Collaborative Research: Shape-Based Imputation and Estimation of Fragmented, Noisy Curves with Application to the Reconstruction of Fossil Bovid Teeth

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Statistical analysis of shape data is relevant to a wide array of fields including biology, anthropology, chemistry, and medicine, to name just a few. Current methods for analyzing shape data generally focus on shapes that are fully observed. This project will develop methods for analyzing shapes that are only partially observed. Traditional missing data techniques are not applicable in the shape setting when 1) shapes are defined by functions and 2) shape-preserving transformations (e.g. translation, rotation, scaling, re-parameterization, etc.) must be accounted for. This project will formalize and harness this perspective for data obtained from fragmentary fossil tooth images of the Family Bovidae (antelopes and buffaloes) as partially observed curves, representing the outlines of imaged objects, possibly with measurement error. The resulting taxonomic identifications will generate more robust estimates of the ecologically sensitive bovids. These improved estimates, in turn, afford novel insight into the paleoenvironmental context of early human evolution. The main goal of this project is to develop statistical methods for the analysis of partially observed shapes (i.e., fragmented curves). This goal will be accomplished via two different approaches: 1) developing computational methods for imputing a fragmented curve by matching and completing it based on a template or donor curve obtained from a sample of fully observed curves, and 2) developing Bayesian model-based methods for estimation and classification of the shape of a noisy, fragmented curve using an empirical prior on the overall shape of the curve. Both approaches will be built using Riemannian geometric tools for shape analysis that ensure proper invariance to shape preserving transformations including translation, scaling (when appropriate), rotation, and re-parameterization. These methods will be developed with the motivating application of the analysis and taxonomic classification of fractured and complete fossil teeth from the Family Bovidae. By generating a proxy for paleoenvironmental conditions, the project seeks to advance our understanding of the relationship between environmental change and hominin evolution.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.