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A discrete fourier transform approach searching for compatible sequences and optimal designs.

In this paper, we apply a new method of evaluating the Discrete Fourier Transform that requires significantly less computational effort. In this evaluation, the Discrete Fourier Transform is defined over the support of the sequence of interest. The method can be applied to search for sequences with zero periodic autocorrelation function. As an example we apply the procedure and we were able to classify weighing matrices W(2n, 9) constructed from two sequences of length n and weight (5, 4) for al 1400 [less than or equal to] n [less than or equal to] 500.

Povzetek: Predstavljena je nova metoda Fourierjeve transformacije.

Keywords: linear models, optimal design, orthogonal designs, discrete Fourier transform, power spectral density, construction

1 Introduction

Sequences with zero or low autocorrelation function have been widely used in Statistics and in particular in the theory of optimal experimental designs. In many cases the experimenter wishes to develop and study an empirical linear regression model of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

It is well known that the least square estimator for the coefficient Vector [beta] is [??] = [(X'X).sup.-I]X'y with covariance matrix Cov([??]) = [delta].sup.2] [(X'X).sup.-I]. Orthogonality of the design matrix X is essential to create models with optimal variance. More details on linear regression analysis and optimal designs can be found in [2, 13]. Sequences with a zero autocorrelation function can be used to generate orthogonal design matrices that achieve the optimal covariance matrix for the estimator of [??]. Such sequences are also known as compatible sequences.

Two level and three level design matrices are commonly used for screening and weighing experiments. Recently, new methods for constructing three level screening designs, from weighing matrices, were proposed in the literature. For example, in [17] the authors used W(n, n - 1) to construct three-level screening designs while their results were generalized to the case of W(n, k) in [6]. The designs constructed by the methods of the above papers, have their main effects orthogonal to each other, orthogonal to any quadratic effects and orthogonal to any two-factors interactions. For quantitative factors, the linear model (1) with such designs can be used for screening out the main effects in the presence of active second order terms, such as two-factor interactions or pure quadratic effects, in the true model.

Such orthogonal designs of experiments can be easily constructed from sequences with zero autocorrelation function. The construction of such sequences is important but the needed computational effort is sometimes enormous, making the search for such desirable designs infeasible. Some known algorithms for developing sequences with zero autocorrelation function and the related optimal ex perimental designs can be found in [1, 8] and the references therein. A method that uses the Discrete Fourier Transform (DKr) was recently developed in the literature (see [3]).

In this paper, we proposed a new evaluation of the DFT that reduces the computational effort. This evaluation can be used in searching for sequences with zero autocorrelation function. It is applicable to sequences with two, three or more levels and the complexity of the method does not depend on the length of the sequences but only on the number of their non zero elements (weight). So, when searching for weighing matrices of order n and weight k, constructed from a number of suitable circulant matrices (sequences), the complexity depends only on k. In this way, for a fixed weight k, one may search for large optimal weighing designs. Moreover, this method can test each of the required sequences independently and decide whether or not this sequence can be a candidate for a set of compatible sequences. As an example, we apply the suggested method to search for sequences with zero periodic autocorrelation function that can be used to classify a special type of weighing matrices.

2 Preliminary Results

A weighing matrix W = W(n, k) is a square matrix with entries 0, +1 having k non-zero entries per row and column and having zero inner product of distinct rows. Hence W satisfies WWT = kin.. The number k is called the weight of W. Weighing matrices have long been studied because of their use in weighing experiments as first studied by Hotelling [10] and later by Raghavarao [12] and others [1, 14].

Given a set of g sequences,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

of length n, the periodic autocorrelation function (abbreviated as PAF) PA(S) is defined, reducing i + s modulo n, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

The set A of the above sequences is called compatible if [[SIGMA].sup.l.sub.j=1], [P.sub.aj] (s) = a, s = 1, 2,..., n - 1. Moreover, if a = 0 then the sequences A are said to have zero periodic autocorrelationfunction (zero PAF).

Notation. We use the following notation throughout this paper.

1. We use 5 to denote -x.

2. We use R = (rij) to denote the n x n back diagonal matrix whose elements satisfy [r.sub.ij] =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. Let A = ([a.sub.0], [a.sub.1], ..., [a.sub.n-l]) where [a.sub.i] [member of] {0, 4-1}. The support of A is the set [SP.sub.A] = {i : [a.sub.i] [empty set] 0}.

