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A diffusion model for bookstein triangle shape

Part of: Stochastic analysis Multivariate analysis Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
∗ Postal address: Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: [email protected]
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Abstract

A stochastic dynamical context is developed for Bookstein's shape theory. It is shown how Bookstein's shape space for planar triangles arises naturally when the landmarks are moved around by a special Brownian motion on the general linear group of invertible (2×2) real matrices. Asymptotics for the Brownian transition density are used to suggest an exponential family of distributions, which is analogous to the von Mises-Fisher spherical distribution and which has already been studied by J. K. Jensen. The computer algebra implementation Itovsn3 (W. S. Kendall) of stochastic calculus is used to perform the calculations (some of which actually date back to work by Dyson on eigenvalues of random matrices and by Dynkin on Brownian motion on ellipsoids). An interesting feature of these calculations is that they include the first application (to the author's knowledge) of the Gröbner basis algorithm in a stochastic calculus context.

Keywords

Bookstein shapeBrownian motioncomputer algebraD. G. Kendall shapevon Mises-Fisher distributionGröbner basis algorithmholomorphic correspondencehyperbolic planemean shapeMinakshisundaram-Pleijel recursion formulaestatistical shapesymbolic Itô calculus

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 317 - 334
Copyright
Copyright © Applied Probability Trust 1998 

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