A technique to construct linear trend free fractional factorial design using some linear codes

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1 International Journal of Statistics and Mathematics Vol. 3(1), pp , February, ISSN: x IJSM Research Article A technique to construct linear trend free fractional factorial design using some linear codes Poonam Singh 1, Puja Thapliyal 2*, Veena Budhraja 3 1 Department of Statistics,Faculty of Mathematical Sciences, University of Delhi,Delhi , India. 2* Department of Statistics, Faculty of Mathematical Sciences, University of Delhi,Delhi , India. 3 Department of Statistics, Sri Venkateswara College, University of Delhi, India. Co-author: veena_budhraja@yahoo.co.in, pbs_93@yahoo.co.in *Corresponding author pujathapliyal98@gmail.com Sometimes, an experimental situation may arise where experimental units are following smooth pattern of trend over time and space. In that situation, to eliminate the effect of uncontrollable variables that are correlated with the time, we use systematic run order instead of randomising the run order. An ordering of treatments, thus obtained is known as Trend free design. This article presents a method for constructing trend free fractional factorial design using parity check matrix of a linear code. The method provides a systematic approach to construct fractional and blocked fractional factorial design with trend free main effects and some two- factor interactions. Keywords: Time count, Linear Codes, Reed Muller Codes, BCH codes,generalised Foldover Scheme INTRODUCTION Fractional Factorial experiments, especially those of exploratory type are often conducted sequentially in a specific run order. In some situations, the experimentation scheme may suffer from some undesirable effects of factors. For instance, in many industrial and agricultural experiments, treatments are applied to experimental units sequentially in time or space, where there may be unknown or uncontrollable variables influencing the experimental process that are highly correlated with the order in which the observations are obtained. Hence, the treatments which are not being influenced by the effect of the trend under study is required. Such type of designs are known as trend free designs. The need of trend free designs was mentioned by Joiner and Campbell (1976) who described an experiment in which the measurement drift with time due to the build up of carbon in a spectrophotometer and also by Freeny and Lai (1997), in an experiment taken from the electronic industry in which a photolithographic polisher showed the tendency drift lower through time. The study of trend free designs was begun by Cox (1951) and has been addressed by different authors like Hill (1960), Daniel and Wilcoxon(1966), Draper and Stoneman (1968), for the construction of trend free design. The important work of Daniel and Wilcoxon (1966) was on constructing trend resistant contrasts for 2 n design and using the contrasts to generate a new run order. Cheng and Jacroux (1988) generalised Daniel and Wilcoxon work for 2 n designs. Coster and Cheng (1988) introduced a generalized foldover method for constructing trend free run order from a sequence of generators in a simple manner. Further, Bailey, Cheng and Kipnis (1992) extended the method to general symmetric and asymmetrical factorial designs. Adekeys and Kuner(2006) compared the run order of 2 k-p design in the presence of time trend. The two level trend robust fractional factorial designs were analysed by Mee and Romanova(2010). In Recent years, Hilow(2012) developed an assignment procedures of constructing minimum cost trend free two level fractional factorial designs.some trend free designs in blocked fractional factorial designs were being studied and investigated by Wu and Soushan(2013). In this article, we present a method to construct the generator matrix for the required designs using the parity check matrix of a linear code. Section 2 gives the preliminaries required. A brief introduction of coding theory is given in section 3. Section 4 presents the construction technique to generate the designs. A method using some linear codes is described and illustrated through examples in section 5. The section 6 presents the construction for trend free blocked fractional factorial designs using the linear codes, mentioned in previous section.

