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

技术进步产生的数据越来越复杂。现代数据对象通常是高维的，可以位于二维、三维甚至四维欧几里得和非欧几里得空间。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。将这些复杂的函数与其他相关的协变量（通常是标量）联系起来通常是有趣的，这是本提案的主要焦点。在文献中，这些模型被称为函数对标量回归模型（功能响应回归模型）或标量对函数回归模型（预测模型）。函数回归模型中的大多数方法学贡献最初是针对第一代函数数据开发的，这些数据由简单的光滑函数组成，通常是一维的。我们在之前的数据去关联和降维思想的基础上，提出了一套新的工具，能够处理更复杂的功能数据中固有的复杂的关联结构，同时处理数据的高维性。我们通过设计一种自适应基策略来处理复杂的关联结构，该策略为函数内的信息借用提供了基础。我们还建议开发一个标准来执行自动降低数据维数的基选择。该方法将嵌入具有收缩先验的贝叶斯框架中，这使我们能够在基空间中获得MCMC样本，这些样本可以通过使用基逆变换轻松地转换为数据空间中的MCMC样本。我们还将开发一种可以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。该计划适用于所有HQP专业水平，并以适合学术界和工业界的方式促进他们的技能学习。这个项目的贡献肯定会对统计方法的发展产生影响，包括最新的数据发展。

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.