BAYESIAN ALIGNMENT OF SIMILARITY SHAPES

Enhanced hyperalignment via spatial prior information

Functional alignment between subjects is an important assumption of functional magnetic resonance imaging (fMRI) group-level analysis. However, it is often violated in practice, even after alignment to a standard anatomical template. Hyperalignment, based on sequential Procrustes orthogonal transformations, has been proposed as a method of aligning shared functional information into a common high-dimensional space and thereby improving inter-subject analysis. Though successful, current hyperalignment algorithms have a number of shortcomings, including difficulties interpreting the transformations, a lack of uniqueness of the procedure, and difficulties performing whole-brain analysis. To resolve these issues, we propose the ProMises (Procrustes von Mises-Fisher) model. We reformulate functional alignment as a statistical model and impose a prior distribution on the orthogonal parameters (the von Mises-Fisher distribution). This allows for the embedding of anatomical information into the estimation procedure by penalizing the contribution of spatially distant voxels when creating the shared functional high-dimensional space. Importantly, the transformations, aligned images, and related results are all unique. In addition, the proposed method allows for efficient whole-brain functional alignment. In simulations and application to data from four fMRI studies we find that ProMises improves inter-subject classification in terms of between-subject accuracy and interpretability compared to standard hyperalignment algorithms.

Procrustes Analysis for High-Dimensional Data

The Procrustes-based perturbation model (Goodall in J R Stat Soc Ser B Methodol 53(2):285-321, 1991) allows minimization of the Frobenius distance between matrices by similarity transformation. However, it suffers from non-identifiability, critical interpretation of the transformed matrices, and inapplicability in high-dimensional data. We provide an extension of the perturbation model focused on the high-dimensional data framework, called the ProMises (Procrustes von Mises-Fisher) model. The ill-posed and interpretability problems are solved by imposing a proper prior distribution for the orthogonal matrix parameter (i.e., the von Mises-Fisher distribution) which is a conjugate prior, resulting in a fast estimation process. Furthermore, we present the Efficient ProMises model for the high-dimensional framework, useful in neuroimaging, where the problem has much more than three dimensions. We found a great improvement in functional magnetic resonance imaging connectivity analysis because the ProMises model permits incorporation of topological brain information in the alignment's estimation process.

Bayesian comparison of protein structures using partial Procrustes distance

An important topic in bioinformatics is the protein structure alignment. Some statistical methods have been proposed for this problem, but most of them align two protein structures based on the global geometric information without considering the effect of neighbourhood in the structures. In this paper, we provide a Bayesian model to align protein structures, by considering the effect of both local and global geometric information of protein structures. Local geometric information is incorporated to the model through the partial Procrustes distance of small substructures. These substructures are composed of β-carbon atoms from the side chains. Parameters are estimated using a Markov chain Monte Carlo (MCMC) approach. We evaluate the performance of our model through some simulation studies. Furthermore, we apply our model to a real dataset and assess the accuracy and convergence rate. Results show that our model is much more efficient than previous approaches.

Size and Shape Analysis of Error-Prone Shape Data

We consider the problem of comparing sizes and shapes of objects when landmark data are prone to measurement error. We show that naive implementation of ordinary Procrustes analysis that ignores measurement error can compromise inference. To account for measurement error, we propose the conditional score method for matching configurations, which guarantees consistent inference under mild model assumptions. The effects of measurement error on inference from naive Procrustes analysis and the performance of the proposed method are illustrated via simulation and application in three real data examples. Supplementary materials for this article are available online.

BAYESIAN ALIGNMENT OF SIMILARITY SHAPES

We develop a Bayesian model for the alignment of two point configurations under the full similarity transformations of rotation, translation and scaling. Other work in this area has concentrated on rigid body transformations, where scale information is preserved, motivated by problems involving molecular data; this is known as form analysis. We concentrate on a Bayesian formulation for statistical shape analysis. We generalize the model introduced by Green and Mardia for the pairwise alignment of two unlabeled configurations to full similarity transformations by introducing a scaling factor to the model. The generalization is not straight-forward, since the model needs to be reformulated to give good performance when scaling is included. We illustrate our method on the alignment of rat growth profiles and a novel application to the alignment of protein domains. Here, scaling is applied to secondary structure elements when comparing protein folds; additionally, we find that one global scaling factor is not in general sufficient to model these data and, hence, we develop a model in which multiple scale factors can be included to handle different scalings of shape components.