LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION

The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische Hochschule Darmstat an University of Warwick A theory of optimum orthogonal fractions is evelope for Fourier regression moels using integer lattice esigns. These provie alternatives to simple gris (prouct esigns) in the case when specifie main effects an interaction terms are require to be analyze. The challenge is to obtain sample sizes which are polynomial in the imension rather than exponential. This is achieve for certain moels with special algorithms base on both algebraic generation an more irect sequential search. 1. Introuction an preliminaries. Bates, Buck, Riccomagno an Wynn (1996) mention that integer lattice esigns are D-optimum in the sense of Kiefer an Wolfowitz (1959) for Fourier regression moels. It has been known for some years that equally space gris (prouct esigns) have this property. In analogy to the situation of polynomial regression, we can reuce the size of the experiment by using a fraction if no or only a limite number of interactions are require to be estimate. Here the lattice will play the role of fractions in the polynomial theory. The alias structure turns out to be raically ifferent with the cyclic group playing an important role via the harmonic nature of the theory. The lattice structure allows us to map a high-imensional moel into a suitable one-imensional moel. For the one-imensional Fourier regression moel of orer m, (1) E Y x = θ 0 + m 2 sin 2πrx θ r + m 2 cos 2πrx φ r r=1 r=1 x 0 1, the equally space esign points on an equiistant gri with, at least, 2m+1 support points is D-optimum in the sense of Kiefer an Wolfowitz [see Hoel (1965)]. Note that these esigns are optimum irrespectively of whether the normalizing factor 2 is inclue in the moel equation (1) or not. A uniform esign with exactly 2m + 1 support points has minimal support; that is, there are exactly as many esign points as there are parameters in the moel. Let A l = α 1 1 l α 1 = 1 be the set of all l-imensional multiinices from 1 1 with unit first entry. Then a complete M-factor interaction moel Receive June 1995; revise November 1996. 1 Work supporte by the UK Engineering an Physical Science Research Council. 2 Work supporte by Grants Ku719/2-1 an Ku719/2-2 of the Deutsche Forschungsgemeinschaft. AMS 1991 subject classifications. 62K05, 62J99. Key wors an phrases. Fourier moels, lattices, orthogonal fractions complexity. 2313

2314 RICCOMAGNO, SCHWABE AND WYNN can be written as follows: E Y x 1 x (2) = θ 0 + M 2 m k1 l=1 k 1 < <k l r 1 =1 m kl r l =1 α A l sin 2π α 1 r 1 x k1 + + α l r l x kl θ k1 k l r 1 r l α + M 2 m k1 l=1 k 1 < <k l r 1 =1 m kl r l =1 α A l cos 2π α 1 r 1 x k1 + + α l r l x kl φ k1 k l r 1 r l α x = x 1 x 0 1. We introuce the notation F m 1 m M for the complete Fourier moel (2) in imensions with one-imensional marginal moels for x 1 x respectively of orers m 1 to m an in which all interactions up to M factors are inclue. In the present paper we place special emphasis on the particular cases of aitive moels F m 1 m 1 without interactions an first orer interaction moels F m 1 m 2, that is, on moels with complete two-factor interactions. Note that the general moel (2) is equivalent to the complete M-factor interaction moel (3) M E Y x 1 x = θ 0 + f k1 x k1 f kl x kl β k1 k l l=1 k 1 < <k l with marginals E Y k x k = θ 0 + f k x k β k which has been investigate, for example, by Schwabe (1996a), in general. Here we consier the marginals f k x k = 2 sin 2πrx k cos 2πrx k r=1 m k. From the stanar formulas for trigonometric functions of sums, it is seen that both moels are linke by an orthogonal transformation matrix which oes not affect the optimality criteria uner consieration. Hoel (1965) showe that prouct esigns are D-optimum for Kronecker prouct type moels (M = ). Thus, if ξ 1 is a D-optimum esign measure for a linear moel E Y 1 x 1 = m 1 i=0 θ if i x 1 on a esign space X 1 an if ξ 2 is D-optimum for a moel E Y 2 x 2 = m 2 i=0 φ ig i x 2 on a space X 2, then the prouct ξ 1 ξ 2 (in the measure-theoretic sense) is D-optimum for the linear moel E Y x 1 x 2 = m 1 m2 i=0 j=0 β ijf i x 1 g j x 2 which inclues all terms of the form f i x 1 g j x 2 on the space X 1 X 2. The same is true for aitive moels without interactions if f 0 = 1 an g 0 = 1 [see Schwabe (1996a)]. In general, the prouct ξ 1 ξ of marginally D-optimum esigns ξ k is D-optimum if a constant term is inclue in each marginal moel [see Schwabe (1996a), Section 6.1]. For the case of Fourier regression, this last restriction is not necessary [see Schwabe (1996b)]. In aitive moels the optimality of prouct esigns has been prove by Rafajłowicz an Myszka (1992) an by Schwabe