2 Singh et al. 073 PRELIMINARIES Fractional Factorial Design: Let D = (q r n-p ) denote a q -p fraction of the q n factorial design in q r incomplete blocks. Let N = q n-p be the number of treatment combinations, K=q n-p-r be the size of each block and B = q r be the number of blocks.the linear model for the q n factorial design is given as (1) Where y is the N 1 column vector of observations, X is the N n design matrix of known constants; N is the n 1 column vector of regression coefficients and e is the N 1 column vector of random errors with zero means and variance σ 2.. When block effects are considered, the model is changed to with B= is N B block matrix with block size K and γ is the corresponding (K 1) block. (2) Definition1: Let denotes the ordered vector of observations, for x = 0,1,2,... be the N 1 vector of trend coefficients and let u i be the contrast for main effect A i ; i=1,2,...n, in the run order. Then the quantity is known as the time count for the main effect A i.. A necessary and sufficient condition for a main effect contrast u to be In general, an N 1 vector u is called trend free if equation(1.1) holds. trend free is that When the assumption of trend effect of different factors of the design is also considered, then this assumption changes the model (2) to (3) Where, t is the (K 1) linear vector and is the trend effect coefficient.the model (3) is a linear model for factorial designs with trend. The first K rows of correspond to K treatment combinations in the principal block, the next K rows to the treatment combinations in the second block and so on. Definition 2: A run order is optimal for the estimation of the factor effects of interest in the presence of nuisance v- degree polynomial trend iff (1.1) (1.2) Where is an Nn matrix of factor effect coefficients and T is the Nv matrix of polynomial trend coefficients. If equation(1.2) is satisfied then the run order is said to be v-trend free. If x is any column of nd t is any column of T then the usual inner product is the time count between x and t. Criterion given in equation (1.2) states that all the time count are zero for optimal run order. Coster and Cheng(1988) introduced Generalised Foldover Scheme to generate systematic run order for full and fractional factorial plans. We give here the definition of Generalised Foldover Scheme (Coster and Cheng(1988)). Definition 3: Generalised Foldover Scheme (GFS): For a factorial design D= (q r n-p ) with n factors A 1, A 2,..A n let G be the generator matrix of design given by

3 Int. J. Stat. Math. 074 Let be a 1 n matrix of zeros. Then the run order of design D is produced by the GFS with respect to the generator sequence is given by where for i = 1,2,,n-p. In the run order, the principal block consists of the first q n-p-r runs; the second block consist of the next q n-p-r runs, and so on. For a factorial design D 1 =(q n-p r ), let be the n factors and G be the generator matrix of design D 1 given by Coster and Cheng (1988) derived following conditions for -trend free effects in GFS. These conditions involve the generator matrix. 1. The main effect of a given factor is trend free if the corresponding letter appears at least ( +1) times in the generator sequence. 2. A 2-factor interaction is -trend free if and only if there are at least ( +1) generator each of which exactly one of the two factors appears at non-zero level. For linear trend free design, the above conditions can be described in the following properties: Property 1: For any factor, if there are at least two non zero elements, then all the main effect components of factor are linear trend free. Property 2:Forany factor,if there are at least two pairs, such that one element is zero and the other element is non zero, then all components of the linear trend free. LINEAR CODES interaction are A linear [n,k,d] q code C over GF(q), where q is prime or prime power, n is the length, k is the dimension and d is the minimum distance, is a k-dimensional subspace of the n-dimensional vector space V(n,q) over GF(q). The dual code C of an [n,k,d] q code C is C = { v V(n,q)/ v.w=0 for all w C}. This is an [n, n-k, d ] q code and an (n-k) n generator matrix H of C is called a parity check matrix of C. If the generator matrix is given in the standard form, a corresponding parity check matrix is given as T H = [ A In k ] Any d -1 columns in generator matrix G of C are linearly independent and any d-1 columns in parity check matrix H are linearly independent. FACTORIAL DESIGNS WITH SOME LINEAR TREND FREE EFFECTS GFS provides a technique to construct linear trend free factional factorial designs using generator matrix, but there is no general method to construct the generator matrix. We use the parity check matrix of a linear [n,k,d] q code to obtain the set of generators to construct the desired designs. Method of Construction 1. Consider the parity check matrix H n-k n of a linear [n,k,d] q code. 2. Partition the matrix H in two submatrix F 1 and F 2, where F 2 consists of the vectors with weight (w) one. 3. Let m denotes the number of columns in F 2. If m <d-1, then select any d-1 columns from H to form a matrix M of order n-k d-1 otherwise select M such that there is at least one column from F 2. The order of M is n-k d Write the transpose of M and the transpose be denoted by MT of order (d-1 n-k). Delete the column with all zero entries and repeated columns to form the matrix G α of order,say (d-1) l 5. Retain G α, if the number of columns (l ) in G α >d-1 Otherwise go to step Apply GFS on G α to obtain the fractional factorialdesign q l-r ; where 1 r (n-k)-(d-1) & (d-1)+1 l n-k. The resultant fractional factorial designis such that some of the main effects and two factor interactions are linear trend free.

