Advertisement

Mathematical Geology

, Volume 32, Issue 5, pp 561–579 | Cite as

Quasi-Symmetry and Reversible Markov Sequences in Sedimentary Sections

  • W. E. Sharp
  • Thomas Markham
Article
  • 83 Downloads

Abstract

Quasi-symmetry can be defined as a purely mathematical property of a matrix—that is, any matrix whose entries are strictly positive possesses quasi-symmetry if it can be written as a product of a diagonal and a symmetric matrix. A unique inverse solution for a quasi-symmetric matrix is readily obtained when the nondiagonal elements of the symmetric and quasi-symmetric matrix are set equal. Then it is shown that a Markov sequence is reversible if and only if it has a quasi-symmetric tally matrix. Because a properly counted Markov sequence must have marginal homogeneity, a simple chi-square test for symmetry on the tally matrix is sufficient to determine if an observed matrix is symmetrical and hence whether the Markov chain is reversible. Applications to sedimentary sequences are illustrated by the use of classical examples and with cyclothem data to determine if the sequence conforms to a reversible or nonreversible Markov process. Should the tally matrix lack marginal homogeneity, it is likely that a sampling bias was introduced by the counting procedure. However, a chi-square test for symmetry on a direct inverse of the tally matrix can be used to determine if the sedimentary sequence conforms to a reversible or a non-reversible Markov process.

quasi-symmetric matrix cyclothems chi-square test 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. Agresti, A., 1990, Categorical data analysis: John Wiley & Sons, New York, 558 p.Google Scholar
  2. Bhapkar, V. P., 1966, A note on the equivalence of two test criteria for hypotheses in categorical data: American Statistical Association Jour., v. 61, p. 228–235.Google Scholar
  3. Bishop, Y. M. M., Fienberg, S. E., and Holland, P. W., 1975, Discrete multivariate analysis-Theory and practice: MIT Press, Cambridge, MA, 557 p.Google Scholar
  4. Bowker, A. H., 1948, A test for symmetry in contingency tables: American Statistical Association Jour., v. 43, p. 572–574.Google Scholar
  5. Caussinus, H., 1965, Contribution á 1'analyse statistique des tableaux de corrélation: Annales de la Faculté des Sciences de 1'Université de Toulouse, v. 24, p. 77–183.Google Scholar
  6. Gingerich, P. D., 1969, Markov analysis of cyclic alluvial sediments: Jour. Sedimentary Petrology, v. 39, p. 330–332.Google Scholar
  7. Haberman, S. J., 1979, Analysis of qualitative data, v. 2-New developments: Academic Press, New York, p. 369–612.Google Scholar
  8. Harbaugh, J.W., and Bonham-Carter, G., 1970, Computer simulation in geology: JohnWiley & Sons, New York, 575 p.Google Scholar
  9. Le Roux, J. P., 1992, Paleoenvironmental interpretation of tabular sandstones in the Beaufort Group of the Karoo Basin, South Africa: South African Jour. Geology, v. 95, no. 5/6, p. 171–180.Google Scholar
  10. McCullagh, P., 1982, Some applications of quasisymmetry: Biometrika, v. 69, p. 303–308.Google Scholar
  11. Plackett, R. L., 1981, The analysis of categorical data: Charles Griffen, London, 206 p.Google Scholar
  12. Richman, D., and Sharp, W. E., 1990,Amethod for determining the reversibility of a Markov sequence: Math. Geology, v. 22, no. 7, p. 749–761.Google Scholar
  13. Sharp, W. E., and Markham, T., 1999, A simple chi-square test for determining the absence of Markov Cyclicity in sedimentary successions even when quasi-symmetry is present: in Lippard, S. J., Naess, A., and Sinding-Larsen, R., eds., Proceedings of IAMG'99, Trondheim, Norway, p. 251–253.Google Scholar
  14. Stuart, A., 1955, A test for homogeneity of the marginal distributions in a two-way classification: Biometrika, v. 42, p. 412–416.Google Scholar
  15. Vistelius, A. B., and Faas, A. V., 1966, The mode of alternation of strata in certain sedimentary rock sections: Acad. Sci. USSR Trans., Earth Science Sections, v. 167, p. 40–42 (v. 164, p. 629-632, 1965).Google Scholar
  16. Weller, J. M., 1930, Cyclical sedimentation of the Pennsylvanian period and its significance: Jour. Geology, v. 38, no. 2, p. 97–135.Google Scholar

Copyright information

© International Association for Mathematical Geology 2000

Authors and Affiliations

  • W. E. Sharp
    • 1
  • Thomas Markham
    • 2
  1. 1.Department of Geological SciencesUniversity of South CarolinaColumbia
  2. 2.Department of MathematicsUniversity of South CarolinaColumbia

Personalised recommendations