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，旨在开发基因组学和生命过程流行病学中重要的公共卫生应用。在全基因组表达和DNA甲基化研究中，定位与临床结果相关的活性基因是很有意义的，例如，利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，这是乳腺癌肿瘤的重要预后指标。在这样的研究中，基因在染色体上的表达谱可以被认为是一个功能预测因子，与临床结果相关的基因是通过其碱基对沿染色体的位置来识别的。该项目的主要目的是开发新的统计推断方法，以发现这些遗传位点，从而确定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发展。虽然功能数据分析在过去十年中已经达到了成熟的发展阶段，但当目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）的情况时，或者在轨迹观察中只有稀疏的时间分辨率的情况下，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具有重要意义。研究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速度轨迹，将作为新方法的应用而发展。要解决的第一个具体问题是表明，在涉及具有分形特征的轨迹的系统中，学习速率是由Hurst参数（即自相似尺度指数）决定的，并表明一种类型的自举学习可以适应分形行为的全部范围。第二个需要解决的具体问题是开发一种用于生成具有分形特性的轨迹（例如，增长率曲线）缺失值的输入方法，并找到一种基于输入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