Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-27T20:15:19.179Z Has data issue: false hasContentIssue false
  • English
  • Français

The geometry of random drift II. The symmetry of random genetic drift

Published online by Cambridge University Press:  01 July 2016

Peter L. Antonelli ,
Jared Chapin ,
G. Mark Lathrop  and
Kenneth Morgan
Peter L. Antonelli
Affiliation:
University of Alberta
Jared Chapin
Affiliation:
University of Alberta
G. Mark Lathrop
Affiliation:
University of Alberta
Kenneth Morgan
Affiliation:
University of Alberta
Get access Rights & Permissions [Opens in a new window]

Abstract

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher–Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

Keywords

MAXIMUM LIKELIHOOD ESTIMATION OF EVOLUTIONARY TREESYULE PROCESSRANDOM GENETIC DRIFTGENERALIZED DISTANCE MEASURESDIFFUSION APPROXIMATIONSTANDARD BROWNIAN MOTIONSPHERICAL BROWNIAN MOTIONRIEMANNIAN GEOMETRYFINITE MARKOV CHAINBHATTACHARYYA-FISHER TRANSFORMATIONCHRISTOFFEL VELOCITY FIELD
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 2 , June 1977 , pp. 250 - 259
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antonelli, P. and Strobeck, C. (1977) The geometry of random drift I. Stochastic distance and diffusion. Adv. Appl. Prob. 9.CrossRefGoogle Scholar
Bhattacharyya, A. (1946) On a measure of divergence between two multinomial populations. Sankhyā 7, 401–406.Google Scholar
Edwards, A. W. F. (1971) Distances between populations on the basis of gene frequencies. Biometrics 27, 873–881.Google Scholar
Eisenhart, L. P. (1926) Riemannian Geometry. Princeton University Press, Princeton, N. J. Google Scholar
Ewens, W. J. (1969) Population Genetics. Methuen, London.Google Scholar
Felsenstein, J. (1973) Maximum-likelihood estimation of evolutionary trees from continuous characters. Amer. J. Hum. Genet. 25, 471–492.Google Scholar
Fisher, R. A. (1930) The distribution of gene ratios for rare mutations. Proc. R. Soc. Edinburgh 60, 204–219.Google Scholar
Jacquard, A. (1974) The Genetic Structure of Populations. Springer-Verlag, New York.Google Scholar
Malyutov, M. B., Passekov, V. F. and Rychov, Y. G. (1972) In The Assessment of Population Affinities in Man, ed. Weiner, J. S. and Huizinga, J. Clarendon Press, Oxford.Google Scholar
Morton, N. E. (ed.) (1973) Genetic Structure of Populations. University Press of Hawaii, Honolulu.Google Scholar