Point Impact and Sparsity in Functional Data Analysis.

描述（由申请人提供）：这是一个开发功能数据分析新方法的项目，旨在针对基因组学和生命历程流行病学中的重要公共卫生应用。在全基因组表达和 DNA 甲基化研究中，人们感兴趣的是定位显示与临床结果相关的活性的基因，例如，使用乳腺癌患者肿瘤的基因表达谱来预测雌激素受体蛋白浓度，这是乳腺肿瘤的重要预后标志物。在此类研究中，染色体上的基因表达谱可以被视为功能预测因子，并且与临床结果相关的基因通过其沿染色体的碱基对位置来识别。该项目的主要目标是开发新的统计推断方法来寻找此类遗传位点，从而识别出可能对诊断和治疗有用的染色体区域。尽管有大量关于基因表达数据的统计文献，但它几乎完全涉及用于检测差异表达基因是否存在的多种测试程序，并且基于表达谱（解释为功能预测因子）定位此类基因的统计方法尚未得到很好的开发。尽管功能数据分析在过去十年中已达到成熟的发展阶段，但当当前可用的方法应用于涉及具有点影响效应（如基因表达）的功能预测因子（或轨迹）的情况下，或者在轨迹观察中仅存在稀疏时间分辨率的情况下时，可能会出现严重的问题。该项目的主要目标是利用轨迹中的分形行为来改进功能数据分析中的统计学习方法。该项目将对理解具有分形行为的各种复杂自适应系统产生重要影响。与心血管风险结果相关的卡路里摄入轨迹和 DNA 甲基化谱以及与神经心理学结果相关的生长率轨迹的研究将作为新方法的应用而发展。要解决的第一个具体问题是证明涉及具有分形特征的轨迹的系统中的学习率由赫斯特参数（即自相似性缩放的指数）决定，并证明一种引导学习可以适应全部的分形行为。要解决的第二个具体问题是开发一种插补方法，用于生成具有分形特性的轨迹（例如增长率曲线）的缺失值，并找到一种基于插补轨迹进行函数回归建模的方法。 1

DESCRIPTION (provided by applicant): This is a project to develop new methods of functional data analysis directed towards important public health applications in genomics and life course epidemiology. In genome-wide expression and DNA methylation studies, it is of interest to locate genes showing activity that is associated with clinical outcomes, e.g., to use gene expression profiles from the tumors of breast cancer patients to predict estrogen receptor protein concentration, an important prognostic marker for breast tumors. In such studies, the gene expression profile across a chromosome can be regarded a functional predictor, and a gene associated with the clinical outcome is identified by its base pair position along the chromosome. The key aim of the project is to develop new methods of statisti- cal inference for finding such genetic loci, leading to the identification of chromosomal regions that are potentially useful for diagnosis and therapy. Although there is extensive statistical literature on gene expression data, it is almost exclusively concerned with multiple testing procedures for detecting the presence of differentially expressed genes, and statistical methods for locating such genes based on expression profiles (interpreted as functional predictors) are not well developed. Although functional data analysis has reached a mature stage of development over the last ten years, serious problems can arise when the currently available methods are applied in situations involving functional predictors (or trajectories) that have point impact effects (as with gene expression), or in situations in which there is only sparse temporal resolution in the observation of the trajectories. The broad objectives of the project are to exploit fractal behavior in the trajectories to improve statistical learning methodology in functional data analysis. The project will have important implications for understanding a wide variety of complex adaptive systems having fractal behavior. Studies of calorie-intake trajectories and DNA methylation profiles related to cardiovascular risk outcomes, and growth rate trajectories related to neuropsychological outcomes, will be developed as applications of the new methodology. The first specific problem to be addressed is to show that the rates of learning in systems involving trajectories with fractal characteristics are determined by the Hurst parameter (i.e., the exponent of self-similarity scaling) and to show that a type of bootstrap learning can adapt to the full range of fractal behavior. The second specific problem to be addressed is to develop an imputation method for generating missing values of trajectories that have fractal properties (e.g., growth rate curves), and to find a way to carry out functional regression modeling based on the imputed trajectories. 1