New functional models for biomedical data

描述（由申请人提供）：非参数功能模型在癌症和其他生物医学研究中有广泛的应用，如生物标志物数据、激素谱、昼夜节律模式和脑电图数据。目前的文献大多侧重于为均值结构建立灵活的非参数模型，而均值周围的随机变化被视为一个讨厌的成分。在许多生物医学应用中，例如本提案中描述的数据，推理的重点是随机变化及其与协变量和实验处理的关系。我们的第一个具体目标是开发方法，以非参数估计未知协方差结构在功能数据分析设置。协方差结构的估计反过来可以用来获得更有效的平均参数估计。我们的第二个目标是将功能数据分析的概念扩展到时间序列数据，其中数据分析的基本单位是时间序列，分析的重点不是平均值，而是协变量如何与随时间的随机变化相关。一个时间序列是由它的光谱唯一定义的，当我们对一组光谱进行平均时，我们可以得到一个群平均光谱，它唯一地定义了一个群平均时间序列。功能线性模型和功能混合效应模型也可以应用于光谱，对协变量和治疗效果进行推理。我们将把我们的方法开发重点放在非平稳时间序列数据上，这在生物医学研究中最常见。我们还将通过构建等效状态空间模型来开发计算效率高的估计程序。我们的第三个具体目标是将功能数据分析的概念推广到密度数据，其中数据分析的基本单位是密度。提出的密度模型允许我们调查协变量或治疗效果，而无需对潜在分布做出明确的假设。所提出的方法是由主要研究者直接或间接参与的临床研究所激发的，并将应用于临床研究。

DESCRIPTION (provided by applicant): Nonparametric functional models have a wide range of applications in cancer and other biomedical research, such as biomarker data, hormone profiles, circadian patterns and EEG data. Most of the current literature focuses on developing flexible nonparametric models for the mean structure while the stochastic variation around the mean is treated as a nuisance component. In many biomedical applications, such as the data described in this proposal, the inferential focus is on the stochastic variation and on how it is related to the covariates and experimental treatment. Our first specific aim is to develop methods to nonparametrically estimate the unknown covariance structure in the functional data analysis setting. The estimate of the covariance structure can in turn be used to obtain more efficient estimates of the mean parameters. Our second aim is to extend the concept of functional data analysis to time series data, in which the basic unit of the data analysis is a time series and the focus on the analysis is not on the mean, but on how the covariates are related to the stochastic variation over time. A time series is uniquely defined by its spectrum and when we average across a group of spectra, we can obtain a group-average spectrum that uniquely defines a group-average" time series. Functional linear models and functional mixed effects models can also be applied to the spectra to make inference on covariates and treatment effects. We will focus our methods development on nonstationary time series data, which are most common in biomedical research. We will also develop computationally efficient estimation procedures through construction of equivalent state space models. Our third specific aim is to generalize the concept of functional data analysis to density data, in which the basic unit of the data analysis is a density. The proposed density models allow us to investigate covariates or treatment effects without making explicit assumptions on the underlying distributions. The proposed methods are motivated by and will be applied to the clinical studies that the principal investigator is directly or indirectly involved.