New Directions in Envelope Models and Methods with Applications to Public Health and Medical Science

多元线性回归（MLR）是一种理解预测因子与响应之间关系的方法，广泛应用于许多学科的估计或预测。随着现代技术的发展，可以测量一个主题越来越多的特征和潜在因素，导致许多当代问题的数据集非常大。在这种情况下，MLR是低效的，因为它无法区分对科学目标有用的信息和可能过多的不相关信息，这些信息可能会掩盖有用的信息。包络模型是统计学中的一个新领域。通过识别不相关的信息，包络分析仅基于有用的信息，因此更有效。在本项目中，将探索新的方向，以扩大包络模型的适用性，特别是在公共卫生和医学科学中的适用性。参与该项目的学生将有机会与其他学科的教师互动，并获得如何跨越传统学科界限运作的重要经验。将开发软件，向统计界提供新的方法。本项目将开发新的模型，丰富包络的范围，使包络法更加灵活，适应更多的实际问题。例如，通常可以从病史、过去经验和其他来源获得先验信息，贝叶斯包络模型将使调查人员能够在分析中纳入先验信息。由于医学研究经常涉及缺失数据，因此将研究处理缺失数据的包络模型。虽然大多数现有的变量选择方法都应用于预测因子，但稀疏包络模型侧重于识别不活跃的个体响应，从而获得更有效和可解释的结果。本项目将包络模型与统计学中的贝叶斯分析、马尔可夫链蒙特卡罗、变量选择和协方差估计等多个分支相结合，将在包络模型领域产生新的理论、方法和算法。

Multivariate linear regression (MLR) is an approach to understand relationships between predictors and responses, and it is broadly applied for estimation or prediction in many disciplines. With the development of modern technology, it is possible to measure more and more characteristics and potential factors for a subject, resulting in very large data sets in many contemporary problems. In such situations, MLR is inefficient because it fails to distinguish between information that is useful to the scientific goal and a likely overabundance of irrelevant information that can obscure the useful information. The envelope model is a new area in statistics. By identifying the irrelevant information, the envelope analysis is based on the useful information only and is therefore more efficient. In this project, new directions will be explored to broaden the applicability of the envelope models, especially the applicability in public health and medical sciences. Students involved in the project will have opportunities to interact with faculty from other disciplines, and gain crucial experience on how to operate across traditional disciplinary boundaries. Software will be developed to make the new methodologies available to the statistical community. This project will develop new models that enrich the area of envelopes, making the envelope method more flexible and adaptive to more practical problems. For example, prior information is often available from medical history, past experience and other sources, the Bayesian envelope model will enable investigators to incorporate prior information in the analysis. As medical studies often involve missing data, envelope models that handle missing data will be studied. While most existing variable selection methods are applied to the predictors, sparse envelope model focuses on identifying inactive individual responses, leading to more efficient and interpretable results. The project connects the envelope model with several branches in statistics, including Bayesian analysis, Markov chain Monte Carlo, variable selection and covariance estimation, which will generate new theory, methods and algorithms in the area of envelope models.