Generalizing the Convex Hull of a Sample: The R Package alphahull

This paper presents the R package alphahull which implements the ñ-convex hull and the ñ-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 ñ-convex hull and the ñ-shape are able to reconstruct non-convex sets. This flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations. The usefulness of the package is illustrated with two small simulation studies on boundary length estimation.

Polynomial volume estimation and its applications

Given a compact set S ⊂ R d 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 simultaneously estimate µ(S) and L(S) from a sample of points inside S is proposed. The basic idea is to assume that S has a polynomial volume, that is, that V (r) := µ{x : d(x, S) ≤ r} is a polynomial in r of degree d, for all r in some interval [0, R). We develop a minimum distance approach to estimate the coefficients of V (r) and, in particular µ(S) and L(S), which correspond, respectively, to the independent term and the first degree coefficient of V (r). The strong consistency of the proposed estimators is proved. Some numerical illustrations are givenThis work has been partially supported by Spanish Grants MTM2016-78751-P (A. Cuevas) and MTM2016-76969-P (B. Pateiro-López)S

A multivariate uniformity test for the case of unknown support

The final publication is available at http://dx.doi.org/10.1007/s11222-010-9222-

Testing uniformity for the case of a planar unknown support

The definitive version is available at http://www3.interscience.wiley.co

Shape classification based on interpoint distance distributions

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 might consider, for every shape, a suitable associated function which, to a large extent, could be used to characterize the shape. This finally leads to identify the shapes with the elements of a quotient space of sets in such a way that all the sets in the same equivalence class share the same identifying function. In this paper, we explore the use of the interpoint distance distribution (i.e. the distribution of the distance between two independent uniform points) for this purpose. This idea has been previously proposed by other authors [e.g., Osada et al. (2002), Bonetti and Pagano (2005)]. We aim at providing some additional mathematical support for the use of interpoint distances in this context. In particular, we show the Lipschitz continuity of the transformation taking every shape to its corresponding interpoint distance distribution. Also, we obtain a partial identifiability result showing that, under some geometrical restrictions, shapes with different planar area must have different interpoint distance distributions. Finally, we address practical aspects including a real data example on shape classification in marine biologyThis work has been partially supported by Spanish Grants MTM2013-44045-P (Berrendero and Cuevas) and MTM2013-41383-P (Pateiro-López

Políticas públicas para reducir los costes de cambio de operador asociados al acceso obligatorio a determinados elementos de la red postal en la UE y efectos de los costes de retención