The discrete Fourier transform (DFT) of a sequence B = ([b.sub.0], [b.sub.1], ..., [b.sub.n-1]) of length n is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [omega] is the primitive n-th root of unity, e 2[pi]i/n. If we take the squared magnitude of each term in the DVF of B, the resulting sequence is called the power spectral density (PSD) of B and will be denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We make use of the following well-known theorem (see [16, chapter 10]).

Theorem 1. Let B be a sequence of length n with elements from the set {0, [+ or -]1}. The PSD of this sequence is equal to the DFT of its periodic autocorrelation function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3 The Support Based Discrete Fourier Transform and Power Spectral Density

The constant value of the PSD of compatible sequences can be easily calculated using the elements of the sequences (see [3]). The following Lemma illustrates an alternative method for calculating this value.

Lemma 1. Let A = {[A.sub.j]: [A.sub.j] = ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ), j = 1,...,g}, be a set of sequences of length n, with zero periodic autocorrelation function. Then we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Using equation (5) and the fact that the sequences have zero PAF we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, when searching for sequences with zero autocorrelation, it is very simple to find the constant value of the PSD since this constant will be the sum of squares of the elements of the sequences.

Let A = ([a.sub.0], [a.sub.1], ..., [a.sub.n-1]) with [a.sub.i] E {0,[+ or -]1} and let [SP.sub.A] be the support of A. The discrete Fourier transform (DFT) of the sequence A can be defined on [SP.sub.A] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [omega] is the primitive n-th root of unity, e 2[pi]i/n.

Lemma 2. Suppose that we have a set A of compatible sequences, as in (2), with PA(S) = a for s = 1, 2, ..., n -1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where k = 0, 1, ... n - 1 and c = [[SIGMA].sup.l.sub.j=l] (0) - a.

Proof. The result is a straightforward generalization of a case proved in [7] for four sequences and thus the proof is omitted.

Remark 1. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = c we have the following. A sequence X should satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] e for all k = 0,1, ..., n - 1 in order to be selected as a candidate for the compatible set A. When searching for sequences with elements from the set {-1, 0, 1} the computational effort of the PSD does not depend on the length but only on the number of non-zero elements (weight) of the sequence.

Example 1. Suppose we wish to search for a weighing design of order 2n and weight w, constructed from two sequences A, B of length n each and 2n [greater than or equal to] w. We need 2n summations and 2n multiplications to calculate the DFr (or the PSD) using the classic definition of DFT but only w summations and [omega] multiplications are needed using the support based definition. If [omega] [less than or equal to] [less than or equal to] n then the new definition is extremely fast (by comparison), while when [omega] = n it will be shown that the needed computational effort of the PSD is reduced in half. Note that the above calculation concerns only one candidate pair of compatible sequences from the total number of possible pairs in the search space. Since the search space in such problems is huge and exponentially increasing with n, it is clear that any reduction in the computational effort at each point of the search space results in a huge reduction of the total computational effort (in absolute terms).

Lemma 3. Suppose we have a set A of l = 2m sequences of length n, as in (2), with elements from the set {-1, 1} and [P.sub.A] (S) = O for s = 1, 2, ..., n - 1. Then the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

is a set of l sequences of length n with elements from the set {-1, O, 1} and [P.sub.B](s) = O for s = 1,2, ..., n-1. The total weight of the new sequences, in B, is nm.

Remark 2. Using Lemma 3, we are able to get the benefits of the new definition of DFT even in the case of sequences with elements from the set {-1, 1}. Suppose that we are interested in searching for four sequences with elements from the set {-1, 1}, zero PAF and length n. By using the proposed definition of DFT we need 2n summations/multiplications for each evaluation of the DFT while the old definition of DFT requires 4n calculations. Generally, each calculation of the DFT will be reduced in half. If recursively t nested evaluations of the DFT are used in the algorithm, then we have a reduction of calculation by a scale factor 1/[2.sup.l].

4 An Illustrating Example

In this section we illustrate the use of the suggested procedure in searching for weighing matrices constructed from suitable circulant matrices (sequences). The computational advantages of this approach, as these were discussed in the previous section, are illustrated through numerical examples. We present in details the case of weighing matrices W(2n, 9) that can be constructed from two circulants.

Weighing matrices W(2n, 9) constructed from two sequences are of great interest but hard to find, since the necessary conditions for their existence are not sufficient (see [9]). It is well known that if there exist a W(2n, k) constructed from two circulant matrices of order n, then k [a.sup.2] + [b.sup.2], where a and b are the row (and column) sums of A and B respectively. The next theorem gives a known construction of weighing matrices by using two sequences with elements from the set {0, 1, -1} and constant PSD (equivalently, by using two circulant matrices [A.sub.1], [A.sub.2] with elements from the set {0, 1,-1} satisfying [A.sub.1][A.sup.T.sub.1] + [A.sub.2][A.sup.T.sub.2] = kI).