4 Singh et al. 075 Remark 1. The total number of possible choices of M matrices is 2. In case d-1 = l in step 5, we get the full factorial design q l with some main effects and two factor interactions Linear trend free. 3. The design of trend freeness of the factors depends upon the weight (w) of the corresponding column in G α. 4. All the main effects will be trend free if the weight(w) of all columns in G α is at least two. Above method of construction can be summarized in following theorem. The proof of the theorem can be obtained from the sequence of steps given in the construction method. The proof(*) is available with the authors. * Theorem1: Existence of a linear [n, k, d] q code implies the existence of q l-r {1 r (n-k)-(d-1) & (d-1)+1 l n-k },fractional factorial design and q l ( l=d-1) full factorial design in which all main effects and some of the two-factor interactions are linear trend free, where (d-1) is number of linearly independent columns in matrix H and l is the number of factors/columns in G α. LINEAR TREND FREE FRACTIONAL FACTORIAL DESIGNS USING DIFFERENT CODES We consider different types of linear codes and generate linear trend free fractional factorial designs using the method of construction described in previous section. For details see Hedayat et.al (1999). a) Reed Muller Codes The r th order binary Reed-Muller code R(r,a) of length n = 2 a, for 0 r a, is the set of all vectors f, where f(i 1,..i a ) is a Boolean function which is a polynomial of degree at most r. For any a and any r, 0 r a, there is a binary r th order RM code R(r,a) with the following properties: Length n = 2 a, dimension k = + + and minimum distance 2 a-r. The parity check matrix of R(r,a) code is the generator matrixof its dual code. The dual of RM(r,a) is RM(a-r-1,a) code. Example 1: Consider a parity check matrix of RM(2,4) H= Any three columns in parity check matrix H matrix are linearly independent. Here d-1=3, and m =5 (in F 2 ) and we get the designs and corresponding generator matrices depending on the columns selected. According to the selection of columns,we have the following cases : i)11 columns with none of the column having weight one and there are possible ways of selection where none of the column is with weight (w) one and the corresponding factorial designs will have all main effects linear trend free. All generated matrices and the table contains generator matrices, their corresponding designs, trend free effects and defining relations are available with th authors.

