New approaches in Functional Data Analysis: inference for incomplete or correlated data and nonlinear methods

函数数据分析(FDA)是统计学的一个分支，涉及对复杂数据对象的分析，例如曲线和曲面，可以被视为随机函数的实现。由于过去几十年来计算机技术的进步，功能数据在许多应用中自然出现，并正变得越来越可访问，从而变得普遍。泛函数据的两个主要特征是：(I)它们属于无限维空间；(Ii)它们具有一定的光滑性。FDA中的经典设置是假设我们有一个随机曲线X的独立实现的集合，并且这些实现中的每一个都是在[0，1]的有限子集上观察到的，并且具有测量误差。分析这类数据的传统方法通常遵循两个步骤：首先是平滑步骤；然后是降维步骤，其中每条曲线由有限数量的平滑基函数的线性组合来近似。然而，FDA的许多有趣的特征，如时间局部化特征和多尺度变化，不能很好地用线性展开来捕捉。我的研究计划的第一个目标是在分析函数数据的非线性方法方面取得方法论上的进步。为了做到这一点，我计划阐述包含时间局部化特征(例如峰值)的数据的协方差函数的估计过程，并探索如何对位于非线性流形中的函数数据进行推断。FDA感兴趣的另一种设置是数据集的每个函数仅在定义[0，1]的整个域的随机子区间上观察的设置。这些数据被称为功能片段，它们的分析非常具有挑战性，因为我们没有关于对角线周围带外的未知协方差函数的统计信息。我在这个领域的目标是开发一种使用低阶矩阵完成技术来估计数据集的每个离散化片段的缺失部分的方法，以及一种进行功能片段注册的方法。传统上，在FDA中，我们使用随机函数的独立实现的样本。然而，在实践中，很可能发生的情况是，观测到的函数在空间上是相关的。我的最后一个目标是探索这类数据的推论。更准确地说，我想阐述一个空间多元函数数据的检验程序，并建立一个函数回归模型，其中响应是计数超过零的计数数据，解释变量是空间相关的函数数据。总而言之，我的研究计划旨在解决FDA三个领域的挑战性目标：(A)分析功能数据的非线性方法，(B)功能片段的分析，以及(C)空间相关功能数据的推断。这一建议的新方法论可能为解决FDA的许多理论和应用难题开辟道路。

Functional Data Analysis (FDA) is a branch of statistics concerned with the analysis of complex data objects, such as curves and surfaces, that can be viewed as realizations of a random function. Functional data arise naturally in many applications, and are becoming more and more accessible and consequently prevalent, due to the advances in computer technology in the last decades. The two main features of functional data are: (i) they belong to an infinite-dimensional space; (ii) they have some smoothness properties. The classical setup in FDA is to suppose that we have a collection of independent realizations of a random curve X and that each of these realizations is observed at a finite subset of [0,1] and with measurement error. Traditional approaches to analyze such data typically follow a two-step procedure: first a smoothing step; further, a dimension reduction step where each curve is approximated by a linear combination of a finite number of smooth basis functions. However, many interesting features in FDA such as time-localized features and multiscale variations are not well-captured by a linear expansion. The first objective of my research program is to make methodological advances to the nonlinear approach for analyzing functional data. In order to do this, I plan to elaborate an estimation procedure of the covariance function for data containing time-localized features (e.g. peaks) and to explore how to do inference for functional data that lie in non-linear manifolds. Another setup of interest in FDA is the one where each function of a dataset is only observed on a random subinterval of the whole domain of definition [0,1]. These data are called functional fragments and their analysis is quite challenging since we have no statistical information on the unknown covariance function outside a band around the diagonal. My objectives in this area are to develop a method to estimate the missing part of each discretized fragment of a dataset using low-rank matrix completion techniques, and a method to do registration of functional fragments. Traditionally, in FDA we work with samples of independent realizations of random functions. However, in practice, it might very well happen that the observed functions are spatially correlated. My last objective is to explore inference for such data. More precisely, I want to elaborate a testing procedure for spatial multivariate functional data and to build a functional regression model where the responses are count data with an excess of zero counts and the explanatory variables are spatially correlated functional data. To conclude, my research program aims to tackle challenging objectives in three areas of FDA: (a) nonlinear approach in analyzing functional data, (b) analysis of functional fragments, and (c) inference for spatially correlated functional data. The novel methodology of this proposal could open the way to solving many difficult problems in FDA, both theoretical and applied.