New Directions in Dimension Reduction

摘要DMS-0204662 PI：李兵这项研究将开发降维的方法，其目的是提高精度和更广泛的应用。具体而言，工作将从三个主要方向进行。(1)经典的公式使回归中的条件密度成为降维的目标。这没有考虑到在许多应用中，主要兴趣集中在条件均值中。此外，经典的公式化需要预测变量之间的同向性，这对于某些问题来说可能过于限制。为了解决这些问题，研究人员建议重新制定的问题，减少维度的预测，因为他们出现在条件均值。这将允许进一步的维数减少，它将提高精度并消除对同性性的要求。(2)在经典公式中，人们不能处理在实践中经常出现的分类预测因子。这项研究将扩大拟议的提法，使其能够处理这种情况。(3)然后，它是可能的和自然的联合收割机这两个新的元素，以进一步发展一个更集中，和较少限制的降维方法的条件手段回归，涉及分类预测。降维方法最初是为了给探索性数据分析提供一个全面的图形工具而引入的。最近，由于计算能力的迅速增长，正在进行积极的开发;这大大增加了收集的数据集的范围和规模。除了作为一种图形方法的重要作用外，降维在识别变量之间的联系（如分类和聚类）的问题中特别有用。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。但现有的降维方法存在一些局限性，如假设预测变量之间具有同质性，不能处理分类预测变量等。这项研究将解决目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.