5 Int. J. Stat. Math. 076 ii) For m=5 and d-1=3, n-m=11.the possible choices of columns selected for generator matrix are + +. Suppose we choose the three columns 4 th, 5 th and 6 th to form the matrix M RM = On transposing we get matrix as G RM in which all three rows d-1 =3 are independent. here d-1 =3 and l=5, Using the method of GFS on generator matrix we get a resolution III, fractional factorial design with defining relation I =A 1 A 3 A 5 = A 1 A 2 A 3 A 4 = A 2 A 4 A 5. Using the properties 1and 2 we observe that main effect component of A 1, A 3, A 5 and the two factor interaction A 1 A 2, A 1 A 3 and A 2 A 4 and A 3 A 4 are linear trend free. Cyclic Codes A linear code over GF(q) is said to be cyclic if whenever (c 0,c 1,..,c n-2,c n-1 ) is a codeword so also is (c 1,c 2,..,c n-1,c 0 ). Cyclic arrays can be described by a single generating vector z = ( z 0 z 1.z n-1 ) such that the generator matrix consists of this vector and its first ( k-1 ) cyclic shifts. The generating vector z is represented by a polynomial z(x) = z 0 + z 1 X + +z n- 1X n-1 which is called a generator polynomial for the code. If a code is cyclic, so is its dual, and the generator polynomial of its dual can be obtained by the following result given in Macwilliam and Sloane (1977). Theorem 2: If C is a cyclic code of length n over GF(q), with generator polynomial z(x), then the dual code C cyclic and has generator polynomial where z * (x) = X deg.z z(x -1 ) is reciprocal polynomial to z(x). is also Example 2: Let = be the generator polynomial for [15,7,5] 2 code. Then the generator polynomial for the dual of this code is given as h (x) = x 7 ( 1+ x -4 +x -6 +x -7 ) = 1+x+x 3 +x 7 Hence the parity check matrix of [15,7,5] 2 code is obtained by writing the coefficients and giving the cyclic shift to the coefficients,as given below H cy = = Any four columns in the H matrix are linearly independent, d-1 = 4 and m=5, n-m= 15-5 = 10 column with none of them of weight(w) one. Then, there are 210 possible selection of such sets of columns from the matrix H. Following the method of construction, we generated all generator matrices from the set of selected columns and listed corresponding fractional factorial designs with their trend free effects and defining relations {available with the authors}.the result of selecting columns 4 th, 5 th, 6 th and 7 th gives the matrix M cy(8 4 M cy(8 4=

6 Singh et al. 077 Deleting column of zeroes and transposing MT= we get generator matrix as G cy of order( (d-1) l) G cy = where l =7 and d-1 =4. Using GFS, a fractional factorial design is obtained with some main effects A 3, A 4, A 5, A 6 and two factor interactions A 1 A 2, A 2 A 3, A 2 A 6, A 2 A 7, A 3 A 4, A 3 A 5, A 3 A 6, A 3 A 7, A 5 A 4, A 6 A 4, A 5 A 6, A 5 A 7 are linear trend free. BCH Codes The BCH codes over GF(q) of length n = q m -1 and designed distance δ is the largest possible cyclic code having zeroes α b,, α b+1,,α b+δ-2 where α є GF( q m ) is the primitive n th root of unity, b is a non negative integer and m is the multiplicative order of q mod n. The parity check matrix of a BCH code with b=1 is given by 2 n n1 1 ( ) ( ) C= n1 1 ( ) ( ) n1 1 ( ) ( ) ( ) where each entry is replaced by the corresponding binary m-tuple. Example 3.: The parity check matrix of [15,5,7] 2 code is H= and we observe here that d-1=6 columns in above matrix are linearly independent. Since the matrix cannot be partitioned into the sub matrices,we consider all 15 columns to get the design and they are Suppose we select columns 8 th,9 th,10 th,11 th,12 th and 13 th from above matrix H.The matrix obtained by selection is M =

7 Int. J. Stat. Math. 078 On transposing and deleting columns of zeroes we get matrix G BCH as G BCH = Applying GFS on generator matrix with l = 10 and d-1 = 6 we get the design fractional factorial design with all main effects linear trend free. Some of the designs that can be generated by selecting different columns of the generator matrix G BCH of the code [15,5,7] 2 along with their defining relations are available with the author. Ternary Golay Code The Golay code were discovered by M.J.E.Golay in late 1940 s. The (unextended) Golay code are examples of perfect codes. A q-nary code that attains the hamming ( or sphere packing) bound i.e. the one which has codewords, is said to be perfect code. Consider the ternary Golay Code [11,6,5] 3 over a ternary alphabet, the relative distance of the codes is as large as it possibly can be for a ternary code, and it satisfies Hamming bound and is therefore a perfect ternary Golay [11,6,5] 3 code. For perfect code its dual distance is same as its covering radius. In terms of design the strength is same as the estimation index of an orthogonal array obtained using this code. Golay codes are unique in the sense that binary or ternary codes with same parameters, can be shown to be equivalent to them. Example 4: Consider the parity check matrix of [11,6,5] 3 and partition it into two submatrices as given in the method in section 4, we get the matrix H as H= Any four columns in matrix H 5 11 are linearly independent d-1 = 4 and m = 5. Thus, 15 sets of column in which none of the columns has weight (w) one. Whereas the total possible selection of columns to formmatrix M is 325. Suppose we choose columns 1 st,2 nd,3 rd and 4 th from matrix H, we get the matrix M =. Next, we generate the matrices of order (4 5) that ensures the linear trend freeness of main effects and following the steps of construction method, we get fractional factorial design. and then fractional factorial design I= A 1 A A 3 A 4 is generated using GFS on matrix G golay. The generated design has all its main effects linear trend free with respect to the properties [1-3].

