估計陡坡樹狀圖

We suggest new approaches to constructing
tree diagrams that describe associations
among points in a scatterplot. Our methods
are based implicitly on gradient estimates. In
our tree diagrams, two data points are
associated with one another if and only if
their respective curves of steepest ascent up
the density or intensity surface lead toward the same mode. The representation, in the
sample space, of the set of steepest ascent
curves corresponding to the data, is called the
gradient tree. It has a regular, octopus-like
structure, and is consistently estimated by its
analogue computed from a nonparametric
estimator which gives consistent estimation
of both the density surface and its derivatives.
We also suggest `forests', in which data are
linked by line segments which represent good
approximations to portions of the population
gradient tree. A forest is a regularization of a
minimum spanning tree. However, forests
use a larger bandwidth for constructing the
density-surface estimate than is implicit in
the MST, with the result that they are
substantially more orderly and are more
readily interpreted.