ON PRINCIPAL HESSIAN DIRECTIONS FOR DATA VISUALIZATION AND DIMENSION REDUCTION - ANOTHER APPLICATION OF STEINS LEMMA
ON PRINCIPAL HESSIAN DIRECTIONS FOR DATA VISUALIZATION AND DIMENSION REDUCTION - ANOTHER APPLICATION OF STEINS LEMMA
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DOI：
10.1080/01621459.1992.10476258
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发表时间：
1992-12-01
影响因子：
3.7
通讯作者：
LI, KC
中科院分区：
文献类型：
--
作者：
LI, KC
现代图形工具增强了我们直接从数据中学习许多东西的能力。有了许多用户友好的图形软件，我们被鼓励比以前更频繁地绘制。与图形直接交互的好处是巨大的。但是，落后于这些高科技进步的是关于策划什么的适当指导的问题。有太多的方向来投影一个高维数据集，无引导的绘图可能是耗时和徒劳的。在最近的一篇文章中，Li基于有效降维（edr）方向的概念建立了一个研究这个问题的统计框架。它们是为了有效地观察和研究高维输入变量与输出变量之间的关系而投影高维输入变量的方向。介绍了一种方法，切片逆回归，并证明是有用的，在寻找edr方向。本文介绍另一种求edr方向的方法。它开始于观察到回归函数的海森矩阵的特征向量有助于研究回归曲面的形状。定义了主Hessian方向（pHd）的符号，其定位回归表面在聚合意义上显示最大曲率的主轴沿着。我们表明，PH的可以用来找到EDR方向。我们进一步使用著名的斯坦引理建议估计。获得了估计pH值的抽样特性。一个显着性检验得出的建议，通过我们的方法发现的一个视图的一致性。讨论了实现这种方法的一些版本，并报告了模拟结果和应用程序的真实的数据。讨论了该方法与探索性投影寻踪的关系。
Modem graphical tools have enhanced our ability to learn many things from data directly. With much user-friendly graphical software available, we are encouraged to plot a lot more often than before. The benefits from direct interaction with graphics have been enormous. But trailing behind these high-tech advances is the issue of appropriate guidance on what to plot. There are too many directions to project a high-dimensional data set and unguided plotting can be time-consuming and fruitless. In a recent article, Li set up a statistical framework for study on this issue, based on a notion of effective dimension reduction (edr) directions. They are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable. A methodology, sliced inverse regression, was introduced and shown to be useful in finding edr directions. This article introduces another method for finding edr directions. It begins with the observation that the eigenvectors for the Hessian matrices of the regression function are helpful in the study of the shape of the regression surface. A notation of principal Hessian directions (pHd's) is defined that locates the main axes along which the regression surface shows the largest curvatures in an aggregate sense. We show that pHd's can be used to find edr directions. We further use the celebrated Stein lemma for suggesting estimates. The sampling properties of the estimated pHd's are obtained. A significance test is derived for suggesting the genuineness of a view found by our method. Some versions for implementing this method are discussed, and simulation results and an application to real data are reported. The relationship of this method with exploratory projection pursuit is also discussed.