8 Singh et al. 079 BLOCKING IN FRACTIONAL FACTORIAL DESIGN WITH SOME LINEAR TREND FREE EFFECTS. When the block size is smaller than the number of treatment combinations in any factorial experiment, the technique of blocking is used to carry out the analysis. The factorial/fractional factorial experiment thus obtained is known as Blocked fractional factorial experiment. When we go for blocking of fractional factorial design, the block structure affects the linear trend-freeness of the effects. We state here the result in continuation with the properties [1-3]. The first h=n-p-r generates the principle block and let z v, h+1 v n-p be the generators of other blocks. Then the following holds: Property 4: For any given v, h+1 v n-p and factor say A 1, suppose z 1v 0. Then (a) All (q-1) main effect components are linear trend-free. (b) If z iv = 0, 2 i n-p, then all components of A 1 A i interaction are linear trend-free. (c) The method described is used to construct Blocked fractional factorial designs with some linear trend free effects. Reed Muller Code: We use here the generator matrix constructed in Example 1 by selecting columns 8 th,9 th and 10 th We get fractional factorial design, here q=2, n=3, p=1, r 1 =1, h=n-p-r 1 =1 independent generators. Using GFS, h=2 generators forms the principle block and remaining generators form the contents of the other block in which main effects A and C are linear trend free. The confounded effect of the design is ABC. Table fractional factorial design with resolution III, I=ABC Block 1: (1), acd, abd, bc Block 2: c, ac, ab, bcd Cyclic Code: We consider here generator matrix constructed in example 2 and by selecting 1 st, 2 nd,5 th and 9 th columns We get G cy = Using GFS fractional factorial design is obtained n=5, p 1 =1, s=2, r 1 =1, h = n-p-r 1 = = 3 independent generators, generates the principal block and other generates the other block contents. Table 1 gives the design generated. Table 2 : blocked fractional factorial design with resolution IV, I = ABCE Block 1: Block 2: (1), ab, bc, ac, acd, bcd, abd, d ce, abce, be, ae, ade, bde, abcde, cde BCH Code: Consider the generator matrix in Example 3 by selecting columns 1 st, 4 th, 5 th, 6 th, 7 th and 8 th as G BCH = fractional factorial design is obtained, here n=9, p=3, r 1 =1 and h = n-p-r 1 = 9-3-1= 5 independent generators forms the principal block and remaining generators the other block. Thus, Blocked fractional factorial.design with defining relation I = DEGJ= ABDEH = BCDF is obtained. Table 3 displays the first 16 runs of the design constituting the principle block and the next 16 the other block and so on.

