ) d , : = (cid:40) ( u , u , . . . , u d ) ∈ R d (cid:12)(cid:12) d (cid:213) i = u i ≤ d + δ d + u i ≥ − δ d + i (cid:41) . By construction, the value of δ in S ( δ ) d , scales the simplex around its centroid S ∗ d = (cid:16) d + , d + , . . . , d + (cid:17) , wherefor δ =
1, we have S ( δ ) d , = S d . Simple depictions of S ( δ ) d , and S ( δ ) d , are given in Figures 49–50.We will numerically assess covering and quantization characteristics for the following two designs. on-lattice covering and quanitization of high dimensional sets 27 ( , ) ( , )( , )( , ) ( / , )( , / ) Fig. 49: S d and S ( δ ) d , with d = δ = . ( , ) ( , )( , )( / , / ) ( / , / )( / , / ) S ∗ d Fig. 50: S d and S ( δ ) d , with d = δ = . Design S1. Z , . . . , Z n are i.i.d. random vectors uniformly distributed in the δ -scaled simplex S ( δ ) d , , where δ ∈ [ , ] is a parameter. Design S2. Z , . . . , Z n are i.i.d. random vectors uniformly distributed in the δ -scaled simplex S ( δ ) d , , where δ ∈ [ , ] is a parameter. To simulate points Y uniformly distributed in the simplex S d , we can simply generate d i.i.d. uniformlydistributed points in [ , ] , add 0 and 1 to the collection of points and take the first d spacings (out of the totalnumber d + Y (cid:48) = δ Y and Y (cid:48)(cid:48) = δ · ( Y − S ∗ d ) + S ∗ d are then uniform in S ( δ ) d , and S ( δ ) d , respectively. This procedure can be easily performed in R using the package ‘uniformly’. δ -effect for d -simplex Using the above procedure, we numerically study characteristics of Designs S1 and S2. In Figures 51–54 weplot C d ( Z n , r ) as a functions of δ ∈ [ , ] across n , r and d for Design S1. The corresponding results for DesignS2 are given in Figures 55–58. In Figures 59–60 and Figures 61–62, we depict E θ ( Z n ) for Designs S1 and S2respectively for different n and d . In each figure we plot values of E θ ( Z n ) for different values of r ; a step in r increase gives the next curve up. Fig. 51: C d ( Z n , r ) for Design S1: d = n = r from 0 .
11 to 0 .
17 increasing by 0 .
02. Fig. 52: C d ( Z n , r ) for Design S1: d = n = r from 0 .
13 to 0 .
19 increasing by 0 . C d ( Z n , r ) for Design S1: d = n = r from 0 .
13 to 0 .
17 increasing by 0 .
01. Fig. 54: C d ( Z n , r ) for Design S1: d = n = r from 0 .
12 to 0 .
15 increasing by 0 . C d ( Z n , r ) for Design S2: d = n = r from 0 .
11 to 0 .
17 increasing by 0 .
02. Fig. 56: C d ( Z n , r ) for Design S2: d = n = r from 0 .
13 to 0 .
19 increasing by 0 . on-lattice covering and quanitization of high dimensional sets 29 Fig. 57: C d ( Z n , r ) for Design S2: d = n = r from 0 .
13 to 0 .
17 increasing by 0 .
01. Fig. 58: C d ( Z n , r ) for Design S2: d = n = r from 0 .
11 to 0 .
14 increasing by 0 . E θ ( Z n ) for Design S1: d = n = E θ ( Z n ) for Design S1: d = n = E θ ( Z n ) for Design S2: d = n = E θ ( Z n ) for Design S2: d = n = From the above figures, we arrive at the following conclusions:• The δ -effect for the simplex is much less prominent than for the cube.• Between Designs S1 and S2, the δ -effect is more apparent for Design S2; for example, compare Figure 60with Figure 62.
10 Appendix: An auxiliary lemma
Lemma 1.
Let δ > , u ∈ R and η u ,δ be a r.v. η u ,δ = ( ξ − u ) , where r.v. ξ ∈ [− δ, δ ] has Beta δ ( α, α ) distributionwith density p α,δ ( t ) = ( δ ) − α Beta ( α, α ) [ δ − t ] α − , − δ < t < δ , α >
0; (37)
Beta (· , ·) is the Beta-function. The r.v. η u ,δ is concentrated on the interval [( max ( , δ − | u |)) , ( δ + | u |) ] . Its firstthree central moments are: µ ( ) u ,δ = E η u ,δ = u + δ α + ,µ ( ) u ,δ = var ( η u ,δ ) = δ α + (cid:20) u + δ α ( α + ) ( α + ) (cid:21) ,µ ( ) u ,δ = E (cid:2) η u ,δ − E η u ,δ (cid:3) = α δ ( α + ) ( α + ) (cid:20) u + δ ( α − ) ( α + ) ( α + ) (cid:21) . In the limiting case α =
0, where the r.v. ξ is concentrated at two points ± δ with equal weights, we obtain: µ ( ) u ,δ = E η u ,δ = u + δ and µ ( k ) u ,δ = [ δ u ] k , µ ( k + ) u ,δ = , for k = , , . . . (38) References
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