Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space |
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Authors: | Frank G Ball Ian L Dryden Mousa Golalizadeh |
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Institution: | (1) Division of Statistics, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK |
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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.
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Keywords: | Complex projective space Complex Watson Computer algebra Diffusion Drift It? calculus It? formula Procrustes Riemannian Shape and size Sphere Stochastic differential equation |
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