LATTICE-BASED D-OPTIMUM DESIGN 2315 (1996a) for other criteria, incluing A-, E- an integrate mean square error optimality. If we use D-optimum esigns with minimal support in the marginal Fourier moels of orer m k, the number of support points for the prouct esign is k=1 2m k + 1 which coincies with the number of parameters in the Kronecker prouct type complete interaction moel (M = ). In general, this number is much larger than the number of parameters in a -imensional moel. For example, if the orers of the marginal moels coincie (m 1 = = m = m), then there are m + 1 parameters in the aitive moel F m m 1 an O 2 parameters in the two-factor interactions moels. It is challenging, then, to obtain D-optimum esign sequences for which the sample size oes not increase much faster in the imension than the number of parameters. In particular, we will look for sequences of D-optimum esigns with a polynomially increasing support rather than the exponentially increasing prouct esigns. The main purpose of the present paper is to show that optimality can be achieve by using lattice esigns for complete Fourier moels instea of prouct esigns. Like the Fourier moels themselves, such esigns have a structure which is cyclic, but for which the aliasing theory is rather ifferent from that for classical polynomial moels. We shall give examples of optimum families of esigns for specifie complete Fourier moels. We consier lattices generate by a single-integer generator which therefore can be ientifie as one-imensional objects. Let g = g 1 g be a - imensional vector with positive integer components an n a positive integer. The vector generates a finite number of line segments in the -imensional hypercube 0 1 ; that is, the set tg 1 mo 1 tg mo 1 t R consists of a finite number of line segments intersecting 0 1. If we ientify the hypercube with the corresponing torus by matching opposite faces, then these line segments form a single close line wrappe aroun the torus. The generate esigns will be supporte on these line segments an will be referre to as lattice esigns. The n-point lattice esign generate by g 1 g is the uniform esign supporte on the gri ( j n g 1 mo 1 j ) ( n g jg1 mo n mo 1 = jg ) mo n n n j = 0 n 1 If g 1 an n are mutually prime, we may assume without loss of generality that g 1 = 1 [see Nieerreiter (1992)]. This relabeling is achieve by a rearrangement of the esign points. Note that the one-imensional uniform esign supporte on n equally space points is a particular case of a lattice esign generate by g = 1. 2. Main statements. In this section we show how the generator g = g 1 g can be chosen for a given moel F m 1 m M such that

