Linear models for statistical shape analysis based on parametrized closed curves

Abstract

We develop a simple, yet powerful, technique based on linear regression models of parametrized closed curves which induces a probability distribution on the planar shape space. Such parametrization is driven by control points which can be estimated from the data. Our proposal is capable to infer about the mean shape, to predict the shape of an object at an unobserved location, and, while doing so, to consider the effect of predictors on the shape. In particular, the model is able to detect possible differences across the levels of the predictor, thus also applicable for two-sample tests. A simple MCMC algorithm for Bayesian inference is also presented and tested with simulated and real datasets. Supplementary material is available online.

Appendix

Algorithm 1

The Gibbs sampling for model (4) when R is fixed is given by the following full conditional distributions

• (1)
  $$\begin{aligned} (\theta _i\mid \ldots ) \sim N_d(\tilde{\theta }_i, \tilde{\Sigma }_{\theta }), \end{aligned}$$

  where $\tilde{\Sigma }_{\theta }=[nZ^T\Omega ^{-1}Z+\Sigma _{\delta }^{-1}]^{-1}$ and $\tilde{\theta _i}=\tilde{\Sigma }_{\theta }[Z^T\Omega ^{-1}Y_i+\Sigma _{\delta }^{-1}\varvec{\theta }X_i].$

• (2)
  $$\begin{aligned} (\Sigma _{j}\mid \ldots ) \sim IW_{2}(\delta _{0}+n, A_{0}+\sum _{i=1}^{n}(Y_{ij}-Z_{ij}\theta _{i})(Y_{ij}-Z_{ij}\theta _{i})^{T}), \end{aligned}$$

  where $Y_{ij}=(y_{j,\textsf {x}}, y_{j,\textsf {y}})^{T}$, $j=1,\ldots ,k$ and $Z_{ij}$ is the matrix defined in (3).

• (3)
  $$\begin{aligned} (\theta _j\mid \ldots ) \sim N(\tilde{\theta }_j,\tilde{\Sigma }_{\theta _j}), \quad j=1,\ldots ,q \end{aligned}$$

  where $\tilde{\Sigma }_{\theta _j}=[\sum _{i=1}^n x_{ij}^2\Sigma _{\delta }^{-1}+ B_0^{-1}]^{-1}$ and

  $$\begin{aligned} \tilde{\theta }_j=\tilde{\Sigma }_{\theta _j}\left[ \sum _{i=1}^n x_{ij}\Sigma _{\delta }^{-1} \theta _i-\sum _{s\ne j}\sum _{i=1}^n x_{ij}x_{is}\Sigma _{\delta }^{-1}\theta _{s}+B_0^{-1}\theta _0 \right] \end{aligned}$$
• (4)
  $$\begin{aligned} (\Sigma _{\delta }\mid \ldots ) \sim IW(\nu +n, D_0+\sum _{i=1}^n( \theta _i-\varvec{\theta }X_i)(\theta _i-\varvec{\theta }X_i)^T). \end{aligned}$$