Bayesian size-and-shape regression modelling

Bayesian shape and size regression modellingStatistical shape analysis is a well-known branch of statistics that aims at inferring on objects' shape. The aim of this work is to implement already known methods, with some slight modifications when needed, and use them to make some simulations on synthetic and real datasets. This branch of statistics involves many other topics in mathematics, such as linear algebra, Markov Chains, MCMC algorithms and calculus. \par At first, some examples are presented: many practical applications of statistical shape analysis come from the biological framework and are useful to justify the need for such flexible methods that will be presented during the work. Secondly, the problem of defining a correct mathematical framework for size-and-shape will be assessed, by reporting and reviewing the available literature, with particular care to Mardia’s work. Having set such a framework, it will be discussed how to derive a stochastic model for size and shape, that will be then used to set up a Bayesian framework to perform statistical analysis of size-and-shape configurations. \par The Bayesian inference is done by means of MCMC algorithms implemented in Julia, using a latent variable approach to correctly isolate the size-and-shape information from the rotational one, pointing out and solving any identification issue of the parameters that might arise. This kind of approach has been already explored in literature and allows for a flexible analysis of the problem. More specifically, synthetic datasets will be generated in various configurations to assess the performances of the model in both the two-dimensional and three-dimensional case. The latter case will be treated with particular care, as some slight modifications are proposed with respect to standard work: we model the rotation information by means of Euler angles and explicitly derive the angles' full conditionals, rather than just using a metropolis step, already proposed in previous studies. This is achieved by further developing some known relations, such as the distribution of the rotations being Matrix Fisher, obtaining two Von-Mises angles and a third angle that can be sampled by using specific accept-reject methods. \par The performances of the model are then assessed by means of specific metrics, such as the Riemannian distance, allowing us to make comparisons between estimates and real data. The work shows how the Bayesian approach can be implemented with little assumptions, leading to remarkable results, with affordable computational efforts, thanks to a proper code implementation.