2316 RICCOMAGNO, SCHWABE AND WYNN the n-point lattice esign is D-optimum. In Section 2.1 we start with an illustrative example. In Section 2.2 the theoretical backgroun is evelope an conitions are elaborate which must be satisfie by g an n. Finally, in Section 2.3 an algorithm is presente to fin families of D-optimum lattice esigns for which the size increases polynomially in the imension. Section 3 collects various applications to aitive moels an to two-factor interaction moels. In the latter D-optimum esigns are investigate for either the main effects only or for the whole parameter vector. Sequences of generators an alternative algebraic methos are etermine by the algorithm of Section 2.3. 2.1. Motivation. A key observation in this paper is that a moel with terms in higher imensions can be mappe into a moel in one imension by exploiting the structure of the lattice. We start with the most simple two-imensional example of the moel F 2 1 1 1 which has five parameters an we use the n-point lattice with generator g = 1 3 an n = 5 support points. For the moel F 2 1 1 1 the moel equation (2) reuces to E Y x 1 t x 2 t = θ 0 + 2 sin 2πt θ 1 + 2 cos 2πt φ 1 + 2 sin 2π3t θ 2 + 2 cos 2π3t φ 2 on the line segments tg 1 mo 1 tg 2 mo 1 = t 3t mo 1 by the perioicity of the trigonometric functions. Hence, the moel can be ientifie with a oneimensional incomplete Fourier moel in the variable t. Moreover, the fivepoint lattice esign generate by g = 1 3 is an equally space esign with five support points on the line segments consiere as a line on the torus. The information matrices coincie in both the original two-imensional moel F 2 1 1 1 an in the erive one-imensional moel on the line segments. Now, it can be checke that the five-point equally space esign is D-optimum for the erive one-imensional moel (see Lemma 1). Hence, the information matrix is a multiple of the ientity which, in turn, proves the D-optimality of the five-point lattice esign for the original moel. Note that the prouct esign requires nine esign points. 2.2. General results. We start by formulating in our terminology the Hoel (1965) result on the optimality of equiistant esigns for one-imensional Fourier moels. Theorem 1. For the one-imensional moel F 1 m 1 the n-point lattice esign generate by the integer g, n 2m + 1, is D-optimum if an only if, within the cyclic group n = 0 1 n 1 +, the carinality of 0 g is 2m + 1 where = r r = m 1 1 m. In other wors, the n-point lattice esign is D-optimum if an only if all members in are istinct an ifferent from 0 mo n. Note that the conition in Theorem 1 is satisfie if an only if all the parameters of the moel are estimable; that is, D-optimality coincies with estimability for one-

LATTICE-BASED D-OPTIMUM DESIGN 2317 imensional lattice esigns. We will see later on that this is also true for higher imensions. When all the entries in the array 0 rg r are istinct in the set Z of all integers, they are also istinct in the cyclic group n if n is sufficiently large (n 2mg + 1). Hence, 2mg + 1 is an upper boun for the minimum sample size require, whereas a lower boun is given by the number of parameters in the moel. Similar bouns are vali in higher imensions. The next result is a generalization of Theorem 1 an is helpful in proving the D-optimality of lattice esigns. Lemma 1. For the incomplete one-imensional Fourier moel, (4) E Y x = θ 0 + q 2 sin 2πr j x θ j + q 2 cos 2πr j x φ j j=1 j=1 where the r j are istinct positive integers an the n-point lattice esign generate by g is D-optimum for the subsystem θ 0 θ 1 φ 1 θ p φ p of parameters, p q, if an only if, within the cyclic group n = 0 1 n 1 +, (i) the carinality of the set 0 g is 2p + 1, where = ±r j j = 1 p, an (ii) r j 0 g for j = p + 1 q. Proof. It is a funamental property of trigonometric functions that n 1 j=0 sin 2π r + s j/n = n 1 j=0 cos 2π r + s j/n = 0 if, an only if, r s mo n. With the stanar formulas on trigonometric functions of sums, this yiels n 1 j=0 sin 2πrj/n sin 2πsj/n = = n 1 j=0 n 1 j=0 sin 2πrj/n cos 2πsj/n cos 2πrj/n cos 2πsj/n = 0 if both r s mo n an r s mo n. Hence, for a esign satisfying (i) an (ii), the information matrix is block iagonal, an the relevant block associate with the parameters of interest equals n times the ientity. This block can be ientifie as the information matrix of a D-optimum esign for the submoel in which only the subsystem θ 0 θ 1 φ 1 θ p φ p of parameters is involve. Thus, by a simple refinement argument, the esign is also D-optimum for the subsystem in the full moel. The converse can be checke by noticing that for r = s mo n or r = s mo n the corresponing rows of the information matrix are linearly epenent an, hence, the parameters cannot be ientifie.

