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Quasi-symmetry and Reversible Markov Sequences in
Sedimentary Sections.1
by
W.E. Sharp 2 and Thomas Markham 3
1 Received __________________; accepted_____________________.
2 Department of Geological Sciences, University of South Carolina,
Columbia, SC 29208 U.S.A.; e-mail: sharp@sc.edu
3Department of Mathematics, University of South Carolina, Columbia,
SC 29208 U.S.A.
Suggested Running Head: Quasi-symmetry
Corresponding Author:
W.E. Sharp
Department of Geological Sciences
University of South Carolina
Columbia, SC 29208 U.S.A.
Phone + 1 803 777 6929
fax + 1 803 777 6610
e-mail sharp@sc.edu
Abstract and Key Words
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. It ...........
..... determine if the sedimentary sequence conforms to a reversible or a
non-reversible Markov process.
KEY WORDS: Quasi-symmetric matrix, cyclothems, Chi-square test
Body of Text
INTRODUCTION
In the study of square-contingency tables cross classified by the same
categories, quasi-symmetry has routinely been
considered as the model to use when the observed table lacks complete symmetry
(e.g. Bishop, Fienberg and Holland, 1975, p.284).
In the alternative view to be presented below, quasi-symmetry will be
considered purely as a mathematical property
of a matrix.
Chi-Square Test
Theorem 1 and Theorem 2 suggest that when the entries in a tally matrix are
subject to random experimental error,
that a Chi-square test may be used to test for quasi-symmetry and hence for
Markov reversibility.
In the special case where the tally matrix A is symmetric, then A will
possess both marginal homogeneity and
quasi-symmetry. This is ......
It had originally been hoped to simply follow the procedures described by
Vistelius and others (1983); however
Example
The following example illustrates the application of Theorem 1. Suppose one has
the observed matrix:
CONCLUSIONS
If counts are made and tabulated of the transitions among various discrete
lithologies within a carefully measured
stratigraphic section, the resulting tally matrix will have matching row and
column sums which are identical or nearly
identical. Such a transition count matrix is
ACKNOWLEDGEMENTS
We want to thank .............. of the Department of Statistics for pointing
out that a Markov process is ergodic and if
counted
REFERENCES
Agresti, A., 1990, Categorical Data Analysis: Wiley, New York, 558 p.
Bhapkar, V.P., 1966, A note on the equivalence of two test criteria for
hypotheses in categorical data:
American Statistical Association Journal, v. 61,
p. 228-235.
Bishop, Y.M.M., Fienberg, S.E. and Holland, P.W., 1975, Discrete Multivariate
Analysis - Theory and Practice:
MIT Press. Cambridge, Massachusetts, 557
p.
de Wijs, H.J. 1974, Method of successive differences applied to mine sampling,
in Jones, M.J., ed.,
Geological, mining and metallurgical sampling:
Institution of Mining and Metallurgy, London, p. 86-89.
Richman, D. and Sharp, W.E., 1990, A method for determining the reversibility of
a Markov sequence:
Mathematical Geology, v. 22, no. 7, p. 749-761.
Vistelius, A.B., Agterberg, J.P., Divi, S.R. and Hogarth, D.D., 1983, A
stochastic model for the crystallization
and textural analysis of a fine-grained
granitic stock near Meech Lake, Gatineau park, Quebec:
Geological Survey of Canada, Paper 81-21, 62 p.
APPENDIX
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