Smoothings, Ridges, and Bumps

In this paper, we seek to formulate a geometrically based alternative to nonlinear, nonparametric regression. Hastie and Stuetzle (1989) introduced the ideas of principal curves and surfaces. These were 1and 2-dimensional non-linear regression-type estimators, which were consistent with respect to the mean, i.e. involved conditional expectations. We have developed a similar notion, but with two additional features. First, we propose to construct mode-like estimators rather than mean-like estimators. Second, the geometrically-based procedures we invoke are easily generalizable to higher dimensions.