9 Int. J. Stat. Math. 080 Table blocked fractional factorial design with resolution IV, I= DEFJ = ABDEH=BCDF Block 1: (1), aej, dfhj, adefh, abefgh, bfghj, abdegj, bdg, bcej, abc, cdefh, abcdfhj, acfghj, cdhj, acdeh, cf, acefj, abcdefj, bcdfg, abcegh, bcghj, bdeh, bdhj, bef, abf, adfg, defgj, aghj, egh. Block 2: abdgh, bdeghj, abfgj, befg, def, adfj, ehj, ah, acdeghj, bcfg, acefg, cfgj, bcdfj, abcdef, bch, abcehj, abcgj, abcdfgh, bcdefghj, cefhj, acfh, bcd, acdj, aeg, gj, adefg. dfgh, bfh, abefhj, bdj, abde. Ternary Golay Code: Consider the generator matrix from Example 4 by selecting columns 1 st,2 nd, 3 rd and 4 th of matrix H as G golay = fractional factorial design is obtained. Here n= 5, p 1 =1 and q=3, r 1 =1,h=n-p-r 1 =3 independent generators are obtained. Thus, principal block is generated by first three generators and remaining two generates the content of the other block. First 3 3 treatment combinations in fractional factorial design (given in Table 4 ) forms the principle block and next 3 3 combinations forms the contents of second block and the remaining the third block contents. The factors are denoted by A,B,C,D and E. Table 4: blocked fractional factorial design with resolution IV, I= ABCDE Block 1: (1), abcde, a 2 b 2 c 2 d 2 e 2, abc 2 d 2, a 2 b 2 e, cde 2, a 2 b 2 cd, c 2 d 2 e, abe 2, ab 2 ce 2, a 2 c 2 d, bd 2 e, a 2 d 2 e 2, bc, ab 2 c 2 de, bc 2 de 2, ab 2 d, a 2 ce, a 2 bc 2 e, b 2 de 2, acd 2, b 2 cd 2 e, ac 2 d 2, b 2 ce, ade, a 2 bcd 2 e 2, b 2 d 2. Block 2: a 2 bde 2, b 2 cd 2, ac 2 e, b 2 c 2 e 2, ad, a 2 bcd 2 e, acd 2 e 2, a 2 bc 2, b 2 de, cde, abc 2 d 2 e 2, a 2 b 2, abe, a 2 b 2 cde 2, c 2 d 2, a 2 b 2 c 2 d 2 e, e 2, abcd, ab 2 c 2 d, a 2 d 2 e, bce 2, a 2 c, ab 2 cd, ab 2 d 2 e 2, bd 2, ab 2 de, a 2 c 2 de 2. Block 3: ab 2 d 2 e, a 2 ce 2, bc 2 d, a 2 c 2 de, bd 2 e 2, ab 2 c, bce, ab 2 c 2 de 2, a 2 d 2, a 2 bcd 2, b 2 d 2 e, ade, b 2 d, acd 2 e, a 2 bc 2 e 2, ac 2, a 2 bde, b 2 cd 2 e 2, c 2 d 2 e 2, ab, a 2 b 2 cde, abcde 2, a 2 b 2 c 2 d 2, e, a 2 b 2 e 2, cd, abc 2 d 2 e. CONCLUSION A technique to construct fractional factorial designs with factors at q level, q>2, with some linear trend free effects using parity check/generator matrix of linear code is developed. Since the generator matrix is not unique in nature, therefore fractional factorial designs with linear trend free effects with same/ different resolution can be easily constructed. REFERENCES Adekeys K, Kunert J (2006). On the Comparison of Run Orders of Unreplicated 2 k-p designs in the presence of a Time Trend. Metrika. 63: Bailey RA, Cheng CS, Kipnis P (1992). Construction of trend resistant factorial designs. Statist. Sinica. 2: Betsumiya K, Harada M (2001). Classification of Formally Self-Dual Even Codes of length to 16. Designs, Codes and Cryptography. 23: Cheng CS, Jacroux M. (1988).On the construction of trend free run-order of two level factorial designs. Journal of American statistical association. 83: Cox DR (1951). Some systematic experimental designs. Biometrica. 38: Coster DC, Cheng CS (1998). Minimum cost trend free run orders of fractional factorial design. The Annals of Statistics. 16: Daniel C, Wilcoxon F (1966). Factorial 2 p-q plans robust against linear and quadratic trends.technometrics. 8: Draper NR, Stoneman DM (1968). Factor changes and linear trends in eight run two level factorial designs. Technometrics. 10:

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