# Quasi-Symmetry and Reversible Markov Sequences in Sedimentary Sections

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## 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.

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