Journal of Statistical Planning and Inference 141 (2011) 1519–1525
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Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Orthogonal designs for computer experiments S.D. Georgiou Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
a r t i c l e in f o
abstract
Article history: Received 12 March 2010 Received in revised form 30 September 2010 Accepted 4 November 2010 Available online 20 November 2010
Computer experiments are becoming increasingly popular surrogates for physical experiments in recent years. In this paper, we propose a general procedure for constructing orthogonal designs with many levels and distinct runs. These designs are suitable for computer experiments. The basic idea of the method is to fold-over suitable orthogonal matrices. The properties of the derived designs are studied and a brief comparison with the designs in the literature is given. & 2010 Elsevier B.V. All rights reserved.
Keywords: Computer experiment Autocorrelation function Integer sequences Orthogonal designs Construction
1. Introduction Computer experiments have become very attractive in the resent years (see for example Fang et al., 2006 or Sacks et al., 1989). A large class of designs for computer experiments includes the well known and commonly used Latin hypercube designs. These designs have n uniformly spaced levels on n runs. Latin hypercube designs are widely studied. Some resent relevant work includes McKay et al. (1979), Tang (1993, 1998), Owen (1994), Ye (1998), Butler (2001), Steinberg and Lin (2006), Bingham et al. (2009), Georgiou (2009), Lin et al. (2009), and Pang et al. (2009). In their recent paper Bingham et al. (2009), relaxed the condition that the number of levels for each factor be identical to the run size and they constructed an extended and useful class of orthogonal and nearly orthogonal designs for computer experiments. Also, Lin et al. (2010) proposed a method for constructing new designs for computer experiments by using the Kronecker product on small designs with specific properties. Under specified circumstances the designs generated in Lin et al. (2010) are orthogonal designs for computer experiments. In this paper, we propose a general method for constructing orthogonal designs suitable for computer experiments. In particular, in Section 2 we recall some known results from the literature that are necessary in the sequence. In Section 3, we present the proposed method. Following Bingham et al. (2009), we relax the condition that the number of levels for each factor should be identical to the run size and we present a construction method that can be used for obtaining orthogonal designs for computer experiments. These designs include n distinct experimental runs and m columns on s uniformly spaced levels (abbreviated as ODðn,sm Þ) without the restrictions on the run size which were needed in Ye (1998) or Steinberg and Lin (2006). 2. Preliminary results An experimental design Dðn,sm Þ with n runs, m factors and s levels will be denoted by an n m matrix X ¼ ½x1 , . . . ,xm , where xj is the jth factor (column vector) and dij is the level of factor j on the ith experimental run. A design T is said to be full D fold-over (or fold-over) of the design D if T ¼ ðD Þ. The levels of a design X are selected to be centered, equally spaced and for E-mail address:
[email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.11.014
