Dimension Reduction and Data Visualization

9803459 Ker-Chau李氏逆回归(SIR)和主海森方向(PHD)是两种新的降维技术。它们的性质已在一般回归设置下进行了研究。这项研究涉及到这些方法在以下几个领域的应用：(1)蒙特卡罗和贝叶斯计算，(2)非线性时间序列，(3)半参数建模，(4)多元结果和函数数据分析，(5)数学/物理/计算机模型的不确定性分析。在(1)中，考虑了重要性抽样和拒收抽样。重要度权重可以作为结果值。目标变成了寻找权重和采样点之间的简化关系的研究。降维可以帮助找到后验分布的模式。在应用SIR/PHD之前，还需要在(2)-(4)中适当地表述问题。尺寸是每个科学领域都可能出现的问题。一般来说，困难在于如何可视化包含多个变量的数据集或具有多个参数的函数的形状。SIR/PHD是无域的降维工具，可以方便地应用于挖掘数据以获取信息至关重要的领域。对于本研究，主要的例子包括地下水建模、从数字通信中的卷积数据序列中恢复信号、以及分析非线性经济时间序列数据。

9803459Ker-Chau LiSliced inverse regression (SIR) and principal Hessian directions (PHD) are two newly established dimension reduction techniques. Their properties have been studied under general regression settings. This research involves the application of these methods to several problems arising from the following areas:(1) Monte Carlo and Bayesian computation,(2) nonlinear time series,(3) semiparametric modeling,(4) multivariate outcome and functional data analysis, and(5) uncertainty analysis of mathematical/physical/computer models.These settings are more complicated. In (1), importance sampling and rejection sampling are considered. The importance weights can be treated as the outcome values. The goal becomes the study of finding a simplified relationship between the weights and the sampled points. Dimension reduction can help find the modes of posterior distributions. Proper formulation of the problem is also needed in (2)-(4) before applying SIR/PHD.Dimensionality is an issue that can arise in every scientific field. Generally speaking, the difficulty lies on how to visualize a data set involving several variables or the shape of a function with several arguments. SIR/PHD are domain-free dimension reduction tools that can be conveniently applied in areas where mining data for information is crucial. For this research, the primary examples include ground water modeling, recovering signals from convoluted data sequences in digital communication, and analyzing nonlinear economic time series data.