Statistical Modelling and Dimension Reduction for Functional Data

9803627Jane-Ling wang曲线或功能数据越来越重要，因为高性能计算设备的增强，以及人们越来越意识到这些数据需要特殊的分析方法。这些数据出现在纵向研究和老龄化研究中。这项研究涉及发展新的非参数方法来分析这些数据。重点是涉及多变量技术和平滑方法的程序。特别感兴趣的是曲线数据的方差分析和回归模型。重点介绍了将曲线数据视为一个或几个随机过程的概括的方法。给定底层过程的Karhunen-Loeve表示，曲线数据的方差分析可以基于估计的主成分或整个曲线。在这种情况下，曲线配准是有趣的。对曲线数据提出了新的回归方法。其中一种回归方法扩展了karhunen-Loeve表示，允许协变量作用于表示的均值函数或协方差函数。另一种回归方法是将多变量数据的切片逆回归（SIR）方法扩展到曲线数据。该方法对曲线数据具有模型灵活性强的优点，在非参数模型拟合阶段之前实现了降维的目的。它计算简单，适合作为功能数据分析的探索性工具。本研究是由研究者在生物寿命和衰老方面的合作研究推动的。近年来，人们对老龄化研究的兴趣日益浓厚，部分原因是其财政和政策影响，部分原因是其内在因素。随着婴儿潮一代在2011年开始步入65岁，老年人口（65岁或以上）将会膨胀，因此其影响是不言而喻的。通过对地中海果蝇（medfly）衰老的大量数据集的分析，研究者一直活跃在这一研究领域。其中一个数据集包含了完整的繁殖历史，即实验中1000只雌性medfly每天产卵的数量。科学问题是如何将生殖特征与蝇类的生存和死亡联系起来。另一组关于营养对雄蝇和雌蝇死亡率影响的数据进行了研究。这里感兴趣的科学问题是将营养和性别影响与衰老联系起来。本研究开发的程序应用于这些和其他medfly数据，以判断方法的实用性和科学价值。它们还有助于解决有关衰老过程的潜在生物学问题。

9803627Jane-Ling WangCurve or functional data are of increasing importance due to enhenced high performance computing facilities and increased awareness that such data require special methods of analysis. Such data arise for example in longitudinal studies and in aging research. This research involves the development of new nonparametric methods to analyze such data. The focus is on procedures which involve both multivariate techniques and smoothing methods. Of particular interest are ANOVA and regression models for curve data. Emphasis is given on methods viewing the curve data as generalizations of one or several stochastic processes. Given a Karhunen-Loeve representation of the underlying processes, ANOVA for curve data can be based on the estimated principal components or on the whole curve. In this context, curve registration is of interest. New regression methods for curve data is also proposed. One of the regression methods extends the karhunen-Loeve representation to allow covariates to act on either the mean function or the covariance functions of the representation. Another regression method extends the sliced inverse regression (SIR) method for multivariate data to curve data. Such an approach for curve data has the advantage of model flexibility and accomplishes the goal of dimension reduction before the nonparametric model fitting stage. It is computationally simple and suitable as an exploratory tool for functional data analysis.This research is motivated by collaborative research of the investigator on biological lifespan and aging. Recently, there is an accelerated interest in aging research partly due to its financial and policy impact and partly due to its intrinsic aspects. The elderly population (age 65 or above) will swell as the babyboomers begin turning 65 in the year 2011 so its impact is self-evident. The investigator has been active in this research area through the analysis of large data sets on the aging of mediterranean fruit flies (medflies). One of these data sets contains the complete reproduction history, in terms of the number of eggs laid daily, for each of the 1,000 female medflies in the experiment. The scientific question is how to relate reproductive traits to survival and mortality of the medflies. Another data set on the nutrition effects to mortality of male and female medflies is studied. Here the scientific question of interest is to relate nutrition and gender effects to aging. The procedures developed in this study are applied to these and other medfly data to judge the practicality and scientific merits of the methods. They also help to address the underlying biological issues about the aging process.