2318 RICCOMAGNO, SCHWABE AND WYNN The following theorem follows irectly from Lemma 1 by embeing the moel in the lattice an taking avantage of the one-imensional structure so obtaine. Theorem 2. In the M-factor interaction moel F m 1 m M, the n- point lattice esign generate by g 1 g is D-optimum for the parameters up to the S-factor interactions S M if an only if the members in the first S + 1 rows of the array 0 r k g k r k k k = 1 r k g k + r l g l r k k an r l l 1 k < l r k1 g k1 + + r km g km r k1 k1 r km km 1 k 1 < < k m r k1 g k1 + + r km g km r k1 k1 r km km 1 k 1 < < k M where k = m k 1 1 m k, are istinct (in the cyclic group n ) an if they are, aitionally ( for S < M), ifferent from those members in the last M S rows in n. The methoology of this paper consists of checking the entries in the various rows of the array in Theorem 2. For example, for the aitive moel F m 1 m 1, we check the first two rows. In some examples we shall seek to be able to estimate the main effects in the presence of possible interactions. For M = 2 it is necessary that the first two rows of the array are istinct an, in aition, the entries of the thir row are istinct from the previous two. Strictly speaking, this yiels D s -optimality in the sense of Kiefer an Wolfowitz (1959) for the subset of main effect parameters. There is no ifficulty in extening the orthogonality results in this way. We have obtaine the results in Section 3 as follows. First, we start from a generator an check whether all the members in the array of Theorem 2 are istinct in the set of all positive integers. If this is true, then they are also istinct in the cyclic group n with n = 2n max + 1, where n max is the largest member in the array. We call this the upper law. After that we look for smaller sizes n, maybe even minimal, such that the members remain istinct within the corresponing cyclic group n. For example, a closer investigation of the array being consiere for aitive moels shows that there are large gaps alreay at the beginning. To reuce the size, we can match the larger members of the array into suitable gaps left by the negative (in Z) members which are concentrate in the upper half. Because of the symmetry of the array with respect to zero, the corresponing negative members will automatically fit into gaps left by the positive (in Z) members in the lower half. Usually the largest gap will be between the maximum member

LATTICE-BASED D-OPTIMUM DESIGN 2319 n max an the next to maximum, say n 2, an in general n max > n 2 + 1. So a size n max + n 2 + 1 is suitable an gives a reuction of, at most, g. A similar rule hols for the other moels. We call this the generalize upper law. If we are intereste only in the main effects, we may procee as follows. If n 3 is the largest member in the first two rows of the array associate with the main effects an n max is the largest member in the whole array escribe in Theorem 2, the generalize upper law for main effects states that a size n max + n 3 + 1 is suitable. Obviously, any size larger than the one given by the upper law or the generalize upper law for main effects is suitable. Remark. We note that ue to invariance consierations [see Kiefer an Wolfowitz (1959) an Pukelsheim (1993), Chapter 13] the D-optimum esigns obtaine are also optimum with respect to any (convex) criterion base on the eigenvalues of the information matrix, incluing A- an E-optimality. This is true for the whole parameter vector as well as for the subsystems of the parameters up to the S-factor interactions (D s -optimality). Moreover, those esigns which are D-optimum for the whole parameter vector result in a minimum value for the integrate mean square error 1 0 1 0 Ŷ x 1 x E Y x 1 x 2 x x 1 where Ŷ is the best linear unbiase preictor. 2.3. An algorithm for the one-step strategy. For every Fourier moel the entries g 1 g of a generator g 1 g have to be ifferent. Without loss of generality, we can look for generators with 1 = g 1 < < g. We present an algorithm for an iterative sequence of generator components obtaine from Theorem 2 by a one-step strategy. A global search for the minimum g is possible, but we shall see that computer results for this simple strategy are in some agreement with theoretical proceures. Also, this algorithm gives a reasonable boun on the increase of the generator components an, hence, of the require size by the upper law as the imension increases. For the generator g 1 g enote by M the set of all members in the array associate with a -factor moel with M-factor interactions. Further assume that we are intereste in all the parameters up to the S-factor interactions, 1 S M, an that the generator g 1 g is suitable. That is to say, the members in the first S + 1 rows of the array (incluing the first row containing only 0) are all istinct, an the members in the remaining last M S rows are istinct from those in the first S + 1 rows, but not necessarily ifferent from each other (within the set Z of all integers). Then S is the set that collects the members of the first S + 1 rows of the array an, in particular, 0 = 0. The following algorithm prouces a sequence of generator components Step 1. Start with = 1 an g 1 = 1. Then g 1 is suitable an S 1 = m 1 1 0 1 m 1, 1 S M.

