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Symbolic computation and the diffusion of shapes of triads

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall
Wilfrid S. Kendall*
Affiliation:
University of Strathclyde
*
∗Present address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape.

The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space.

Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.

Keywords

SYMBOLIC MANIPULATIONCOMPUTER ALGEBRAREDUCESTATISTICS OF SHAPECLIFFORD-GREEN THEOREMSEMIMARTINGALEDIFFUSIONBROWNIAN MOTIONORNSTEIN–UHLENBECK PROCESSITO CALCULUSSTOCHASTIC INTEGRAL
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 775 - 797
Copyright
Copyright © Applied Probability Trust 1988 

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References

Ambartzumian, R. V. (1984) Factorization in integral and stochastic geometry. In Stochastic Geometry, Geometric Statistics, Stereology, edited by Ambartzumian, R. V. and Weil, W., Leipzig, Teubner.Google Scholar
Bookstein, F. L. (1986) Size and shape spaces for landmark data in two dimensions. Statistical Science 1, 181–242.Google Scholar
Clifford, P. and Green, N. J. B. (1985) Distances in Gaussian point sets. Math. Proc. Camb. Phil. Soc. 97, 515–524.CrossRefGoogle Scholar
Clifford, P., Green, N. J. B. and Pilling, M. J. (1987) Statistical models of chemicals kinetics in liquids (with discussion). J. R. Statist. Soc. B49, 266–300.Google Scholar
Elworthy, K. D. (1982) Stochastic Differential Equations on Manifolds. Cambridge University Press.CrossRefGoogle Scholar
Fitch, J. P. (1985) Solving algebraic problems with REDUCE. J. Symb. Comp. 1, 211–227.CrossRefGoogle Scholar
Goldstein, M. (1981) Revising previsions; a geometric interpretation. J. R. Statist. Soc B43, 105–130.Google Scholar
Hall, G. R. (1982) Acute triangles in the n-ball. J. Appl. Prob. 19, 712–715.CrossRefGoogle Scholar
Ikeda, N., and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Kendall, D. G. (1977) The diffusion of shape. Adv. Appl. Prob. 9, 428–430.CrossRefGoogle Scholar
Kendall, D. G. (1984) Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.CrossRefGoogle Scholar
Kendall, D. G. and Kendall, W. S. (1980) Alignments in two-dimensional random sets of points. Adv. Appl. Prob. 12, 380–424.CrossRefGoogle Scholar
Mannion, D. (1988) A Markov chain of triangle shapes Adv. Appl. Prob. 20, 348–370.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales, Volume 2. Wiley, Chichester.Google Scholar
Silverman, B. W. and Young, G. A. (1987) The bootstrap: to smooth or not to smooth? Biometrika 74, 469–479.CrossRefGoogle Scholar
Small, C. G. (1983) Characterization of distributions from maximal invariant statistics. Z. Wahrscheinlichkeitsth. 63, 517–527.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S., and Mecke, J. (1987) Stochastic Geometry and Its Applications. Wiley, Chichester.Google Scholar