On a Grouping Method for Constructing Mixed Orthogonal Arrays

Author(s)
Chung-Yi Suen

ABSTRACT

Mixed orthogonal arrays of strength two and size*s*^{mn} are constructed by grouping points in the finite projective geometry PG(mn-1, s). PG(mn-1, s) can be partitioned into [(*s*^{mn}-1)/(*s*^{n}-1)](n-1)-flats such that each (n-1)-flat is associated with a point in PG(m-1, *s*^{n}). An orthogonal array *L*_{smn}((*s*^{n})^{(smn-)(sn-1)} can be constructed by using (*s*^{mn}-1)/( *s*^{n}-1) points in PG(m-1, *s*^{n}). A set of (st-1)/(s-1) points in PG(m-1, *s*^{n}) is called a (t-1)-flat over GF(s) if it is isomorphic to PG(t-1, s). If there exists a (t-1)-flat over GF(s) in PG(m-1, *s*^{n}), then we can replace the corresponding [(st-1)/(s-1)] sn-level columns in *L*_{smn}((*s*^{n})^{(smn-)(sn-1)} by (*s*^{mn}-1)/( *s*^{n}-1) *s*^{t} -level columns and obtain a mixed orthogonal array. Many new mixed orthogonal arrays can be obtained by this procedure. In this paper, we study methods for finding disjoint (t-1)-flats over GF(s) in PG(m-1, *s*^{n}) in order to construct more mixed orthogonal arrays of strength two. In particular, if m and n are relatively prime then we can construct an *L*_{smn}((*s*^{m})^{smn-1/sm-1-i(sn-1)/ (s-1)}( *s*^{n})^{ i(sm-1)/ s-1})
for any 0__<__*i*__<__(*s*^{mn}-1)(*s*-1)/( *s*^{m}-1)( *s*^{n}-1) New orthogonal arrays of sizes 256, 512, and 1024 are obtained by using PG(7,2), PG(8,2), and PG(9,2) respectively.

Mixed orthogonal arrays of strength two and size

KEYWORDS

Finite field; Finite projective geometry; (t-1)-flat over GF(s) in PG(m-1,*s*^{n} ); Geometric orthogonal array; Matrix representation; Minimal polynomial; Orthogonal main-effect plan; Primitive element; Tight.

Finite field; Finite projective geometry; (t-1)-flat over GF(s) in PG(m-1,

Cite this paper

C. Suen, "On a Grouping Method for Constructing Mixed Orthogonal Arrays,"*Open Journal of Statistics*, Vol. 2 No. 2, 2012, pp. 188-197. doi: 10.4236/ojs.2012.22022.

C. Suen, "On a Grouping Method for Constructing Mixed Orthogonal Arrays,"

References

[1] R. C. Bose and K. A. Bush, “Orthogonal Arrays of Strength Two and Three,” The Annals of Mathematical Statistics, Vol. 23, No. 4, 1952, pp. 508-524. doi:10.1214/aoms/1177729331

[2] R. L. Plackett and J. P. Burma, “The Design of Optimum Multifactorial Experiments,” Biometrika, Vol. 33, No. 4, 1946, No. 4, pp. 305-325.

[3] C. R. Rao, “Factorial Experiments Derivable from Combinatorial Arrangements of Arrays,” Journal of Royal Statistical Society (Supplement), Vol. 9, No. 1, 1947, pp. 128-139. doi:10.2307/2983576

[4] C. R. Rao, “Some Combinatorial Problems of Arrays and Applications to Design of Experiments,” In: J. N. Srivastava, Ed., A Survey of Combinatorial Theory, North- Holland, Amsterdam, 1973, pp. 349-359.

[5] J. C. Wang and C. F. J. Wu, “An Approach to the Construction of Asymmetrical Orthogonal Arrays,” Journal of American Statistical Association, Vol. 86, No. 414, 1991, pp. 450-456. doi:10.2307/2290593

[6] C. F. J. Wu, R. C. Zhang and R. Wang, “Construction of Asymmetrical Orthogonal Array of Type OA(sk, sm(sr1)n1 ??? (srt)nt),” Statistica Sinica, Vol. 2, No. 1, 1992, pp. 203- 219.

[7] A. Dey and C. K. Midha, “Construction of Some Asymmetrical Orthogonal Arrays,” Statistics & Probability Letters, Vol. 28, No. 3, 1996, pp. 211-217. doi:10.1016/0167-7152(95)00126-3

[8] Y. S. Zhang, Y. Q. Lu and S. Q. Pang, “Orthogonal Arrays Obtained by Orthogonal Decomposition of Projection Matrices,” Statistica Sinica, Vol. 9, No. 2, 1999, pp. 595-604.

[9] C. Suen and W. F. Kuhfeld, “On the Construction of Mixed Orthogonal Arrays of Strength Two,” Journal of Statistical Planning and Inference, Vol. 133, No. 2, 2005, pp. 555-560. doi:10.1016/j.jspi.2004.03.018

[10] A. S. Hedayat, N. J. A. Sloane and J. Stufken, “Orthogonal Arrays,” Springer, New York, 1999. doi:10.1007/978-1-4612-1478-6

[11] S. Addelman, “Orthogonal Main Effect Plans for Asymmetrical Factorial Experiments,” Technometrics, Vol. 4, No. 1, 1962, pp. 21-46. doi:10.2307/1266170

[12] C. F. J. Wu, “Construction of 2m4n Deigns via a Grouping Scheme,” Annals of Statistics, Vol. 17, No. 4, 1989, pp. 1880-1885. doi:10.1214/aos/1176347399

[13] E. M. Rains, N. J. A. Sloane and J. Stufken, “The Lattice of N-Run Orthogonal Arrays,” Journal of Statistical Planning and Inference, Vol. 102, No. 2, 2002, pp. 477- 500. doi:10.1016/S0378-3758(01)00119-7

[14] C. Suen, A. Das and A. Dey, “On the Construction of Asymmetric Orthogonal Arrays,” Statistica Sinica, Vol. 11, No. 1, 2001, pp. 241-260.

[15] J. W. P. Hirschefld, “Projective Geometries over Finite Fields,” Oxford University Press, Oxford, 1979.

[16] C. Suen and A. Dey, “Construction of Asymmetric Orthogonal Arrays through Finite Geometries,” Journal of Statistical Planning and Inference, Vol. 115, No. 2, 2003, pp. 623-635. doi:10.1016/S0378-3758(02)00165-9

[1] R. C. Bose and K. A. Bush, “Orthogonal Arrays of Strength Two and Three,” The Annals of Mathematical Statistics, Vol. 23, No. 4, 1952, pp. 508-524. doi:10.1214/aoms/1177729331

[2] R. L. Plackett and J. P. Burma, “The Design of Optimum Multifactorial Experiments,” Biometrika, Vol. 33, No. 4, 1946, No. 4, pp. 305-325.

[3] C. R. Rao, “Factorial Experiments Derivable from Combinatorial Arrangements of Arrays,” Journal of Royal Statistical Society (Supplement), Vol. 9, No. 1, 1947, pp. 128-139. doi:10.2307/2983576

[4] C. R. Rao, “Some Combinatorial Problems of Arrays and Applications to Design of Experiments,” In: J. N. Srivastava, Ed., A Survey of Combinatorial Theory, North- Holland, Amsterdam, 1973, pp. 349-359.

[5] J. C. Wang and C. F. J. Wu, “An Approach to the Construction of Asymmetrical Orthogonal Arrays,” Journal of American Statistical Association, Vol. 86, No. 414, 1991, pp. 450-456. doi:10.2307/2290593

[6] C. F. J. Wu, R. C. Zhang and R. Wang, “Construction of Asymmetrical Orthogonal Array of Type OA(sk, sm(sr1)n1 ??? (srt)nt),” Statistica Sinica, Vol. 2, No. 1, 1992, pp. 203- 219.

[7] A. Dey and C. K. Midha, “Construction of Some Asymmetrical Orthogonal Arrays,” Statistics & Probability Letters, Vol. 28, No. 3, 1996, pp. 211-217. doi:10.1016/0167-7152(95)00126-3

[8] Y. S. Zhang, Y. Q. Lu and S. Q. Pang, “Orthogonal Arrays Obtained by Orthogonal Decomposition of Projection Matrices,” Statistica Sinica, Vol. 9, No. 2, 1999, pp. 595-604.

[9] C. Suen and W. F. Kuhfeld, “On the Construction of Mixed Orthogonal Arrays of Strength Two,” Journal of Statistical Planning and Inference, Vol. 133, No. 2, 2005, pp. 555-560. doi:10.1016/j.jspi.2004.03.018

[10] A. S. Hedayat, N. J. A. Sloane and J. Stufken, “Orthogonal Arrays,” Springer, New York, 1999. doi:10.1007/978-1-4612-1478-6

[11] S. Addelman, “Orthogonal Main Effect Plans for Asymmetrical Factorial Experiments,” Technometrics, Vol. 4, No. 1, 1962, pp. 21-46. doi:10.2307/1266170

[12] C. F. J. Wu, “Construction of 2m4n Deigns via a Grouping Scheme,” Annals of Statistics, Vol. 17, No. 4, 1989, pp. 1880-1885. doi:10.1214/aos/1176347399

[13] E. M. Rains, N. J. A. Sloane and J. Stufken, “The Lattice of N-Run Orthogonal Arrays,” Journal of Statistical Planning and Inference, Vol. 102, No. 2, 2002, pp. 477- 500. doi:10.1016/S0378-3758(01)00119-7

[14] C. Suen, A. Das and A. Dey, “On the Construction of Asymmetric Orthogonal Arrays,” Statistica Sinica, Vol. 11, No. 1, 2001, pp. 241-260.

[15] J. W. P. Hirschefld, “Projective Geometries over Finite Fields,” Oxford University Press, Oxford, 1979.

[16] C. Suen and A. Dey, “Construction of Asymmetric Orthogonal Arrays through Finite Geometries,” Journal of Statistical Planning and Inference, Vol. 115, No. 2, 2003, pp. 623-635. doi:10.1016/S0378-3758(02)00165-9