New Directions in Dimension Reduction

DMS-0204662PI：Bing Li这项研究将开发降维方法，旨在提高精度和更广泛的应用范围。具体地说，这项工作将从三个主要方向进行。(1)经典公式将回归中的条件密度作为降维目标。这没有考虑到在许多应用中，主要兴趣集中在条件平均值上。此外，经典公式要求预报器之间具有同方差，这对某些问题可能过于严格。为了解决这些问题，研究人员建议将问题重新表述为降低预测因子的维度，因为它们出现在条件平均值中。这将允许进一步的降维，它将提高精度，并消除对同方差的要求。(2)在经典公式中，人们不能处理在实践中经常出现的直言预测。这项研究将扩大拟议的提法，使其能够处理此类情况。(3)将这两个新元素结合起来，进一步发展一种更有针对性、更少限制的降维方法，用于涉及类别预测因素的回归的条件均值，是可能的，也是自然的。最初引入降维方法是为了为探索性数据分析提供一个全面的图形化工具。最近，由于计算能力的快速增长，正在进行积极的开发；这极大地增加了收集的数据集的范围和维度。除了作为一种图形方法的重要作用外，降维在分类和聚类等变量之间的联系的问题中尤其有用。当数据点的维度超过数据点的总数时，这也是有用的，这通常是许多科学数据集的情况，例如基因表达数据。但现有的降维方法有几个局限性，例如假设预测值之间是齐次的，不能处理分类预测值。这项研究将解决当前方法的这些局限性。

AbstractDMS-0204662PI: Bing LiThis research will develop methods in dimension reduction, which aim at increased accuracy and a wider spectrum of applications. Specifically, the work will proceed in three main directions. (1) The classical formulation makes the conditional density in regression the target for dimension reduction. This does not take into consideration that in many applications the primary interest centers in the conditional mean. Moreover, the classical formulation requires homoskedasticity among predictors, which can be too restrictive for some problems. To address these issues the investigator proposes to reformulate the problem as reducing the dimensions of the predictors as they appear in the conditional mean. This will allow further dimension reduction, it will improve accuracy and remove the requirement for homoskedasticity. (2) Within the classical formulation one cannot handle categorical predictors, which occur frequently in practice. This research will broaden the proposed formulation so that it can handle such cases. (3) It is then possible and natural to combine these two new elements to further develop a more focused, and less restricted dimension reduction method for conditional means for regressions involving categorical predictors. The methods of dimension reduction were introduced originally to provide a comprehensive graphical tool for exploratory data analysis. Recently, active developments are under way due to the rapid growth of computing power; this has dramatically increased the scope and dimensions of the collected data sets. Besides its important role as a graphic method, dimension reduction is particularly useful in problems where interest lies in identifying connections among the variables, such as classification and clustering. It is also useful when the dimension of a data point exceeds the total number of data points, which is typically the case for many scientific data sets, such as gene expression data. But the available dimension reduction methods have several limitations, such as assuming homogeneity between predictors and not be able to handle categorical predictors. This research will tackle these limitations of the current methodology.