This paper is divided into two parts. The first part is devoted to a study of absorption in a genetic population model and the second part to introducing a new method of matrix inversion.
The author begins Part I by presenting the haploid population model under consideration, a constant-size population in which mutation is absent. Using the approach or Malecot, the problem is formulated in terms of an absorbing Markov chain. A brief review or earlier work by Maleoot, Fisher, Wright, and Feller is given before proceeding to the study of the distribution or time taken for the population to consist of only one gene type.
The first method used by the author to determine the probability function for first passage time requires the eigenvectors or the transition matrix. A blocking transformation is used to show that only the eigenvectors corresponding to the even eigenvalues are needed to find the distributions. It is also shown that these eigenvectors are symmetric. In order to simplify computation of the needed vectors, a transformation which triangularizes the transition matrix is presented. This transformation also leads to a simple derivation of the eigenvalues and is used to derive the distributions for populations of size two through nine. Although the general solution is not obtained, expressions for the first seven eigenvectors are listed along with general results concerning the triangularized form.
Next, the author attacks the problem by developing the theory or moments or the distribution and applies this theory to determine the means and variances. These are tabulated for population sizes 2(1)9 and 10(10)50. Then, by assuming a large population size, a diffusion process continuous in space and time is used to approximate the Markov chain. Expressions for the mean and variance are derived and are tabulated for the same values as in the moment approach. A comparison of the entries of these two tables gives support to the theory that the diffusion process works as the population size becomes infinite but with the gene frequency ratio kept fixed. Also, it demonstrates that diffusion theory works better for a gene frequency ratio near one-half than for values near zero or one and further that this theory consistently over-estimates the absorption time moments. The author notes, however, that the point of primary importance is the indication that diffusion theory works even with fixed gene number, that is, even in the tails of the distribution. He also states that the percentage error decreases at a faster rate for fixed gene frequency ratio than for fixed gene number and concludes the first part by suggesting a rate of decrease of percentage error for constant gene frequency.
In Part II, the author begins by noting the importance of the inverse matrix in Markov chain theory. The new inversion technique given is basically one which generates the inverse with a minimum number of determinantal operations. The method is thoroughly described and it is applied to some problems encountered in the first part of the paper. The paper ends with a comparison of the new method with well-known techniques presently in use which shows that the method proposed is especially efficient for small order matrices. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/106358 |
| Date | January 1962 |
| Creators | Knox, Samuel Roscoe |
| Contributors | Statistics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | 112 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20377022 |
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