Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space

Frank G. Ball, Ian L. Dryden () and Mousa Golalizadeh
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Frank G. Ball: University of Nottingham
Ian L. Dryden: University of Nottingham
Mousa Golalizadeh: University of Nottingham

Methodology and Computing in Applied Probability, 2008, vol. 10, issue 1, 1-22

Abstract: Abstract We discuss Brownian motion and Ornstein–Uhlenbeck processes specified directly in planar shape space. In particular, we obtain the drift and diffusion coefficients of Brownian motion in terms of Kendall shape variables and Goodall–Mardia polar shape variables. Stochastic differential equations are given and the stationary distributions are obtained. By adding in extra drift to a reference figure, Ornstein–Uhlenbeck processes can be studied, for example with stationary distribution given by the complex Watson distribution. The triangle case is studied in particular detail, and some simulations given. Connections with existing work are made, in particular with the diffusion of Euclidean shape. We explore statistical inference for the parameters in the model with an application to cell shape modelling.

Keywords: Complex projective space; Complex Watson; Computer algebra; Diffusion; Drift; Itô calculus; Itô formula; Procrustes; Riemannian; Shape and size; Sphere; Stochastic differential equation; 60D05; 60J60; 60J65 (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s11009-007-9042-6

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