2320 RICCOMAGNO, SCHWABE AND WYNN Step 2. If the generator g 1 g is suitable, then etermine the set + M = + S 1 containing all sums of members of M an of its first S rows, respectively. Then a all integers for which any multiple up to orer 2m +1 is inclue in + ++, that is, = 2m +1 r=1 1/r + Z. Step 3. Let g +1 be a positive integer not inclue in ++. Then the generator g 1 g g +1 is suitable an S S +1 = r +1 rg +1 + S 1 for S = 1 M. Step 4. Set = + 1 an go to Step 2. For S = 1 we are only intereste in the main effects. For S = 2 we are intereste in the main effects an, aitionally, all two-factor interactions. Finally, for S = M 1 we are intereste in all but the M-factor interactions an for S = M we are intereste in all parameters. In the particular situation of equal orer of Fourier regression in all factors, the constructions in the algorithm guarantee that g = O M+S 1, as, because the carinality of S increases like O S an hence the carinality of ++ increases, at most, as O M+S 1. Consequently, if g +1 is chosen as the smallest positive integer not belonging to ++, the iterative sequence of generator components an the corresponing sizes increase polynomially. This is to be compare to exponential growth for generators of the form 1 a a 2 a 1 (power generators) an generators efine by linear recursions (Fibonacci-type generators). Note that the total number of parameters increases polynomially like M an the number of parameters of interest increases like S. 3. Examples an constructions. The esign generate by the power generator ( 1 ) g 1 g = 1 2m 1 + 1 2m 1 + 1 2m 2 + 1 2m k + 1 with k=1 2m k + 1 supporting points is optimum for all M-factor interaction moels F m 1 m M, 1 M. However, the number of esign points is, in general, much larger than the number of parameters, only being the same in the case M =. 3.1. Aitive moel. For aitive moels we have only to check the first two rows of the array given in Theorem 2. 3.1.1. Power-type generators. In the aitive Fourier moel F m m 1 with equal orers of the Fourier regressions in each component, the upper law gives a sample size of n = 2m 2m + 1 1 + 1 for the preceing power generator which increases exponentially in the imension while the number of parameters is linear in. While noticing that there is an unnecessary gap in the array between m an 2m + 1, we obtain the power generator k=1 g 1 g = 1 m + 1 m + 1 2 m + 1 1

