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.

技术的进步产生了越来越复杂的数据。现代数据对象通常是高维的，可以位于 2D、3D 甚至 4D 欧几里德和非欧几里德空间中。此类功能的示例可以出现在各种场景中，例如可穿戴设备、成像记录、医学成像研究、眼球跟踪设备、定制仪器等。将这些复杂函数与其他感兴趣的协变量（通常是标量）相关联通常是有意义的，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（函数响应回归模型）或函数标量回归模型（预测模型）。函数回归模型中的大多数方法论贡献最初是针对第一代函数数据开发的，该数据由简单的平滑函数（通常是一维）组成。 我们在之前的数据去相关和降维思想的基础上，推出了一套新的工具，能够处理更复杂的函数数据中固有的复杂的相关结构，同时解决数据高维问题。我们通过设计自适应基础策略来处理复杂的相关结构，该策略为在函数内借用信息提供基础。我们还建议制定一个标准来执行自动降低数据维度的基础选择。该方法将嵌入到具有收缩先验的贝叶斯框架中，这使我们能够获得基础空间中的 MCMC 样本，这些样本可以通过使用基础逆变换轻松转换为数据空间中的 MCMC 样本。我们还将开发一种可扩展到大型数据集的算法，以及一个易于访问和开源的软件包。拟议的计划适合所有 HQP 专业水平，并以适合学术界和工业界的方式促进他们学习技能。该计划所做出的贡献肯定会对统计方法的进步产生影响，以纳入最新的数据进展。