Foundations of Dimension Reduction and Graphics

回归分析是研究响应变量如何随着一个或多个预测变量在其可能值上的变化而变化的一般领域。回归是统计分析中应用最广泛的领域之一，用于监控装配线的性能，确定社会创新的成功或失败，并根据传递的数据预测未来的结果。回归分析有着悠久的历史，至少可以追溯到200年前。针对特定类型问题的无数方法（例如，例如，在一个实施例中，其中响应是存活时间或二元变量的问题）。 这个项目中提出的工作以一种非常普遍的方式看待回归。它基于两个问题。首先，通过使用图可以了解多少相关性？第二个问题是：在不对手头问题的性质做出任何限制性假设的情况下，回归方法能走多远？在过去的十年中，第一个问题已经取得了实质性的进展，总结在两本书中，库克（1998年a）的理论总结和库克和韦斯伯格（1999年a）通过图形回归的应用方法。必须理解理论和应用问题，以开发基于图形的回归方法。第二个问题很重要，因为所有现有的通过图形进行回归的方法都是基于一些假设，通常涉及预测因子的分布。本项目拟制定的方法将克服一开始就作出假设所带来的限制。特别地，不需要预测因子必须至少近似线性相关的假设。此外，该方法可以扩展到定性预测因子等。

Regression analysis is the general area of study of how a response variable changes as one or more predictors are varied over their possible values. Regression is one of the most widely applied areas in statistical analysis, and is used for monitoring the performance of assembly lines, for determining the success or failure of social innovations, to predict the future outcomes based on passed data. Regression analysis has a long history, dating back at least 200 years. A myriad of methods for specific types of problems (e. g., problems in which the response is a survival time, or a binary variable) have been developed. The work proposed in this project looks at regression in a very general way. It is founded on asking two questions. First, how much can be learned about dependence through using graphs? And the second question: how far can one push regression methodology without making any limiting assumptions about the nature of the problem at hand?Over the last decade, substantial progress has been made on the first of these questions, summarized in two books, a theoretical summary of the area in Cook (1998a) and an applied approach to regression through graphics in Cook and Weisberg (1999a). Both theoretical and applied issues must be understood to develop methodology for regression based on graphics. The second question is important because all the existing methodology for regression through graphics is based on a few assumptions, generally concerning the distribution of the predictors. The methodology to be developed in this project will overcome the limitations that are imposed by making assumptions at the outset. In particular, the assumption that predictors must be at least approximately linearly related is not required. In addition, the method can be extended to qualitative predictors like factors.