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

函数数据分析的新方法：不完整或相关数据的推理以及非线性方法

• 批准号: RGPIN-2020-07235
• 负责人: Descary, MarieHélène
• 金额: $ 1.31万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759318
• 关键词: New; approaches; Functional; Data; Analysis

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.

功能数据分析（FDA）是统计学的一个分支，涉及复杂数据对象的分析，如曲线和曲面，可以被视为随机函数的实现。由于近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，并且变得越来越容易访问并因此变得普遍。函数数据的两个主要特征是：(i)它们属于无限维空间；（ii）它们具有一些平滑特性。FDA的经典设置是假设我们有一个随机曲线X的独立实现集合，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。分析此类数据的传统方法通常遵循两个步骤：首先是平滑步骤；进一步，一个降维步骤，其中每条曲线由有限个光滑基函数的线性组合近似。然而，FDA中许多有趣的特征，如时间局部特征和多尺度变化，并没有被线性展开很好地捕捉到。我的研究计划的第一个目标是在分析功能数据的非线性方法上取得方法学上的进步。为了做到这一点，我计划详细阐述包含时间局部特征（例如峰）的数据的协方差函数的估计过程，并探索如何对非线性流形中的函数数据进行推理。FDA中另一个有趣的设置是数据集的每个函数仅在整个定义域的随机子区间上观察[0,1]。这些数据被称为功能片段，它们的分析相当具有挑战性，因为我们没有对角线周围带外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方法，使用低秩矩阵补全技术来估计数据集的每个离散片段的缺失部分，以及一种方法来做功能片段的注册。传统上，在FDA中，我们使用随机函数的独立实现样本。然而，在实践中，很可能发生观察到的函数在空间上是相关的。我的最后一个目标是探索对这些数据的推断。更准确地说，我想详细说明空间多元功能数据的测试程序，并建立一个功能回归模型，其中响应是计数数据，计数超过零，解释变量是空间相关的功能数据。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)分析功能片段，(c)空间相关功能数据的推断。该建议的新方法可以为解决FDA的许多理论和应用难题开辟道路。