LATTICE-BASED D-OPTIMUM DESIGN 2321 with minimum base m + 1 for which the upper law gives an exponentially increasing size n = 2m m + 1 1 + 1. More generally, for F m 1 m 1 we can use the generalize power generator g 1 g = ( 1 ) 1 m 1 + 1 m 1 + 1 m 2 + 1 m k + 1 with size n = 2m 1 k=1 m k + 1 + 1. 3.1.2. Recursively efine generators. From Theorem 2 we see that a 1 -imensional generator g 1 g 1 is suitable if an only if 0 an all r k g k, r k k, k = 1 1, are istinct in Z. If we choose the next component g greater than the largest member of the preceing set, then the resulting generator g 1 g prouces an array in which again all members are istinct. At each step the largest member will be m g. As it is reasonable to choose g as small as possible in orer to reuce the gaps, we obtain the following linear recursion formula: g = m 1 g 1 + 1 with an initial value g 1 = 1 as usual. Accoring to the upper law, the corresponing size is n = 2m g + 1. In particular, for the aitive moels F m m 1 with equal orers in the components, we get g = m 1 / m 1 an a size n = 2m +1 m 1 / m 1 in case m 2. This size is smaller than the size of the power generator esigns, but it is still exponentially increasing. For m = 1 the generator sequence simplifies to g = which increases linearly an the corresponing esign has minimal support (n = 2 + 1). 3.1.3. One-step generators an linear generators. For the moel F 1 1 1 the one-step strategy obviously prouces the same sequence g = of generator components as the recursive scheme given previously which is optimum in the sense that a minimal size n can be achieve which is equal to the number of parameters. For the moel F 2 2 1 the algorithm of Section 2.3 prouces the generator sequence 1 3 4 5 7 9 11 12 13 15 16 17 19 20 21 23 25 27 28 29 with corresponing minimal sizes 10 13 17 23 29 34 37 41 47 49 53 59 61 65 71 77 83 87 89 for 2 which was foun by exhaustive search. We notice that all o numbers are inclue in the sequence of generator components. Hence, g 2 1 an the corresponing size increases linearly in the imension. The last observation suggests a simple alternative sequence of generator components g = 2 1 k=1

2322 RICCOMAGNO, SCHWABE AND WYNN It can be prove that these linearly generate esigns are suitable an that the minimum obtainable size is n = 4 + 2. Note that this size is, in fact, smaller than the size obtaine for the one-step strategy if = 5 6 7 The construction of such linear generators can be extene irectly to the aitive moel F m m 1 of general orer m by g = 1 m + 1 an the size n = 2 1 m 2 + 2m + 1 obtaine by the upper law increases linearly in. For the aitive moel F 3 3 1 with thir-orer Fourier regression, the sequence g = 3 2 prouces minimum obtainable sizes of n = 17 for = 2 an n = 9 for 3. In the general case of unequal orers m k, we use the generating proceure g k = k 1 m + 1 where m = max k m k. The moel is a submoel of F m m 1. Hence, the entries in the corresponing array are istinct an the generate esign is D-optimum if n is sufficiently large (n 2 max k m k g k + 1). 3.2. Moels with two-factor interactions. For the complete two-factor interaction moel F m 1 m 2, we have to check the first three lines of the array given in Theorem 2. The power generator introuce in the beginning of Section 3 prouces a sample size which is slightly smaller than the size for a prouct-type esign. In the following we consier moels F m m 2 with equal orers in the components an we will be intereste in either the main effects (S = 1) or the whole parameter vector (S = 2). We compare the performances of the minimum power generator, the recursively efine generators an the onestep generator. 3.2.1. The moel F 1 1 2. First, we stuy inference on the parameters associate with the main effects. The minimum power generator is 1 3 3 1 an the size increases exponentially with base 3. An iterative sequence of generator components is given by the recursion formula g = 2g 1 + 1. Hence, g = 2 1 an the size is increasing exponentially with base 2. The one-step algorithm prouces a linear generator with components g = 2 1 an a linearly increasing size n = 6 4. For inference on the whole parameter vector, we observe again that the minimum power generator prouces an exponentially increasing size with base 3. An iterative sequence of generator components is efine by the Fibonaccitype recursion g = 2g 1 + g 2 + 1 with initial values g 1 = 1, g 2 = 3. Hence, g = 1+ 2 +1 + 1 2 +1 2 /4 by stanar methos for nonhomogeneous linear recursions. This gives an approximate size n 1 + 2 +1 / 2 by the upper law. Finally, the algorithm of Section 2.3 prouces a one-step sequence of generator components which starts as follows: 1 3 8 18 30 43 67 90 122 161