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simplicity integer-valued. This class of designs includes the well known and commonly used family of Latin hypercube designs where in this case s is equal to n. The key question, as in constructing a Latin hypercube design, is how to mate the levels for the different factors. There are several variations on how to space the levels ‘uniformly’ for each factor. The simplest scheme, and the one that we will employ in this paper, is to take the levels to be ( (s 1)/2, . . ., 1, 0, 1, . . ., (s 1)/2) when s is odd and ( s/2, . . ., 1, 1, . . ., s=2Þ when s is even. All levels (except zero; if exist) should be equally replicated in each column so that the design will be mean orthogonal. In regression analysis, it is desirable to include orthogonal independent variables in a regression model, so that the estimates of the factors and interaction coefficients would be uncorrelated. Usually, a polynomial model, of degree k with m factors, is fitted. This model is of the form X X X Y ¼ b0 þ bi x i þ bi1 i2 xi1 xi2 þ þ bi1 ik xi1 . . . xik þ e, ð1Þ irm
i1 r i2 r m
i1 r r ik r m
where xi are the independent variables, bi are the linear effects of xi, bi1 ...it is the effect of the t-order interaction of xi1 . . . xit . Obviously bii corresponds to the quadratic effect of factor xi while bi1 i2 , for i1 ai2 , is the second order interaction of factors xi1 xi2 . Following Steinberg and Lin (2006) we recall some evaluation criteria defined by calculating the alias matrices for fitting a first order model when second order effects may be present. Suppose that X is a design with n runs, m factors and s levels. Let X1 be the regression matrix for the first order model, including a column of ones and the m columns of X. Let Xint be the n m(m 1)/2 matrix with all the possible two-factor interactions and Xquad be the n m matrix with all the pure quadratic terms. The alias matrices for the first order model associated with the two-factor interactions and the pure quadratic terms are given by T ¼ Aint ¼ ðX1T X1 Þ1 X1T Xint
ð2Þ
Q ¼ Aquad ¼ ðX1T X1 Þ1 X1T Xquad ,
ð3Þ
and
respectively. Designs that are suitable for screening are expected to have relatively small absolute values in these bias matrices. Circulant matrices will be extensively used for the proposed construction. A matrix is said to be circulant if each row vector is rotated one element to the right relative to the preceding row vector. A circulant matrix A ¼ circðBÞ is fully specified by one vector B, which appears as the first row of the matrix. The remaining rows of A are each cyclic permutations of the vector B with offset equal to the row index. The last column of A is the vector B in reverse order, and the remaining columns are each cyclic permutations of the last column. Note that different sources define the circulant matrix in different ways, for example with the coefficients corresponding to the first column rather than the first row of the matrix. Circulant matrices with a different direction of shift are called back-circulant. If no confusion is caused the circulant matrix and the corresponding generator row vector will be abbreviated by the same letter, i.e. A ¼ circðAÞ. The elements of the information matrix of a circulant matrix can be calculated by the periodic autocorrelation function of its first row. Let A ¼ fAj : Aj ¼ ðaj,0 ,aj,1 , . . . ,aj,n1 Þ, j ¼ 1, . . . ,‘g be a set of ‘ row vectors of length n. The periodic autocorrelation function PA(s) (abbreviated as PAF) is defined, reducing i+s modulo n, as PA ðsÞ ¼
‘ X n1 X
aj,i aj,i þ s ,
s ¼ 0,1, . . . ,n1,
ð4Þ
j¼1i¼0
while the non-periodic autocorrelation function NA(s) (abbreviated as NPAF) is defined as NA ðsÞ ¼
‘ ns1 X X
aj,i aj,i þ s ,
s ¼ 0,1, . . . ,n1:
ð5Þ
j¼1 i¼0
The set of row vectors A is said to have zero PAF if PA ðsÞ ¼ 0,8s ¼ 1,2, . . . ,n1 and zero NPAF if NA ðsÞ ¼ 0,8s ¼ 1,2, . . . ,n1. For the results of this paper generally zero PAF is sufficient. However zero NPAF vectors imply zero PAF vectors exist and employ some useful multiplication techniques. Hence zero NPAF can give more general results. An example of vectors with zero NPAF are the famous Golay sequences. Golay sequences is the set A ¼ fA1 ,A2 g of two row vectors, of length n, with elements from the set f1,1g, that have zero NPAF. These sequences are known to exist for lengths 2a 10b 26c , where a, b, c are non-negative integers. More details on Golay sequences can be found in Borwein and Ferguson(2003). Throughout the paper Rk denotes the back diagonal identity matrix of order k and circðBÞ is the circulant matrix with first row B. The following theorem is very useful in the construction of orthogonal matrices. Theorem 1 (Geramita and Seberry, 1979, Theorem 4.49). Suppose there exist four circulant matrices A, B, C, D of order n satisfying AAT þBBT þCC T þ DDT ¼ fIn :
S.D. Georgiou / Journal of Statistical Planning and Inference 141 (2011) 1519–1525
Then the Goethals–Seidel array 0 CRn A BRn B BRn A DT Rn B GS ¼ B @ CRn DT Rn A DRn
C T Rn
BT Rn
DRn
1521
1
C Rn C C C BT Rn A T
A
is an orthogonal matrix of order 4n. The circulant matrices needed in Theorem 1 can be easily constructed by using the following corollary. Corollary 1. If there are four row vectors A, B, C, D of length n with zero periodic autocorrelation function, then these vectors can be used as the first rows of circulant matrices which can be used in Theorem 1 to form an orthogonal matrix of order 4n. Following Kharaghani (2000) a set fA1 ,A2 , . . . ,A2k g of square real matrices is said to be amicable if k X
ðA2i1 AT2i A2i AT2i1 Þ ¼ 0:
ð6Þ
i¼1
We need the following array from Kharaghani (2000). Theorem 2 (Kharaghani, 2000). Let fAi g8i ¼ 1 be an amicable set of circulant matrices of order n, satisfying the Kharaghani array 0 1 A2 A4 Rn A3 Rn A6 Rn A5 Rn A8 Rn A7 Rn A1 B A A1 A3 Rn A4 Rn A5 Rn A6 Rn A7 Rn A8 Rn C B C 2 B C T T T T B A4 Rn A3 Rn C A A A R A R A R A R n n n n 1 2 7 5 8 6 B C B A R T T T T A4 Rn A2 A1 A7 Rn A8 Rn A5 Rn A6 Rn C B C 3 n B C H¼B AT8 Rn AT7 Rn A1 A2 AT4 Rn AT3 Rn C B A6 Rn A5 Rn C B C A6 Rn AT7 Rn AT8 Rn A2 A1 AT3 Rn AT4 Rn C B A5 Rn B C T T T B A8 Rn A7 Rn AT Rn A5 Rn A4 Rn A3 Rn A1 A2 C 6 @ A A7 Rn A8 Rn AT5 Rn AT6 Rn AT3 Rn AT4 Rn A2 A1
P8
i¼1
Ai ATi ¼ fIn . Then
is an orthogonal matrix of order 8n. In the next section we develop some methods for constructing designs suitable for computer experiments. 3. The proposed methods In this section, the use of circulant matrices with special properties is suggested for constructing designs for computer experiments. In the following theorem we describe a general construction method for orthogonal designs by using a square orthogonal matrix D and its full fold-over design. Theorem 3. Let D be an orthogonal matrix of order n. If each of its columns has s levels, equally replicated, from the set (i) f 71, . . . , 7 ð2s3Þ, 7 ð2s1Þg, then there exists an orthogonal design ODð2n,ð2sÞn Þ with 2n runs, n factors and 2s levels. (ii) f 71, . . . , 7 ðs1Þ, 7 sg, then there exists an orthogonal design ODð2n þ 1,ð2s þ 1Þn Þ with 2n +1 runs, n factors and 2s +1 levels. Proof. Each column of the orthogonal matrix D has s levels, equally replicated (say n/s times each), and so P DT D ¼ ðn=sÞ si ¼ 1 ð2s2iþ 1Þ2 In . D Þ. It is obvious from the definition of X that each column of X consists of the 2s equally spaced and equally (i) Set X ¼ ðD replicated levels of the set f2s þ 1,2s þ 3, . . . ,3,1,1,3, . . . ,2s3,2s1g. Moreover, s X D X T X ¼ ðDT DT Þ ð2s2i þ 1Þ2 In : ¼ DT D þ DT D ¼ 2ðn=sÞ D i¼1
Thus, X is the desirable orthogonal design with 2n runs, n factors and 2s levels. (ii) Set 0 1 D B C X ¼ @ 0n A: D The rest of the proof is similar to the proof of Theorem 3(i).
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Corollary 2. Let D be an orthogonal matrix of order 4n constructed from four circulant matrices as in Theorem 1. Moreover, if each of its columns has s levels, equally replicated, from the set f 7 1, 7 3, . . . , 7 ð2s3Þ, 7ð2s1Þg, then there exists an orthogonal design ODð8n,ð2sÞ4n Þ with 8n runs, 4n factors and 2s levels.
Proof. The proof is straightforward from Theorem 3(i).
&
Corollary 3. Let D be an orthogonal matrix of order 4n constructed from four circulant matrices as in Theorem 1. Moreover, if each of its columns has s levels, equally replicated, from the set f 7 1, 7 2, . . . , 7 ðs1Þ, 7sg, then there exists an orthogonal design ODð8n þ 1,ð2s þ 1Þ4n Þ with 8n +1 runs, 4n factors and 2s+ 1 levels (including zero). Proof. The proof is straightforward from Theorem 3(ii).
&
Corollary 4. Let D be an orthogonal matrix of order 8n constructed from eight circulant matrices as in Theorem 2. Moreover, if each of its columns has s levels, equally replicated, from the set f 7 1, 7 3, . . . , 7 ð2s3Þ, 7ð2s1Þg, then there exists an orthogonal design ODð16n,ð2sÞ8n Þ with 16n runs, 8n factors and 2s levels. Proof. The proof is straightforward from Theorem 3(i).
&
Corollary 5. Let D be an orthogonal matrix of order 8n constructed from four circulant matrices as in Theorem 2. Moreover, if each of its columns has s levels, equally replicated, from the set f 7 1, 7 2, . . . , 7 ðs1Þ, 7sg, then there exists an orthogonal design ODð16n þ 1,ð2s þ1Þ8n Þ with 16n+ 1 runs, 8n factors and 2s +1 levels (including zero). Proof. The proof is straightforward from Theorem 3(ii). a
b
c
&
d
Theorem 4. Let t ¼ 2 10 26 34 , where a, b, c are non-negative integers, d= 0 or 1, and set ‘ ¼ 2t þ1. Then there exist the following orthogonal designs, suitable for computer experiments: (i) ODð8‘,84‘ Þ. (ii) ODð8‘ þ 1,94‘ Þ. Proof. Set g ¼ 2a 10b 26c and t ¼ 34d g. Let a ¼ ða11 ,a12 , . . . ,a1g Þ and b ¼ ða21 ,a22 , . . . ,a2g Þ be two Golay sequences of length g. If d = 1 then define the row vectors A1 ¼ ða, a, a, a , a , a , b , a , b , b, b , b, a, b , b , b, b, a, b, b , a, b, b , b, b, a, a, a , b, b , a, b, b, aÞ, A2 ¼ ðb, a , a , b, a, a , b , b, b, a , a , a, a , b, a, a , b, a , a , a, a, b, a , a, a , a, b , a, b , b , b , b, b, bÞ, where x stands for x, else, if d =0, define A1 ¼ a and A2 ¼ b. Thus A1 and A2 are the desirable vectors of length t with zero PAF. (i) Set 2t
t
tþ1
zfflfflfflffl}|fflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ A ¼ 1ð1,0, . . . ,0 Þ þ 3ð0,A1 ,0, . . . ,0 Þ þ5ð0, . . . ,0 ,A2 Þ, B ¼ 5ð1,0, . . . ,0Þ7ð0,A1 ,0, . . . ,0Þ þ1ð0, . . . ,0,A2 Þ, C ¼ 3ð1,0, . . . ,0Þ þ1ð0,A1 ,0, . . . ,0Þ þ7ð0, . . . ,0,A2 Þ, D ¼ 7ð1,0, . . . ,0Þ þ 5ð0,A1 ,0, . . . ,0Þ3ð0, . . . ,0,A2 Þ: These are four row vectors of length 2t +1 with elements from the set f 71, 73, 75, 77g. With routine calculations we can prove that the above vectors have zero PAF and can be used in Corollary 2 to give an ODð8‘,84‘ Þ. (ii) Set 2t
t
tþ1
zfflfflfflffl}|fflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ A ¼ 1ð1,0, . . . ,0 Þ þ 2ð0,A1 ,0, . . . ,0 Þ þ3ð0, . . . ,0 ,A2 Þ, B ¼ 3ð1,0, . . . ,0Þ4ð0,A1 ,0, . . . ,0Þ þ1ð0, . . . ,0,A2 Þ, C ¼ 2ð1,0, . . . ,0Þ þ1ð0,A1 ,0, . . . ,0Þ þ4ð0, . . . ,0,A2 Þ, D ¼ 4ð1,0, . . . ,0Þ þ 3ð0,A1 ,0, . . . ,0Þ2ð0, . . . ,0,A2 Þ: These are four row vectors of length 2t +1 with elements from the set f 71, 72, 73, 74g. With routine calculations we can prove that the above vectors have zero PAF and can be used in Corollary 3 to give an ODð8‘ þ1,94‘ Þ. &
S.D. Georgiou / Journal of Statistical Planning and Inference 141 (2011) 1519–1525
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Remark 1. From Theorem 4 we obtain the following orthogonal designs for computer experiments: ODðn,8n=2 Þ and ODðn þ1,9n=2 Þ, where t ¼ 2a 10b 26c 34d , a, b, c are non-negative integers, d = 0 or 1, and n ¼ 8‘ ¼ 8ð2t þ 1Þ. The first 20 values of n are n = 24, 40, 72, 136, 168, 264, 328, 424, 520, 552, 648, 840, 1032, 1096, 1288, 1608, 1672, 2056, 2184, 2568. Example 1. Let A1 ¼ ð1,1,1,1Þ and A2 ¼ ð1,1,1,1Þ be two Golay sequences of length t = 4. (i) Set A ¼ 1ð1,0,0,0,0,0,0,0,0Þ þ 3ð0,1,1,1,1,0,0,0,0Þ þ5ð0,0,0,0,0,1,1,1,1Þ ¼ ð1,3,3,3,3,5,5,5,5Þ, B ¼ 5ð1,0,0,0,0,0,0,0,0Þ7ð0,1,1,1,1,0,0,0,0Þ þ 1ð0,0,0,0,0,1,1,1,1Þ ¼ ð5,7,7,7,7,1,1,1,1Þ, C ¼ 3ð1,0,0,0,0,0,0,0,0Þ þ 1ð0,1,1,1,1,0,0,0,0Þ þ7ð0,0,0,0,0,1,1,1,1Þ ¼ ð3,1,1,1,1,7,7,7,7Þ, D ¼ 7ð1,0,0,0,0,0,0,0,0Þ þ 5ð0,1,1,1,1,0,0,0,0Þ3ð0,0,0,0,0,1,1,1,1Þ ¼ ð7,5,5,5,5,3,3,3,3Þ: These are four row vectors of length 9 with elements from the set f 71, 73, 75, 77g. With simple calculations we can show that the above vectors have zero NPAF (thus zero PAF) and can be used in Corollary 2 to give an ODð72,836 Þ. (ii) Set A ¼ 1ð1,0,0,0,0,0,0,0,0Þ þ 2ð0,1,1,1,1,0,0,0,0Þ þ3ð0,0,0,0,0,1,1,1,1Þ ¼ ð1,2,2,2,2,3,3,3,3Þ, B ¼ 3ð1,0,0,0,0,0,0,0,0Þ4ð0,1,1,1,1,0,0,0,0Þ þ 1ð0,0,0,0,0,1,1,1,1Þ ¼ ð3,4,4,4,4,1,1,1,1Þ, C ¼ 2ð1,0,0,0,0,0,0,0,0Þ þ 1ð0,1,1,1,1,0,0,0,0Þ þ4ð0,0,0,0,0,1,1,1,1Þ ¼ ð2,1,1,1,1,4,4,4,4Þ, D ¼ 4ð1,0,0,0,0,0,0,0,0Þ þ 3ð0,1,1,1,1,0,0,0,0Þ2ð0,0,0,0,0,1,1,1,1Þ ¼ ð4,3,3,3,3,2,2,2,2Þ: These are four row vectors of length 9 with elements from the set f 71, 72, 73, 74g. With simple calculations we can show that the above vectors have zero NPAF (thus zero PAF) and can be used in Corollary 3 to give an ODð73,936 Þ.
