A Geometric Approach to Pairwise Bayesian Alignment of Functional Data Using Importance Sampling

We present a Bayesian model for pairwise nonlinear registration of functional data. We use the Riemannian geometry of the space of warping functions to define appropriate prior distributions and sample from the posterior using importance sampling. A simple square-root transformation is used to simplify the geometry of the space of warping functions, which allows for computation of sample statistics, such as the mean and median, and a fast implementation of a $k$-means clustering algorithm. These tools allow for efficient posterior inference, where multiple modes of the posterior distribution corresponding to multiple plausible alignments of the given functions are found. We also show pointwise $95\%$ credible intervals to assess the uncertainty of the alignment in different clusters. We validate this model using simulations and present multiple examples on real data from different application domains including biometrics and medicine

Distribution on Warp Maps for Alignment of Open and Closed Curves

Alignment of curve data is an integral part of their statistical analysis, and can be achieved using model- or optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in $\mathbb{R}^d, d=1,2,3$, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of $[0,1]$ and the unit circle $\mathbb{S}^1$ that is (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on $[0,1]$, the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided

Numerical inversion of SRNFs for efficient elastic shape analysis of star-shaped objects.

The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics. This approach is based on defining new mathematical representations of parameterized surfaces, including the square root normal field (SRNF), and then using the L2 norm to compare their shapes. Past work is based on using the pullback of the L2 metric to the space of surfaces, performing statistical analysis under this induced Riemannian metric. However, if one can estimate the inverse of the SRNF mapping, even approximately, a very efficient framework results: the surfaces, represented by their SRNFs, can be efficiently analyzed using standard Euclidean tools, and only the final results need be mapped back to the surface space. Here we describe a procedure for inverting SRNF maps of star-shaped surfaces, a special case for which analytic results can be obtained. We test our method via the classification of 34 cases of ADHD (Attention Deficit Hyperactivity Disorder), plus controls, in the Detroit Fetal Alcohol and Drug Exposure Cohort study. We obtain state-of-the-art results

Bayesian sensitivity analysis with the Fisher–Rao metric

We propose a geometric framework to assess sensitivity of Bayesian procedures to modelling assumptions based on the nonparametric Fisher–Rao metric. While the framework is general, the focus of this article is on assessing local and global robustness in Bayesian procedures with respect to perturbations of the likelihood and prior, and on the identification of influential observations. The approach is based on a square-root representation of densities, which enables analytical computation of geodesic paths and distances, facilitating the definition of naturally calibrated local and global discrepancy measures. An important feature of our approach is the definition of a geometric ϵ-contamination class of sampling distributions and priors via intrinsic analysis on the space of probability density functions. We demonstrate the applicability of our framework to generalized mixed-effects models and to directional and shape data

Spatially Penalised Registration of Multivariate Functional Data

Registration of multivariate functional data involves handling of both cross-component and cross-observation phase variations. Allowing for the two phase variations to be modelled as general diffeomorphic time warpings, in this work we focus on the hitherto unconsidered setting where phase variation of the component functions are spatially correlated. We propose an algorithm to optimize a metric-based objective function for registration with a novel penalty term that incorporates the spatial correlation between the component phase variations through a kriging estimate of an appropriate phase random field. The penalty term encourages the overall phase at a particular location to be similar to the spatially weighted average phase in its neighbourhood, and thus engenders a regularization that prevents over-alignment. Utility of the registration method, and its superior performance compared to methods that fail to account for the spatial correlation, is demonstrated through performance on simulated examples and two multivariate functional datasets pertaining to EEG signals and ozone concentrations. The generality of the framework opens up the possibility for extension to settings involving different forms of correlation between the component functions and their phases