Theorem 2. (Geramita and Seberry (1979), Theorem 4.46) If there exist two circulant matrices [A.sub.1], [A.sub.2] of order n with elements from the set {0, 1, -1}, satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then there exists a W(2n, k).

The construction of the corresponding weighing matrix is achieved by using either of the following two arrays

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The case of weighing matrices constructed from two circulants A and B having ([parallel]S[P.sub.A][parallel], [parallel]S[P.sub.A][parallel])[parallel][SP.sub.B][parallel] (9, 0), which is actually the case where a circulant weighing matrix exists, was resolved in [15]. In [15] it was shown that a circulant weighing matrix W(n, 9) exist if and only if n is multiple of 13 or 24 (i.e. 13 [parallel] n or 24 [parallel] n). This case implies that there exists one sequence with elements from the set {0, 1,-1} having weight 9 and zero PAF. Thus, a circulant weighing matrix W(n, 9) exists and a weighing matrix W(2n, 9), constructed from two circulant matrices, exists (take one to be the matrix with all elements zero and the other to be the circulant weighing matrix W(n, 9) as it is given in [15]). For more details on this case see [15]. If (([parallel]S[P.sub.A][parallel], [parallel]S[P.sub.A][parallel])[parallel][SP.sub.B][parallel] (9) = (8, 1) and PA(S) + PB(S) = 0, [for all] s = 1, 2, ..., n - 1, and there exists a W(2n, 9), constructed from two circulant matrices, such that 9 = [a.sup.2] + [1.sup.2] which is not possible since 8 is not a perfect square. So the case (8, 1) is not permitted.

Two pairs of sequence are said to be equivalent if the one can be constructed from the other by some transformations. More specifically, we recall the following definition.

Definition 1. We say that two pairs of (0, [+ or -] l) sequences ((A,B) and (C,D)) of length n are equivalent if f one can be obtained from the other by applying some of the following transformations.

1. Multiply one or both sequences of the pair by - 1.

2. Reverse one or both sequences of the pair.

3. Take circulant permutation of one or both sequences of the pair.

4. Multiply the elements of the support of both sequences by l (l, n) = 1.

We call the corresponding weighing matrices, constructed from the two circulant matrices whose first rows are the sequences (A, B) and (C, D), equivalent. More details on weighing matrices constructed from two circulant matrices can be found in [11].

In this section we classify the weighing matrices W(2n, 9) for 400 [less than or equal to] n [less than or equal to] 500 constructed from two sequences of weight 5 and 4 respectively. Results for n [less than or equal to] 100 were presented in [4], for n [less than or equal to] 400 in [5], and the results for 400 [less than or equal to] n [less than or equal to] 500 are new and given in Table 4. One representative of each order was known, and were presented in [5]. Thus, in Table 4, we only present the number of inequivalent solutions for each order. All inequivalent sequences and the corresponding weighing designs are available on request.

In the next example we illustrate numerically the computational gain from the suggested approach in the case of n = 500.

Example 2. Following Example l, we need just 9 calculations for each evaluation of the PSD in contrast to the 1000 calculations needed for the old definition. Note that we have about 5004 sequences in the search space and thus the proposed algorithm require 2 x 10 (10) while the old definition needs 4 x 10 (12) simple calculations. So, the time needed is approximately 1 day by applying the old definition and just 10 minutes with the new evaluation. As an extreme example consider a large search space (for n=100000) where the required search time using the old definition is more than a year (infeasible). The reduction of time (simple computations) will be by a scale factor 9/2n (i.e., 999.99% less) and thus the required time will be just a few hours.

5 Discussion

In this paper, we proposed a new evaluation of the DFT that reduces the computational effort. This evaluation can be used in searching for sequences with zero autocorrelation function. It is applicable to sequences with two, three or more levels and the complexity of the method does not depend on the length of the sequence but only on the number of non-zero elements (weight). So, when searching for weighing matrices of order n and weight w, constructed from a number of suitable circulant matrices (sequences), the complexity depends only on w. In this way, for fixed weight w, one may search for large optimal weighing designs. Moreover, this method can test each of the required sequences independently and decide whether or not this sequence can be a candidate for a set of compatible sequences. As an example, we applied the suggested method to search for sequences with zero periodic autocorrelation function that can be used to classify a special type of weighing matrices W(2n, 9).