4. Properties of the constructed designs The following two theorems that were proved in Georgiou (2009) are also valid for the designs constructed in this paper. Theorem 5. Let X ¼ ðx1 , . . . ,xm Þ be an orthogonal design with 2n runs, m factors and 2s levels constructed by the proposed fold-over method. Then, any quadratic effect of a factor is orthogonal to all the main effects in the constructed orthogonal design. Theorem 6. Let X ¼ ðx1 , . . . ,xm Þ be an orthogonal design with 2n runs, m factors and 2s levels constructed by the proposed fold-over method. Then, any two-factor interaction is orthogonal to all the main effects in the constructed orthogonal design. So, the orthogonal designs generated by the presented fold-over method have all their main effects orthogonal to any quadratic effect and to any two-factor interaction. Remark 2. Theorems 5 and 6 are valid for all the designs constructed by Theorems 3 and 4 or by Corollaries 2–5. The results presented in this section indicate that the estimates of quadratic effects and two-factor interactions are uncorrelated with the estimates of the linear effects but this does not forbid the quadratic effects and the two-factor interactions to be correlated to each other. 5. Orthogonal designs with n runs, m factors and s levels Table 1 gives the parameters of some orthogonal designs that can be used for computer experiments. These designs are constructed by the methods of this paper. In the first column (Design) we show the parameters of constructed orthogonal factors design in the form ODðruns,levels Þ. In the last two columns we describe how the design is constructed by presenting the theorem we used and the needed vectors. In the second column (Ref) we indicate whether the design is new or how a design with the same parameters can be constructed by a method in the literature. By writing 1 in the ‘Ref’ column, we mean that an orthogonal design with same parameters can be constructed by the multiplication method of Bingham et al. (2009). Thirteen new orthogonal designs are constructed. Infinitely many new orthogonal designs can be obtained from Theorem 4 or by computer search, but for practical reasons only a small number of them is presented in Table 1. 6. Discussion In this paper we proposed some new methods for constructing orthogonal designs for computer experiments. In the main method of the paper we suggest the use of an orthogonal square matrix and its full fold-over design for obtaining the desirable result. A practical and commonly used method for building orthogonal matrices is by using four or eight circulant matrices
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Table 1 Some orthogonal designs ODðn,sm Þ, for computer experiments, constructed by the proposed methods. Design
Ref.
Method
Needed vectors
OD(16,8 )
1
Corollary 4
A1 = (1), A2 = (3), A3 = (5), A4 = (7) A5 = (1), A6 = (3), A7 = (5), A8 = (7)
ODð17,98 Þ
1
Corollary 5
8
A1 = (1), A2 = (2), A3 = (3), A4 = (4) A5 = (1), A6 = (2), A7 = (3), A8 = (4)
ODð24,812 Þ ODð25,912 Þ ODð24,1212 Þ
New
New
New
Theorem 4,
A = (1,3,5), B =( 5, 7,1),
Corollary 2
C =( 3,1,7), D= ( 7,5, 3)
Theorem 4,
A = (1,2,3), B =( 3, 4,1),
Corollary 3
C =( 2,1,4), D= ( 4,3, 2)
Corollary 2
A = (1, 3,5), B= (3,5, 7), C =(7, 9, 9), D = (11,11, 1)
ODð25,1312 Þ
New
Corollary 3
A = (3,6,3), B =(4, 4, 5), C =(5, 6, 1), D = (1, 2, 2)