[Abstract] Purpose: Our objective is to examine the switching costs that exist in the European postal service, where an ambitious process of opening up to competition has been carried out since 1997, as well as the different types of retention costs. Theoretical Framework: The theoretical framework is determined by the sectoral regulatory framework regarding the liberalization of services of general economic interest in the EU, the experience of postal operators in recent decades and the changes experienced in the demand for postal services in the current context of profound technological change in the field of communication. Design/Methodology/Approach: This research work is carried out through a systematic review of the relevant specialized literature, the study of the policy decisions adopted in the EU, the changes in the strategies of postal operators and the rigorous analysis of the evolution of the demand of the postal services in a context of rapid and profound change. Finding: Inadequate regulation of the access to some elements of postal infrastructure or services within the scope of the universal postal service, such as the postcode system, address database, post office boxes, delivery boxes, re-direction and retour to sender service, can generate switching costs and thus limiting competition, with negative effects on the consumer surplus and social welfare. Standardization and access to these elements may be regulated separately Research, Practical & Social Implications: This article proposes solutions to the technical functions of coordination between operators. Our proposals are in line with sectoral directives aimed at strengthening competition in the long term in the postal market. Originality/Value: A correct classification of the change costs applied to the liberalized postal service is made. Likewise, a classification of retention costs. We must underline the progress that our proposals represent to reduce the costs of switching to elements of the postal network whose access is mandatory.[Resumo] Objetivo: Nosso objetivo é examinar os custos de mudança existentes no serviço postal europeu, onde um ambicioso processo de abertura à concorrência foi realizado desde 1997, bem como os diferentes tipos de custos de retenção. Estrutura Teórica: A estrutura teórica é determinada pela estrutura regulatória setorial referente à liberalização dos serviços de interesse econômico geral na UE, pela experiência dos operadores postais nas últimas décadas e pelas mudanças ocorridas na demanda por serviços postais no contexto atual de profundas mudanças tecnológicas no campo da comunicação. Projeto/Metodologia/Abordagem: Este trabalho de pesquisa é realizado por meio de uma revisão sistemática da literatura especializada relevante, do estudo das decisões políticas adotadas na UE, das mudanças nas estratégias dos operadores postais e da análise rigorosa da evolução da demanda dos serviços postais em um contexto de mudanças rápidas e profundas. Conclusões: A regulamentação inadequada do acesso a alguns elementos da infraestrutura postal ou serviços no âmbito do serviço postal universal, como o sistema de código postal, banco de dados de endereços, caixas postais, caixas de entrega, serviço de redirecionamento e retorno ao remetente, pode gerar custos de mudança e, portanto, limitar a concorrência, com efeitos negativos sobre o excedente do consumidor e o bem-estar social. A padronização e o acesso a esses elementos podem ser regulamentados separadamente Pesquisa, Implicações Práticas e Sociais: Este artigo propõe soluções para as funções técnicas de coordenação entre as operadoras. Nossas propostas estão de acordo com as diretrizes setoriais que visam fortalecer a concorrência a longo prazo no mercado postal. Originalidade/Valor: É feita uma classificação correta dos custos de mudança aplicados ao serviço postal liberalizado. Da mesma forma, uma classificação dos custos de retenção. Devemos destacar o progresso que nossas propostas representam para reduzir os custos de mudança para elementos da rede postal cujo acesso é obrigatório.[Resumen] Objetivo: Nuestro objetivo es examinar los costes de cambio de operador existentes en el servicio postal europeo, donde se ha llevado a cabo un ambicioso proceso de apertura a la competencia desde 1997, así como los distintos tipos de costes de retención. Marco Teórico: El marco teórico viene determinado por la estructura regulatoria sectorial relativa a la liberalización de los servicios de interés económico general en la UE, la experiencia de los operadores postales en las últimas décadas y los cambios que se han producido en la demanda de servicios postales en el actual contexto de profundos cambios tecnológicos en el ámbito de la comunicación. Diseño/Metodología/Enfoque: Este trabajo de investigación se realiza mediante una revisión sistemática de la literatura especializada relevante, el estudio de las decisiones políticas adoptadas en la UE, los cambios en las estrategias de los operadores postales y el análisis riguroso de la evolución de la demanda de servicios postales en un contexto de rápidos y profundos cambios. Conclusiones: Una regulación inadecuada del acceso a algunos elementos de la infraestructura postal o servicios del servicio postal universal, como el sistema de códigos postales, la base de datos de direcciones, los buzones, los buzones de entrega, el servicio de redirección y la devolución al remitente, puede generar costes de cambio y, por tanto, limitar la competencia, con efectos negativos sobre el excedente del consumidor y el bienestar social. La normalización y el acceso a estos elementos pueden regularse por separado. Investigación, Implicaciones Prácticas y Sociales: Este artículo propone soluciones para las funciones técnicas de coordinación entre operadores. Nuestras propuestas se ajustan a las orientaciones sectoriales destinadas a reforzar la competencia a largo plazo en el mercado postal. Originalidad/Valor: Se realiza una correcta clasificación de los costes de cambio de operador aplicados al servicio postal liberalizado. Igualmente, una clasificación de los costes de retención. Hay que destacar el avance que suponen nuestras propuestas en la reducción de los costes de cambio de operador para los elementos de la red postal cuyo acceso es obligatorio

On statistical properties of sets fulfilling rolling-type conditions

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity and rolling condition. First, the relations between these shape conditions are analyzed. Second, we obtain for the estimation of sets fulfilling a rolling condition a result of "full consistency" (i.e., consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P-uniformity class (in Billingsley and Topsoe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an r-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (a.s.) to that of the underlying support. Fifth, the above results are applied to get new consistency statements for level set estimators based on the excess mass methodology (Polonik, 1995)

Generalizing the convex hull of a sample: The R package alphahull

This vignette 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 flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations