Summary: Modern 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 ibid. 86, No. 414, 316-342 (1991; Zbl 0742.62044), the author 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.