LATTICE-BASED D-OPTIMUM DESIGN 2323 an the corresponing sample sizes are 9 23 53 97 147 221 315 425 567 (for 2) obtaine by the upper law. 3.2.2. The moel F 2 2 2. Next, we consier the complete twofactor interaction moel F 2 2 2 with marginal Fourier regressions of orer 2, for which n max = 2g + 2g 1 an n 3 = 2g. We start by making inference on the parameters associate with the main effects (incluing the constant term θ 0 ). This is the D s -optimality mentione in Section 2.2. The minimum power generator is given by g 1 g = 1 5 5 2 5 1 with a size increasing exponentially with base 5. A recursively efine, Fibonacci-type sequence of generator components is given by g = 2g 1 + 2g 2 + 1 with initial values g 1 = 1, g 2 = 5. This has solution g = 2 1 + 3 + 1 3 1 /3 an the sample size given by the generalize upper law for main 2 effects equals approximately n 2 1+ 3 +1 / 3 an increases exponentially in the imension. The one-step iterative sequence of generator components starts as follows: (5) 1 5 13 17 37 41 49 53 109 113 121 125 For this sequence the iterative generating law was foun g = g 1 + 4α 1 with g 1 = 1 an with α 1 = 1 an α 2 j = 3α 2 j 1 1 α k = α 2 j for k = 2 j q q o The next terms in the sequence are 145 149 157 161 325 329 given by both the one-step algorithm an the iterative sequence. The ifference sequence α k, k = 1 2 gives 1 2 1 5 1 2 1 14 1 2 1 5 1 2 1 41 1 2 1 5 1 2 1 14 1 2 1 5 1 2 1 122 1 2 1 5 1 2 1 14 1 2 1 5 1 2 1 41 1 2 1 5 1 2 1 14 1 2 1 5 1 2 1 365 which shows a nice self-similar structure. It can be prove that the iterative sequence prouces suitable generators. Stanar methos yiel the solution α 2 j = 1 2 3j + 1 an g 2 j +1 = 4 3 j + 1. This, finally, implies that g increases

2324 RICCOMAGNO, SCHWABE AND WYNN like γ where γ = log 3 / log 2 1 5850. The number γ 1 = log 2 / log 3 is the Hausorff imension of the Cantor set an the reason for this unexpecte connection can be seen by stuying the structure of the sequence. For inference on the whole parameter vector, the minimum power generator prouces a size which increases exponentially with base 5. Alternatively, we present the following Fibonacci-type sequence of generator components g = 4g 1 + 2g 2 + 1 with initial values g 1 = 1, g 2 = 5. Hence, the generator sequence is g = 3+2 6 2+ 6 + 3 2 6 2 6 6 /30 an the size is approximately equal to 4 + 6 2 + 6 /5 by the upper law. We finish this section with the one-step sequence of generator components (6) 1 5 23 60 77 173 222 409 535 634 935 1182 1361 1497 an the corresponing sizes 25 113 333 549 1001 1581 2525 3777 4677 6277 8469 10173 11533 for 2 obtaine by the upper law. 3.2.3. The general moel F m m 2. Finally, we investigate the two-factor interaction moel F m m 2 with marginal Fourier moels of orer m, for which n max = m g + g 1 an n 3 = mg. Again, we start by making inferences on the main effects. The minimum power generator given by 1 2m+1 2m+1 1 prouces a size which is exponentially increasing with base 2m + 1. The following iterative, Fibonaccitype sequence: g = m g 1 + g 2 + 1 with initial values g 1 = 1, g 2 = 2m + 1 gives g ( m 1 m 2 )( m + m2 + 4m 2m 1 m2 + 4m 2 an by the generalize upper law for main effects the sizes ( n m2 3 + 5m + 2 )( m + m2 + 4m 2m 1 m2 + 4m 2 ) ) 1 increase exponentially in the imension. In generalization of the case m = 2, we obtain another iterative sequence of generator components which is increasing at most polynomially: with initial value g 1 = 1 where g = g 1 + 2mα 1 α 2 j = m + 1 α 2 j 1 1

LATTICE-BASED D-OPTIMUM DESIGN 2325 with initial value α 1 = 1 an α k = α 2 j for k = 2 j q q o The ifference sequence again shows a self-similar structure. As before, we obtain α 2 j = m 1 m + 1 j + 1 m an g 2 j = 2m m + 1 j + 1 This implies that g increases like γ where γ = log m + 1 / log 2. However, for m > 2 the iterative sequence iffers from the sequence generate by the one-step algorithm. Turning to inference on the whole parameter vector, for the minimum power generator we obtain the size 2m g + g 1 + 1 = 4m m + 1 2m + 1 2 + 1 by the upper law. An iterative sequence of generator components is g = m 2g 1 + g 2 + 1 with initial values g 1 = 1, g 2 = 2m + 1. Hence, g with the approximate size n ( 1 m 3m 1 m + 1 + 1 ) ( ) m + m m + 1 2 ( m 2 + 3m 1 m + 1 m ) ( m + m m + 1 ) We conclue our presentation with some figures representing the behavior of the one-step generator sequence for the two-factor interaction moels, both for inference on the main effects an on the whole parameter vector. For the main effects in F 2 2 2, Figure 1 shows the logarithm of α, the first ifferences of the generators ivie by 4, over the imension for the sequence (5) up to = 180. The self-similar structure mentione previously is again evient. Figure 2 shows the logarithm of the generator of the same sequence over log an gives an estimate slope of 1 7. For inference on the whole parameter vector in F 2 2 2, Figure 3 shows the logarithm log g of the generators for the one-step sequence 6 over the logarithm log of the imension giving an estimate slope of γ 2 72. We have been unable to fin an iterative sequence which yiels this sequence or to link it to a Cantor-like set. The two log log plots inicate polynomial growth γ of the size n in with γ approximately equal to the estimate slopes. Note that the estimate 1.7 for the analysis of the main effects slightly overestimates γ = log 3 / log 2 1 5850.

2326 RICCOMAGNO, SCHWABE AND WYNN Fig. 1. Main effect of the moel F 2 2 2 log α over. Fig. 2. Main effect of the moel F 2 2 2 log g over log.

LATTICE-BASED D-OPTIMUM DESIGN 2327 Fig. 3. All effects of the moel F 2 2 2 log g over log. REFERENCES Bates, R. A., Buck, R. J., Riccomagno, E. an Wynn, H. P. (1996). Experimental esign an observation for large systems. J. Roy. Statist. Soc. Ser. B 58 77 94. Brown, L., Olkin, I., Sacks, J. an Wynn, H., es. (1985). Jack Carl Kiefer Collecte Papers. Springer, New York. Hoel, P. G. (1965). Minimax esigns in two imensional regression. Ann. Math. Statist. 36 1097 1106. Kiefer, J. an Wolfowitz, J. (1959). Optimum esigns in regression problems. Ann. Math. Statist. 30 271 294. Nieerreiter, H. (1992). Ranom Number Generation an Quasi-Monte Carlo Methos. SIAM, Philaelphia. Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. Rafajłowicz, E. an Myszka, W. (1992). When prouct type experimental esign is optimal; brief survey an new results. Metrika 39 321 333. Schwabe, R. (1996a). Optimum Designs for Multi-Factor Moels. Lecture Notes in Statist. 113. Springer, New York. Schwabe, R. (1996b). Moel robust experimental esign in the presence of interactions: the orthogonal case. Tatra Mt. Math. Publ. 7 97 104. Department of Statistics University of Warwick Coventry CV4 7AL Unite Kingom E-mail: emr@stats.warwick.ac.uk Fachbereich Mathematik Technische Hochschule Darmstat Darmstat 64 289 Germany E-mail: schwabe@mathematik.th-armstat.e Department of Statistics University of Warwick Coventry CV4 7AL Unite Kingom E-mail: hpw@stats.warwick.ac.uk