The reduction of the computational effort could lead to many new numerical results and help to resolve other open cases for weighing matrices. Moreover, the support based approach may be used for theoretical investigation of weighing matrices and other optimal designs. The same approach can be applied in many other cases where circulant matrices are used for the construction of optimal designs (see [2, 8, 12]).

Received: May 12, 2012

6 Acknowledgments

We thank the Editor and anonymous referees for their useful remarks which led to an improvement in the content and preparation of the article.

References

[1] Craigen, R. and Kharaghani, H. Orthogonal designs, in: C.J. Colbourn and J.H. Dinitz (eds.), Handbook of Combinatorial Designs, 2 ed., CRC Press, Boca Raton, FL, 2007, pp. 273-280.

[2] V.V. Fedorov, Theory of Optimal Experiments, Academic Press, New York, 1972.

[3] R.J. Fletcher, M. Gysin and J. Seberry, Application of the discrete Fourier transform, to the search for generalised Legendre pairs and Hadamard matrices, Australas. J. Combin., 23 (2001), 75-86.

[4] S. Georgiou and C. Koukouvinos, New infinite classes of weighing matrices, Sankhya Ser. B, 64 (2002), 26-36.

[5] S. Georgiou, Signed differences for weighing designs, Sankhya Ser. B, 72 (2010), 107-121.

[6] S.D. Georgiou, S. Stylianou, and M. Aggarwal, Efficient three-level screening designs using weighing matrices, (submitted).

[7] S. Georgiou, C. Koukouvinos and S. Stylianou, On good matrices, skew Hadamard matrices and optimal designs, Computational Statistics and Data Analysis, 41 (2002), 171-184.

[8] A.V. Geramita, and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York, 1979.

[9] J. Horton and J. Seberry, When the necessary conditions are not sufficient: sequences with zero autocorrelation function, Bulletin 1CA, 27 (1999), 51-61.

[10] H. Hotelling, Some improvements in weighing and other experimental techniques, Ann. Math. Stat., 16 (1944), 294-300.

[11] C. Koukouvinos and J. Seberry, New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function--a review, J. Statist. Plann. Inference, 81 (1999), 153-182.

[12] D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, New York, 1971.

[13] G.A.F. Seber and A.J. Lee, Linear Regression Analysis, Wiley, New York, 2003.

[14] J. Seberry and M. Yamada, Hadamard matrices, sequences and block designs, in Contemporary Design Theory--a Collection of Surveys, eds J.H. Dinitz and D.R. Stinson, Wiley, New York, 1992, pp. 431-560.

[15] Yoseph Strassler, The Classification of Circulant Weighing Matrices of Weight 9, PhD Thesis, Bar-Ilan University, Ramat-Gan, 1997.

[16] S.A. Tretter, Introduction to Discrete-time Signal Processing, Wiley, New York, 1976.

[17] L. Xiao, D.K.J. Lin, and E Bai, Constructing Definitive Screening Designs Using Conference Matrices, Journal of Quality Technology, 44 (2012), 2-8.

S.D. Georgiou Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece E-mail: stgeorgiou@aegean.gr

K. Drosou and C. Koukouvinos Department of Mathematics National Technical University of Athens Zografou 15773, Athens, Greece E-mail: drosou.kr@gmail.com, ckoukouv@math.ntua.gr
Table 1: Number N of inequivalent solutions for the construction
of W(2n, 9), when ([absolute value of SPA], [absolute value of
SPB]) = (5,4) and 400 [less than or equal to] n [less than or equal
to] 500.

n   400   403   405   406   407   408   410   413
N   16    5     5     8     5     8     5     1
n   414   415   416   418   420   424   425   427
N   2     1     10    12    27    3     6     1
n   429   430   432   434   435   437   440   441
N   10    5     7     7     6     6     20    3
n   442   444   445   448   450   451   455   456
N   8     2     1     15    10    6     7     9
n   459   460   462   464   465   468   469   470
N   3     11    17    6     5     7     1     4
n   472   473   475   476   480   481   483   484
N   3     6     7     13    20    5     5     8
n   485   488   490   492   493   494   495   496
N   1     3     11    3     4     9     10    5
n   497   500
N   1     11
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Author:Georgiou, S.D.; Drosou, K.; Koukouvinos, C.
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Date:Dec 1, 2012
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