Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials

Author(s)
Peter Zizler

Affiliation(s)

Department of Mathematics/Physics and Engineering, Mount Royal University, Calgary, Canada.

Department of Mathematics/Physics and Engineering, Mount Royal University, Calgary, Canada.

ABSTRACT

Let *G* be a non-abelian
group and let l^{2}*(G)* be a
finite dimensional Hilbert space of all complex valued functions for which the
elements of *G* form the (standard) orthonormal basis. In
our paper we prove results concerning *G*-decorrelated
decompositions of functions in *l*^{2}*(G)*. These *G*-decorrelated
decompositions are obtained using the *G*-convolution either by
the irreducible characters of the group *G* or by an orthogonal
projection onto the matrix entries of the irreducible representations of the
group *G*. Applications of these *G*-decorrelated
decompositions are given to crossover designs in clinical trials, in particular
the William’s 6×3 design with 3 treatments. In our example, the
underlying group is the symmetric group *S*_{3}.

KEYWORDS

Non-Abelian Fourier Transform, Group Algebra, Irreducible Representation, Irreducible Character,*G*-Circulant Matrix,
*G*-Decorrelated Decomposition,
Crossover Designs in Clinical Trials

Non-Abelian Fourier Transform, Group Algebra, Irreducible Representation, Irreducible Character,

Cite this paper

Zizler, P. (2014) Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials.*Applied Mathematics*, **5**, 917-927. doi: 10.4236/am.2014.56087.

Zizler, P. (2014) Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials.

References

[1] Davis, P.J. (1979) Circulant Matrices. John Wiley and Sons.

[2] Dummit, D.S. and Foote, R.N. (1999) Abstract Algebra. John Wiley and Sons.

[3] Morrison, K.E. (1998) A Generalization of Circulant Matrices for Non-Abelian Groups. Private Communication.

http://www.math.uh.edu/molshan/ftp/pub/proceed_cd.pdf

[4] Zizler, P. (2013) On Spectral Properties of Group Circulant Matrices. Pan American Mathematical Journal, 23, 1-23.

[5] Foote, R., Mirchandani, G., Rockmore, D., Healy, D. and Olson, T. (2000) A Wreath Product Group Approach to Signal and Image Processing: Part I—Multiresolution Analysis. IEEE Transaction in Signal Processing, 48, 102-132.

http://dx.doi.org/10.1109/78.815483

[6] Healy, D., Mirchandani, G., Olson, T. and Rockmore, D. (1996) Wreath Products for Image Processing. Proceedings of the ICASSP, 6, 3582-3586.

[7] Malm, E.J. (2005) Decimation-in-frequency Fast Fourier Transforms for the Symmetric Group. Thesis, Department of Mathematics, Harvey Mudd College.

[8] Maslen, D. and Rockmore, D. (1997) Generalized FFTs—A Survey of Some Recent Results. In: Finkelstein, L. and Kantor, W., Eds., Proceedings of the DIMACS Workshop on Groups and Computation, 183-237.

[9] Rockmore, D.N. (2003) Recent Progress and Applications in Group FFTs. NATO Advanced Study Institute on Computational Noncommutative Algebra and Applications.

[10] Sagan, B.E. (1991) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Wadsworth & Brooks/Cole, Pacific Grove.

[11] Terras, A. (1999) Fourier Analysis on Finite Groups and Applications. Cambridge.

[12] Stankovic, R.S., Moraga, C. and Astola, J.T. (2005) Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design. IEEE Press, John Wiley and Sons.

[13] Karpovsky, M.G. (1977) Fast Fourier Transforms on finite Non-Abelian Groups. IEEE Transaction on Computers, C-26, 1028-1030.

http://dx.doi.org/10.1109/TC.1977.1674739

[14] Senn, S. (2002) Cross-Over Trials in Clinical Research. 2nd Edition, Wiley.

http://dx.doi.org/10.1002/0470854596

[15] Chow, S.-C. and Liu, J.-P. (2013) Design and Analysis of Clinical Trials: Concepts and Methodologies. Wiley.

[1] Davis, P.J. (1979) Circulant Matrices. John Wiley and Sons.

[2] Dummit, D.S. and Foote, R.N. (1999) Abstract Algebra. John Wiley and Sons.

[3] Morrison, K.E. (1998) A Generalization of Circulant Matrices for Non-Abelian Groups. Private Communication.

http://www.math.uh.edu/molshan/ftp/pub/proceed_cd.pdf

[4] Zizler, P. (2013) On Spectral Properties of Group Circulant Matrices. Pan American Mathematical Journal, 23, 1-23.

[5] Foote, R., Mirchandani, G., Rockmore, D., Healy, D. and Olson, T. (2000) A Wreath Product Group Approach to Signal and Image Processing: Part I—Multiresolution Analysis. IEEE Transaction in Signal Processing, 48, 102-132.

http://dx.doi.org/10.1109/78.815483

[6] Healy, D., Mirchandani, G., Olson, T. and Rockmore, D. (1996) Wreath Products for Image Processing. Proceedings of the ICASSP, 6, 3582-3586.

[7] Malm, E.J. (2005) Decimation-in-frequency Fast Fourier Transforms for the Symmetric Group. Thesis, Department of Mathematics, Harvey Mudd College.

[8] Maslen, D. and Rockmore, D. (1997) Generalized FFTs—A Survey of Some Recent Results. In: Finkelstein, L. and Kantor, W., Eds., Proceedings of the DIMACS Workshop on Groups and Computation, 183-237.

[9] Rockmore, D.N. (2003) Recent Progress and Applications in Group FFTs. NATO Advanced Study Institute on Computational Noncommutative Algebra and Applications.

[10] Sagan, B.E. (1991) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Wadsworth & Brooks/Cole, Pacific Grove.

[11] Terras, A. (1999) Fourier Analysis on Finite Groups and Applications. Cambridge.

[12] Stankovic, R.S., Moraga, C. and Astola, J.T. (2005) Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design. IEEE Press, John Wiley and Sons.

[13] Karpovsky, M.G. (1977) Fast Fourier Transforms on finite Non-Abelian Groups. IEEE Transaction on Computers, C-26, 1028-1030.

http://dx.doi.org/10.1109/TC.1977.1674739

[14] Senn, S. (2002) Cross-Over Trials in Clinical Research. 2nd Edition, Wiley.

http://dx.doi.org/10.1002/0470854596

[15] Chow, S.-C. and Liu, J.-P. (2013) Design and Analysis of Clinical Trials: Concepts and Methodologies. Wiley.