Abstract
This paper describes some of the basic applications of the algebraic theory of canonical decomposition to the analysis of data. The notions of structured data and symmetry studies are discussed and applied to demonstrate their role in well known principles of analysis of variance and their applicability in more general experimental settings.
Similar content being viewed by others
References
M. Viana, Symmetry Studies—an Introduction, IMPA Institute for Pure and Applied Mathematics: Rio de Janeiro, Brazil, 2003.
M. Viana, Lecture Notes on Symmetry Studies, EURANDOM, Eindhoven University of Technology: Eindhoven, The Netherlands, 2005 (2005-027, Electronic version http://www.eurandom.tue.nl).
J.-P. Serre, Linear Representations of Finite Groups, Springer: New York, 1977.
M. Viana, and D. Richards, “Contemporary mathematics—algebraic methods in statistics and probability.” In M. Viana and D. Richards (eds.), vol. 287, American Mathematical Society: Providence, RI, 2001
M. Risse, “Democracy and social choice: A response to Saari,” John F. Kennedy School of Government. Faculty Research Working Papers Series, RWP03-023, 2003.
R. A. Fisher, “The theory of confounding in factorial experiments in relation to the theory of groups,” Annals of Eugenics vol. 11 pp. 341–353, 1942.
R. A. Fisher, “A system of confounding for factors with more than two alternatives, giving completely orthogonal cubes and higher powers,” Annals of Eugenics vol. 12 pp. 283–290, 1945.
P.v.d. Ven, and A. Di Bucchianico, Factorial Designs and Harmonic Analysis on Finite Abelian Groups, EURANDOM: Eindhoven University of Technology, Eindhoven, The Netherlands, 2006 (2006-023, Electronic version http://www.eurandom.tue.nl).
V. Lakshminarayanan, and M. Viana, “Dihedral representations and statistical geometric optics I: Spherocylindrical lenses,” Journal of the Optical Society of America. A, Optics and Image Science vol. 22(11) pp. 2483–2489, 2005.
M. Viana, and V. Lakshminarayanan, “Dihedral representations and statistical geometric optics II: Elementary instruments,” Journal of Modern Optics, vol. 54(4) pp. 473–485, 2007.
E. O’Neill, Introduction to Statistical Optics, Dover: New York, 1963.
M. Viana, and V. Lakshminarayanan, “Data analytic aspects of chirality,” Symmetry vol. 16(14), 2005.
H. Eghbalnia, J. Townsend, and A.H. Assadi, Symmetry, Features and Information. In E. Bayro-Corrochano (ed.) Springer: New York, 2005.
M. Viana, “Symmetry studies and decompositions of entropy,” Entropy vol. 8(2) pp. 88–109, 2006.
C. Campbell, “The refractive group,” Optometry and Vision Science vol. 74 pp. 381–387, 1997.
M. Viana, “Invariance conditions for random curvature models,” Methodology and Computing in Applied Probability vol. 5 pp. 439–453, 2003.
M. Viana, Structured Data—An Introduction to the Study of Symmetry in Applications, Cambridge University Press: New York, 2007, in press.
M. L. Eaton, Multivariate Statistics—A Vector Space Approach, Wiley: New York, 1983.
R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley: New York, 1982.
K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd edn., Cambridge University Press: New York, 2002.
B. Bai, and L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” Journal of the Optical Society of America. A, Optics and Image Science vol. 22(4) pp. 654–661, 2005.
D. C. Harris, and M. D. Bertolucci, Symmetry and Spectroscopy—An Introduction to Vibrational and Electronic Spectroscopy, Oxford University Press: New York, 1978.
W. Faris, “Review of Roland Omnès, the interpretation of quantum mechanics,” Notices of the American Mathematical Society vol. 43(11) pp. 1328–1339, 1996.
R. Omnès, The Interpretation of Quantum Mechanics, Princeton Press: Princeton, NJ, 1994.
P. Cartier, “A mad day’s work: from Grothendiek to Connes and Kontsevich- the evolution of concepts of space and symmetry,” Bulletin (New Series) of the American Mathematical Society vol. 38(4) pp. 389–408, 2001.
W. J. Youden, Statistical Methods for Chemists, Wiley: New York, 1951.
G. James, and M. Liebeck, Representations and Characters of Groups, Cambridge University Press: Cambridge, 1993.
E. Wit, and P. McCullagh, The extendibility of statistical models. In M. Viana, and D. Richards, (eds.) Contemporary Mathematics—Algebraic Methods in Statistics and Probability, vol. 287 pp. 327–340, American Mathematical Society: Providence, RI, 2001.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Viana, M. Symmetry Studies for Data Analysis. Methodol Comput Appl Probab 9, 325–341 (2007). https://doi.org/10.1007/s11009-007-9022-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-007-9022-x