"Functional and High-dimensional Data Analysis: Regularization, Representation and Regression"

《函数式和高维数据分析：正则化、表示和回归》

• 批准号: 341333-2012
• 负责人: Yao, Fang
• 金额: $ 2.19万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572233
• 关键词: Functional; dimensional; Data; Analysis; Regularization

Large complex data collected in modern scientific experiments impose tremendous challenges for traditional statistical methods due to high-dimensional and complicated structures. Functional data analysis (FDA) has emerged as a promising field that employs realizations of latent or observed stochastic processes as model units, whose applications are commonly found in many fields, such as longitudinal studies, microarray data, brain images, internet investigations, etc. A fundamental issue of modeling functional data arises from the "curse of dimensionality", as functional objects are conceptually framed as infinite-dimensional processes. This naturally relates to the recently revolutionized area of high-dimensional model selection and regularization. The long-term objective of this research program is to (1) develop a wide range of flexible yet practical functional modeling frameworks to accommodate the increased complexity; (2) investigate suitable regularization and inferential methods tailored for such models, combining the strength of high-dimensional techniques with functional approaches. Specifically, the research program focuses on two core areas of FDA. For the first area of functional data representation, a new regularization framework determining the key feature of dimensionality will be investigated and is intended to be generally applicable for both pre-fixed basis (e.g., splines and wavelets) and data-driven basis (e.g., eigenbasis). Further research will be conducted for correlated functional observations, motivated by studies on genetically related diary cattle. In the second area of functional regression, I expand the scope to embrace high-dimensional predictors, which results in semiparametric structures that call for efficient regularization and inferential approaches. This is expected to set ground for a variety of semiparametric functional regression models, and generate a broad impact on both FDA and high-dimensional fields.

现代科学实验中收集的大量复杂数据由于其高维、复杂的结构给传统的统计方法带来了巨大的挑战。功能数据分析（FDA）是一个新兴的研究领域，它以潜在的或可观测的随机过程为模型单元，在纵向研究、微阵列数据、脑图像、互联网调查等领域有着广泛的应用。因为功能对象在概念上被框定为无限维过程。这自然与最近革命性的高维模型选择和正则化领域有关。 该研究计划的长期目标是：（1）开发广泛的灵活而实用的功能建模框架，以适应日益增加的复杂性;（2）研究适合此类模型的正则化和推理方法，将高维技术的优势与功能方法相结合。具体而言，该研究计划侧重于FDA的两个核心领域。对于函数数据表示的第一个领域，将研究确定维度的关键特征的新的正则化框架，并且该框架旨在通常适用于预先固定的基础（例如，样条和小波）和数据驱动的基础（例如，特征基）。通过对遗传相关奶牛的研究，将对相关功能观察进行进一步研究。在函数回归的第二个领域，我扩大了范围，包括高维预测，这将导致半参数结构，要求有效的正则化和推理方法。这有望为各种半参数函数回归模型奠定基础，并对FDA和高维领域产生广泛影响。