Modelling modern data objects: statistical methods for high-dimensionality and intricate correlation structures

现代数据对象建模：高维和复杂相关结构的统计方法

• 批准号: RGPIN-2020-06941
• 负责人: FerreiraMiranda, Michelle
• 金额: $ 1.31万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741505
• 关键词: Modelling; modern; data; objects; statistical

Advances in technology have been generating data with increased complexity. Modern data objects are often high-dimensional and can lay in 2D, 3D and even 4D Euclidean and non-Euclidean spaces. Examples of such functions can arise in a wide range of scenarios, such as wearable devices, imaging recordings, medical imaging studies, eye-tracking devices, custom made instruments, and others. It is usually of interest to associate these complex functions to other covariates of interest, often scalars, which is the main focus of this proposal. In the literature, these models are known as function-on-scalar regression models (functional response regression models) or scalar-on-function regression models (predictive models). Most of the methodological contributions in functional regression models were initially developed for the first generation functional data which consists of simple smooth functions, typically one dimensional. We build up on previous ideas of data decorrelation and dimensionality reduction to bring a new set of tools that are able to handle the intricate correlation structures inherent in more complex functional data, while simultaneously addressing data high-dimensionality. We handle the complex correlation structures by devising an adaptive basis strategy that provides the foundation for borrowing information within functions. We also propose to develop a criteria to perform basis selection that automatically reduces data dimensionality. The methodology will be embedded in a Bayesian framework with shrinkage priors, that allows us to obtain MCMC samples in the basis space that are easily converted into MCMC samples in the data space through the use of basis inverse transforms. We will also develop an algorithm that will scale up to large datasets and a software package that will be easily accessible and open source. The proposed program is suitable for all HQP expertise level and facilitate their learning of skills in a way that is appropriate for both academia and industry. The contributions that will result from this program will definitely have an impact in advancing the statistical methods to include the latest data advancement.

技术的进步一直在产生复杂程度越来越高的数据。现代数据对象通常是高维的，可以位于2D、3D甚至4D欧几里得和非欧几里德空间中。这种功能的例子可以出现在广泛的场景中，例如可穿戴设备、成像记录、医学成像研究、眼睛跟踪设备、定制的仪器等。通常感兴趣的是将这些复杂函数与其他感兴趣的协变量相关联，通常是标量，这是本提案的主要焦点。在文献中，这些模型被称为标量函数回归模型(功能反应回归模型)或标量函数回归模型(预测模型)。函数回归模型中的大部分方法学贡献最初是针对第一代函数数据而开发的，第一代函数数据由简单的光滑函数组成，通常是一维的。我们建立在以前的数据去相关性和降维思想的基础上，带来了一套新的工具，能够处理更复杂的功能数据中固有的复杂相关性结构，同时解决数据的高维问题。我们通过设计一种自适应的基本策略来处理复杂的关联结构，该策略为在函数内借用信息提供了基础。我们还建议开发一种执行基础选择的标准，该标准自动降低数据维度。该方法将嵌入到一个具有收缩先验的贝叶斯框架中，该框架允许我们在基空间中获得MCMC样本，这些样本通过使用基逆变换很容易地转换为数据空间中的MCMC样本。我们还将开发一种算法，该算法将扩展到大型数据集，并开发一个易于访问和开源的软件包。拟议的计划适用于所有HQP专业水平，并促进他们以一种适合学术界和工业界的方式学习技能。这项计划将产生的贡献肯定会对推进统计方法产生影响，使其包括最新的数据进步。