ODð32,1616 Þ
1
Corollary 2
1
Corollary 3
A = (13,13,15, 15), B= (1,1, 3,3), C =(5, 5, 7, 7), D = (9, 9, 11, 11)
16
ODð33,17 Þ
A = (3, 3, 4, 4), B= (5, 5, 6, 6), C =(7,7, 8,8), D = (1,1, 2,2)
ODð32,1616 Þ
1
Corollary 4
A1 = (1,3), A2 = (5,7), A3 = (9,11), A4 = (13,15),
ODð33,1716 Þ
1
Corollary 5
A1 = (5, 8), A2 = (6, 7), A3 = (1, 2), A4 = (3, 4),
A5 = (1, 3), A6 = (5, 7), A7 = (9, 11), A8 =(13, 15)
A5 = (5,6), A6 = (7,8), A7 = (1, 2), A8 = (3,4) 20
ODð40,8 Þ ODð41,920 Þ ODð40,2020 Þ
New
New
New
Theorem 4,
A = (1,3,3,5, 5), B= ( 5, 7, 7,1, 1),
Corollary 2
C =( 3,1,1,7, 7), D = ( 7,5,5, 3,3)
Theorem 4,
A = (1,2,2,3, 3), B= ( 3, 4, 4,1, 1),
Corollary 3
C =( 2,1,1,4, 4), D = ( 4,3,3, 2,2)
Corollary 2
A = (7, 9,9, 11, 11), B= (13, 13, 15, 15, 17), C =(19, 17, 19,1,1), D =(3,3, 5,5, 7)
ODð41,2120 Þ
New
Corollary 3
A = (6,8, 7, 7,8), B= (9, 9,10,10,1), C =(1, 2, 2,3, 3), D = (4, 4, 5, 5, 6)
ODð48,1624 Þ
New
Corollary 4
A1 = (1,9,5), A2 = (13, 1, 3), A3 = (1,9,5), A4 = (15,11, 13), A5 = (7, 5, 13), A6 = (3, 7,11), A7 = (7,15,11), A8 = (3,15, 9)
ODð49,1724 Þ
New
Corollary 5
A1 = (5,1, 4), A2 = (1, 6,3), A3 = (2,4, 7), A4 = (2,5,2), A5 = (7,6,3), A6 = (8,8, 7), A7 = (3, 6, 5), A8 = (4,8, 1)
ODð48,2424 Þ
New
Corollary 4
A1 = (9,1,23), A2 = (3,3,9), A3 = (1, 21,5), A4 =(15,11, 7), A5 = (19, 17, 13), A6 = (13, 17, 15), A7 = (5,19,23), A8 = (21, 7, 11)
ODð49,2524 Þ
New
Corollary 5
A1 = (8, 9, 10), A2 = (11,12, 1), A3 = (2, 3,4), A4 = (5, 6,7), A5 = (8, 9, 10), A6 = (11,12, 1), A7 = (2, 3,4), A8 = (5, 6,7)
ODð56,1428 Þ
New
Corollary 2
A = (1, 1,1,1, 3, 3, 3), B= (7,7, 9,9,9, 9, 11), C =(3,5,5,5,5, 7, 7), D = ( 11,11, 11,13,13,13,13)
with zero periodic autocorrelation function. By applying the suggested methods we obtain many new orthogonal designs for computer experiments. Also we provide the construction of an infinite class of orthogonal designs for computer experiments. The properties of the designs, derived by the methods of this paper, are studied. We showed that each of the constructed designs has the property that the estimates of quadratic effects (or two-factor interactions) are uncorrelated with the estimates of the linear effects.
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In the results, given in Table 1, some designs with the same size and same number of factors, but with different numbers of levels per factor are presented. The practitioner will have to experimentally choose between two such designs depending on the problem under study. Most of the times, a computer experiment can be performed with different but suitable choices of factor levels. Further investigation on the choice of the required levels of a design is outside of the scope of this paper. The proposed method can also be expanded and used for the construction of near orthogonal designs as in Bingham et al. (2009). In such case the correlation of the derived designs would be pre-calculated by the periodic function of the used row vectors. Another interesting and maybe promising generalization would be to further study and construct supersaturated Latin hypercube designs as these were initially introduced in Butler (2005). The above interesting expansions are out of the scope of this paper and further investigation towards that area is required.
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