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Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space

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  • Frank G. Ball

    (University of Nottingham)

  • Ian L. Dryden

    (University of Nottingham)

  • Mousa Golalizadeh

    (University of Nottingham)

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.

Suggested Citation

  • Frank G. Ball & Ian L. Dryden & Mousa Golalizadeh, 2008. "Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 1-22, March.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:1:d:10.1007_s11009-007-9042-6
    DOI: 10.1007/s11009-007-9042-6
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    References listed on IDEAS

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    1. Goodall, Colin & Mardia, Kanti V., 1992. "The noncentral Bartlett decompositions and shape densities," Journal of Multivariate Analysis, Elsevier, vol. 40(1), pages 94-108, January.
    2. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    3. Amaral, G.J.A. & Dryden, I.L. & Wood, Andrew T.A., 2007. "Pivotal Bootstrap Methods for k-Sample Problems in Directional Statistics and Shape Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 695-707, June.
    4. K. V. Mardia & I. L. Dryden, 1999. "The complex Watson distribution and shape analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(4), pages 913-926.
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