Methods for Statistical Learning

拟议的研究位于现代统计学习和“传统”统计思想的交叉点，如实验的设计和分析以及不确定性量化。在统计学习中，数据被用来训练监督学习器，这是一种灵活的统计模型，它使用输入变量的值来预测响应变量。统计技术，如贝叶斯建模，使得量化监督学习的不确定性成为可能。实验的设计和分析为模型的训练提供了一种收集相关数据的方法。******这个研究项目涉及新的监督学习模型的发明和应用。它将推广贝叶斯加性回归树（BART）模型，使其适用于更广泛的数据类型，并为响应变量为数字的情况合并新的结构。广义数据类型将包括两个以上类的分类，通过多项回归。附加结构将包括单调性、线性和混合效应线性模型的结合以及位置和分散的联合建模，可以采用常规误差模型，也可以采用更灵活的误差模型。与最初的BART模型一样，统计不确定性框架将以一种可扩展到大数据问题的方式实现。BART模型的一个特别重点将是计算机实验的顺序设计和分析。顺序设计算法可以利用BART的灵活性，并在顺序设计准则的评估中使用量化不确定性的能力。******本研究计划的另一个方向将是将计算机实验和经典实验设计的全套工具和思想应用于通过模拟评估统计模型性能的问题。几乎所有提出新统计模型的研究都依赖于模拟实验来研究统计模型在现实场景中的性能，这是不适合理论研究的。然而，大多数实验都没有采用统计设计或分析方法。这项研究将带来全套的实验设计，包括鲜为人知的方法，如田口的鲁棒参数设计，模拟实验。它将开发一种系统的方法，研究人员可以用它来了解一系列变化来源的表现。******开发的许多方法将适用于大数据问题，并且通常会受到实际应用的启发。这些方面将为学生提供良好的培训机会，使他们在完成学业时成为“数据科学家”。

The proposed research lies at the intersection of modern statistical learning and "traditional" statistical ideas such as design and analysis of experiments and uncertainty quantification. In statistical learning, data are used to train a supervised learner, which is a flexible statistical model that predicts a response variable using the values of input variables. Statistical techniques, such as Bayesian modelling, make it possible to quantify uncertainty about the supervised learner. Design and analysis of experiments provide a way to collect relevant data for training the model. ******This research program concerns the invention and application of novel supervised learning models. It will generalize the Bayesian Additive Regression Trees (BART) model, making it applicable to a wider variety of data types and incorporating new structure for the case where the response variable is numeric. Generalized data types will include classification with more than two classes, via multinomial regression. Additional structure will include monotonicity, combination with linear and mixed effect linear models and joint modelling of location and dispersion, with either a normal error model or a more flexible error model. As with the original BART model, a framework for statistical uncertainty will be implemented in a way that scales to large data problems. A particular focus of the BART model will be the sequential design and analysis of computer experiments. A sequential design algorithm can exploit the flexibility of BART and use the ability to quantify uncertainty in evaluation of a sequential design criterion.******Another direction for this research program will be the application of the full suite of tools and ideas from computer experiments and classical design of experiments to the problem of evaluating the performance of statistical models through simulation. Nearly all research that presents a new statistical model relies on simulation experiments to study the performance of statistical models in realistic scenarios that are not amenable to theoretical study. Yet most experiments fail to employ either statistical design or analysis methods. This research will bring the full suite of experimental design, including lesser known methods such as Taguchi's robust parameter design, to simulation experiments. It will develop a systematic approach that researchers can use to gain understanding of performance over a range of sources of variation.******Many of the methods developed will be applicable to big data problems, and will often be inspired by real applications. These aspects will provide excellent training opportunities for students, developing them as "data scientists" by the time they complete their studies.