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

形状数据的统计分析与广泛的领域相关，包括生物学，人类学，化学和医学，仅举几例。目前分析形状数据的方法通常侧重于完全观察到的形状。该项目将开发分析仅部分观察到的形状的方法。当1)形状由函数定义，2)必须考虑形状保持变换（例如平移、旋转、缩放、重新参数化等）时，传统的缺失数据技术不适用于形状设置。该项目将把从牛科（羚羊和水牛）的牙齿化石图像中获得的数据作为部分观察到的曲线来形式化和利用这一视角，这些曲线代表了成像物体的轮廓，可能存在测量误差。由此产生的分类鉴定将对生态敏感的牛科动物产生更可靠的估计。这些改进的估计，反过来又为早期人类进化的古环境背景提供了新的见解。该项目的主要目标是开发用于分析部分观察到的形状（即破碎曲线）的统计方法。这一目标将通过两种不同的方法来实现：1)开发计算方法，通过根据从完全观察曲线样本中获得的模板或供体曲线进行匹配和完成，来输入碎片化曲线；2)开发基于贝叶斯模型的方法，使用曲线整体形状的经验先验来估计和分类有噪声的碎片化曲线的形状。这两种方法都将使用黎曼几何工具进行形状分析，以确保形状保持转换的适当不变性，包括平移、缩放（适当时）、旋转和重新参数化。这些方法将随着牛科断裂和完整牙齿化石的分析和分类的应用而得到发展。通过生成古环境条件的代理，该项目旨在促进我们对环境变化与人类进化之间关系的理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.