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The geometry of random drift I. Stochastic distance and diffusion

Published online by Cambridge University Press:  01 July 2016

Peter L. Antonelli  and
Curtis Strobeck
Peter L. Antonelli
Affiliation:
University of Alberta
Curtis Strobeck*
Affiliation:
University of Sussex
*
∗Now at the University of Alberta.
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Abstract

A stochastic distance measure is defined for a general diffusion process on a parameter space X. This distance is defined by where (gij) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij) is the fundamental tensor of the Riemannian space (X, gij), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift with n alleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher to n alleles. The geometry associated with the diffusion equation for random drift with n alleles is that of a part of an (n − 1)-sphere of radius two. We also show that the diffusion equation for random drift is not spherical Brownian motion, although it is approximated by it near the centroid of frequency space.

Keywords

STOCHASTIC DISTANCERANDOM GENETIC DRIFTDIFFUSION PROCESSRIEMANNIAN GEOMETRYGENERALIZED LAPLACIANSPHERICAL BROWNIAN MOTIONCHRISTOFFEL VELOCITY FIELDFISHER'S ANGULAR TRANSFORMATIONMULTINOMIAL DISTRIBUTIONMAHALANOBIS DISTANCE MEASUREEDWARDS AND CAVALLI-SFORZA DISTANCE MEASURE
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 2 , June 1977 , pp. 238 - 249
Copyright
Copyright © Applied Probability Trust 1977 

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