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Principal Component Analysis for Riemannian Functional Data and Bayes Classification

机译：黎曼函数数据和贝叶斯分类的主成分分析

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摘要

Functional data, or samples of smooth random functions observed over a continuum, have drawn extensive interest over the past 20 years. Classical linear functional data have been modeled in infinite-dimensional Hilbert spaces, where the infinite-dimensionality calls for dimension reduction techniques in theoretical and practical analysis. The infinite-dimensionality at the same time may offer information that benefits the classification tasks. Modern functional data often have more complex structural properties in addition to the infinite dimensionality, such as nonlinear geometrical constraints, which makes the data objects no longer live in a Hilbert space. Developing methods that respect and exploit the underlying data structure is key to effective analysis of such complex functional data.;We consider an intrinsic Riemannian functional principal component analysis (RFPCA) for smooth Riemannian manifold-valued functional data. RFPCA is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to linear tangent spaces around the Frechet mean function. One can then obtain low-dimensional representations on the tangent spaces through a classical functional principal component analysis (FPCA), and on the original manifold by pulling back the tangent space representations through Riemannian exponential maps. Theoretical analysis yields error bounds for the tangent-space approximation and a uniform central limit theorem for the mean function, which implies root-n uniform convergence rates for this and other model components. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the unit sphere.;Constructing Bayes classifiers for infinite dimensional functional data is difficult due to the fact that probability density functions do not exist for functional data. We approach this problem by considering density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified. The density ratios are then factored into density ratios of individual projection scores, reducing the classification problem to obtaining a series of one-dimensional nonparametric density estimates. A study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as perfect classification. The proposed classifiers also perform favorably in finite sample applications including spectral data, imaging data for attention deficit hyperactivity disorder patients, yeast gene expression data, as well as in simulations.
机译：在过去的20年中，功能数据或连续观察到的平滑随机函数样本引起了广泛的关注。经典的线性功能数据已在无穷维希尔伯特空间中建模，无穷维要求在理论和实践分析中采用降维技术。同时，无限维可以提供有益于分类任务的信息。除了无穷维之外，现代功能数据通常还具有更复杂的结构属性，例如非线性几何约束，这使得数据对象不再生活在希尔伯特空间中。开发尊重和利用基础数据结构的方法是有效分析此类复杂功能数据的关键。我们考虑了平滑黎曼流形值功能数据的内在黎曼函数主成分分析（RFPCA）。通过首先通过Riemannian对数映射将流形值数据映射到Frechet均值函数周围的线性切线空间来执行RFPCA。然后，可以通过经典的功能主成分分析（FPCA）在切空间上获得低维表示，而在原始流形上可以通过Riemannian指数图拉回切空间表示来获得低维表示。理论分析得出切线空间逼近的误差界和均值函数的统一中心极限定理，这意味着该模型和其他模型组件的根n均匀收敛速度。我们的应用程序包括一个用于分析纵向成分数据的新颖框架，该框架通过将纵向成分数据映射到单位球面上的轨迹来实现。;由于不存在概率密度函数，因此难以构造用于无限维功能数据的贝叶斯分类器功能数据。我们通过考虑对要分类的组通用的本征函数序列上的投影的密度比来解决此问题。然后将密度比率计入各个投影得分的密度比率中，从而减少了分类问题，从而获得了一系列一维非参数密度估计。对建议的分类器在大样本限制下的渐近行为的研究表明，在某些条件下，错误分类率收敛为零，这种现象已被称为完美分类。拟议的分类器在有限的样本应用中也表现出色，包括光谱数据，注意力缺陷多动障碍患者的成像数据，酵母基因表达数据以及模拟。

著录项

  • 作者

    Dai, Xiongtao.;

  • 作者单位

    University of California, Davis.;

  • 授予单位 University of California, Davis.;
  • 学科 Statistics.
  • 学位 Ph.D.
  • 年度 2018
  • 页码 99 p.
  • 总页数 99
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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