Beatriz Pateiro-López

University of Santiago de Compostela | USC · Departamento de Estadística e Investigación Operativa

Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical analysis of some of these measures of non-convexity of a set S, by dealing with their estimation based on a s...

Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standard assumption. Relevant results in this field, such as rates of convergence and confidence sets,...

Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical analysis of some of these measures of non-convexity of a set $S$, by dealing with their estimation based on a...

This paper deals with the class imbalance problem in the context of the automatic selection of the best storage format for a sparse matrix with the aim of maximizing the performance of the Sparse Matrix Vector Multiplication (SpMV) on GPUs. Our classification method uses Convolutional Neural Networks (CNNs), and proposes several solutions to mitiga...

We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume that the domain has a boundary with positive reach. We propose a data-splitting approach to correct the bias of the plug-in estimator based on the sample α-convex hull. We show that this simple estimator ach...

Given a compact set S⊂Rd we consider the problem of estimating, from a random sample of points, the Lebesgue measure of S, μ(S), and its boundary measure, L(S) (as defined by the Minkowski content of ∂S). This topic has received some attention, especially in the two-dimensional case d=2, motivated by applications in image analysis. A new method to...

We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume the domain has a boundary with positive reach. We propose a data splitting approach to correct the bias of the plug-in estimator based on the sample alpha-convex hull. We show that this simple estimator achi...

According to Kendall (1989), in shape theory, The idea is to filter out effects resulting from translations, changes of scale and rotations and to declare that shape is “what is left”. While this statement applies in principle to classical shape theory based on landmarks, the basic idea remains also when other approaches are used. For example, we m...

We study the problem of estimating a compact set $S\subset \mathbb{R}^d$ from a trajectory of a reflected Brownian motion in $S$ with reflections on the boundary of $S$. We establish consistency and rates of convergence for various estimators of $S$ and its boundary. This problem has relevant applications in ecology in estimating the home range of...

The medial axis and the inner parallel body of a set CC are different formal translations for the notions of the “central core” and the “bulk”, respectively, of CC. On the basis of their applications in image analysis, both notions (and especially the first one) have been extensively studied in the literature, from different points of view. A modif...

This work presents the implementation in R of the -shape of a finite set of points in the three-dimensional space ℝ3. This geometric structure generalizes the convex hull and allows to recover the shape of non-convex and even non-connected sets in 3D, given a random sample of points taken into it. Besides the computation of the α-shape, the R packa...

In this work we deal with the problem of estimating the support S of a probability distribution under shape restrictions. The shape restriction we deal with is an extension of the notion of convexity named α-convexity. Instead of assuming, as in the convex case, the existence of a separating hyperplane for each exterior point of S, we assume the ex...

A new test is proposed for the hypothesis of uniformity on bi-dimensional supports. The procedure is an adaptation of the "distance to boundary test" (DB test) proposed in Berrendero, Cuevas, & Vázquez-Grande (2006). This new version of the DB test, called DBU test, allows us (as a novel, interesting feature) to deal with the case where the support...

A test for the hypothesis of uniformity on a support S⊂ℝ d is proposed. It is based on the use of multivariate spacings as those studied in Janson (Ann. Probab. 15:274–280, 1987). As a novel aspect, this test can be adapted to the case that the support S is unknown, provided that it fulfils the shape condition of λ-convexity. The consistency proper...

A new projection-based definition of quantiles in a multivariate setting is proposed. This approach extends in a natural way to infinite-dimensional Hilbert and Banach spaces. Sample quantiles estimating the corresponding population quantiles are defined and consistency results are obtained. Principal quantile directions are defined and asymptotic...

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r -convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain a result of ‘full consistency’ (i.e. consis...

This paper presents the R package alphahull which implements the α-convex hull and the α-shape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the α-convex hull and the α-shape are able to reconstruct non-convex sets. This flexib...

The problem of estimating the surface area, L 0, of a set G d has been extensively considered in several fields of research. For example, stereology focuses on the estimation of L 0 without needing to reconstruct the set G. From a more geometrical point of view, set estimation theory is interested in estimating the shape of the set. Thus, surface a...

The problem of estimating the Minkowski content L0(G) of a body G ⊂ Rd is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L0(G). This estimator is based on the informatio...

The problem of estimating the Minkowski content L0(G) of a body G ⊂ ℝ d is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L0(G). This estimator is based on the informati...

Univariate partially linear regression models have been widely discussed in recent years. In this paper, we consider a multivariate partially linear regression model under independent errors, where the response variable is d-dimensional. We obtain the asymptotic bias and variance for both the parametric and the nonparametric components. Moreover, w...