43 research outputs found
On boundary detection
Given a sample of a random variable supported by a smooth compact manifold
, we propose a test to decide whether the boundary of
is empty or not with no preliminary support estimation. The test statistic
is based on the maximal distance between a sample point and the average of its
-nearest neighbors. We prove that the level of the test can be estimated,
that, with probability one, its power is one for large enough, and that
there exists a consistent decision rule. Heuristics for choosing a convenient
value for the parameter and identifying observations close to the
boundary are also given. We provide a simulation study of the test
Set Estimation Under Biconvexity Restrictions
A set in the Euclidean plane is said to be biconvex if, for some angle
, all its sections along straight lines with inclination
angles and are convex sets (i.e, empty sets or
segments). Biconvexity is a natural notion with some useful applications in
optimization theory. It has also be independently used, under the name of
"rectilinear convexity", in computational geometry. We are concerned here with
the problem of asymptotically reconstructing (or estimating) a biconvex set
from a random sample of points drawn on . By analogy with the classical
convex case, one would like to define the "biconvex hull" of the sample points
as a natural estimator for . However, as previously pointed out by several
authors, the notion of "hull" for a given set (understood as the "minimal"
set including and having the required property) has no obvious, useful
translation to the biconvex case. This is in sharp contrast with the well-known
elementary definition of convex hull. Thus, we have selected the most commonly
accepted notion of "biconvex hull" (often called "rectilinear convex hull"): we
first provide additional motivations for this definition, proving some useful
relations with other convexity-related notions. Then, we prove some results
concerning the consistent approximation of a biconvex set and and the
corresponding biconvex hull. An analogous result is also provided for the
boundaries. A method to approximate, from a sample of points on , the
biconvexity angle is also given
An optimal aggregation type classifier
We introduce a nonlinear aggregation type classifier for functional data
defined on a separable and complete metric space. The new rule is built up from
a collection of arbitrary training classifiers. If the classifiers are
consistent, then so is the aggregation rule. Moreover, asymptotically the
aggregation rule behaves as well as the best of the classifiers. The
results of a small si\-mu\-lation are reported both, for high dimensional and
functional data
Surface and length estimation based on Crofton's formula
We study the problem of estimating the surface area of the boundary of a
sufficiently smooth set when the available information is only a set of points
(random or not) that becomes dense (with respect to Hausdorff distance) in the
set or the trajectory of a reflected diffusion.
We obtain consistency results in this general setup, and we derive rates of
convergence for the iid case or when the data corresponds to the trajectory of
a reflected Brownian motion.
We propose an algorithm based on Crofton's formula, which estimates the
number of intersections of random lines with the boundary of the set by
counting, in a suitable way (given by the proposed algorithm), the number of
intersections with the boundary of two different estimators: the Devroye--Wise
estimator and the -convex hull of the data. \r
On the notion of polynomial reach: a statistical application
The volume function V(t) of a compact set S\in R^d is just the Lebesgue
measure of the set of points within a distance to S not larger than t.
According to some classical results in geometric measure theory, the volume
function turns out to be a polynomial, at least in a finite interval, under a
quite intuitive, easy to interpret, sufficient condition (called ``positive
reach'') which can be seen as an extension of the notion of convexity. However,
many other simple sets, not fulfilling the positive reach condition, have also
a polynomial volume function. To our knowledge, there is no general, simple
geometric description of such sets. Still, the polynomial character of
has some relevant consequences since the polynomial coefficients carry some
useful geometric information. In particular, the constant term is the volume of
S and the first order coefficient is the boundary measure (in Minkowski's
sense). This paper is focused on sets whose volume function is polynomial on
some interval starting at zero, whose length (that we call ``polynomial
reach'') might be unknown. Our main goal is to approximate such polynomial
reach by statistical means, using only a large enough random sample of points
inside S. The practical motivation is simple: when the value of the polynomial
reach , or rather a lower bound for it, is approximately known, the polynomial
coefficients can be estimated from the sample points by using standard methods
in polynomial approximation. As a result, we get a quite general method to
estimate the volume and boundary measure of the set, relying only on an inner
sample of points and not requiring the use any smoothing parameter. This paper
explores the theoretical and practical aspects of this idea