Beatriz Pateiro-López

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 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. 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 Topsøes (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 (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).

A multivariate uniformity test for the case of unknown support

A test for the hypothesis of uniformity on a support S⊂ℝd 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 properties of this test are analyzed and its performance is checked through a small simulation study. The numerical problems involved in the practical calculation of the maximal spacing (which is required to obtain the test statistic) are also discussed in some detail.

Quantiles for finite and infinite dimensional data

A new projection-based definition of quantiles in a multivariate setting is proposed. This approach extends in a natural way to infinite-dimensional Hilbert spaces. The directional quantiles we define are shown to satisfy desirable properties of equivariance and, from an interpretation point of view, the resulting quantile contours provide valuable information when plotting them. Sample quantiles estimating the corresponding population quantiles are defined and consistency results are obtained. The new concept of principal quantile directions, closely related in some situations to principal component analysis, is found specially attractive for reducing the dimensionality and visualizing important features of functional data. Asymptotic properties of the empirical version of principal quantile directions are also obtained. Based on these ideas, a simple definition of robust principal components for finite and infinite-dimensional spaces is also proposed. The presented methodology is illustrated with examples throughout the paper.

Length and surface area estimation under smoothness restrictions

The problem of estimating the Minkowski content L 0(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 L 0(G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator.

Surface area estimation under convexity type assumptions

The problem of estimating the surface area, L 0, of a set G⊂ℝ 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 area estimation can be seen as a further step where the emphasis is placed on an important geometric characteristic of G. The Minkowski content is an attractive way to define L 0 that has been previously used in the literature on surface area estimation. Pateiro-López and Rodríguez-Casal [B. Pateiro-López and A. Rodríguez-Casal, Length and surface area estimation under smoothness restrictions, Adv. Appl. Prob. 40(2) (2008), pp. 348–358] proposed an estimator, L n , for L 0 under convexity type assumptions. In this paper, we obtain the L 1-convergence rate of L n .

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 biology.

On the estimation of the medial axis and inner parallel body

The medial axis and the inner parallel body of a set C are different formal translations for the notions of the “central core” and the “bulk”, respectively, of C. 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 modified version of the medial axis, called λ-medial axis, has been recently proposed in order to get better stability properties. The estimation of these relevant subsets from a random sample of points is a partially open problem which has been considered only very recently. Our aim is to show that standard, relatively simple, techniques of set estimation can provide natural, consistent, easy-to-implement estimators for both the λ-medial axis Mλ(C) and the inner parallel body Iλ(C) of C. The consistency of these estimators follows from two results of stability (i.e. continuity in the Hausdorff metric) of Mλ(C) and Iλ(C) obtained under some, not too restrictive, regularity assumptions on C. As a consequence, natural algorithms for the approximation of the λ-medial axis and the λ-inner parallel body can be derived. The whole approach could be useful for some practical problems in image analysis where estimation techniques are needed.

Predicting species distributions in new areas or time periods with alpha-shapes

Abstract Statistical models relating species distributions to environmental data are now commonly applied to predict where invasive species may become established or how range limits may shift under climate change. As species absences can originate from factors other than an unsuitable environment (e.g. dispersal constraints), the models that discriminate between occupied and unoccupied environments are likely to underestimate potential ranges. However, the techniques that “envelope” the occupied environments (i.e. profile techniques) usually rely on simple convex estimators (e.g. elliptical or rectangular shapes), which tend to overestimate these ranges. Here we describe alpha-shapes, a profile-type technique that relaxes the assumption of convexity. By using native range data for the invasive African clawed frog, we demonstrate how this technique can be used to model climatic envelopes of variable complexity. In particular, we compared predictions from an envelope maximizing discrimination between presences and absences, an envelope tightly enclosing all occupied climatic combinations (i.e. the minimum bounding envelope) and an “expert-based” generalization of the previous. In addition, we also use this technique to identify climatic combinations that are outside the climatic space of the study area (i.e. non-analog climates). The envelope accounting for the absences of the African clawed frog achieved a high discrimination ability (true skill statisticsxa0=xa00.71), but failed to predict many of the areas in which the species occurs. Predictions based on the minimum bounding envelope encompassed all species occurrences while still providing a sharp delineation of its distribution range. The generalized version of the previous envelope also captured all occurrences, but predicted a wider extent of suitable areas. We also found that most parts of the world present climatic conditions that are non-analog to those of our study area. Although conceptually more suitable for predicting species distributions across space and time than presence–absence models, profile techniques are frequently overlooked because of their inability to fit flexible envelopes. Here, we demonstrate that alpha-shapes are a transparent and intuitive profile-type technique that has this flexibility.

Minimax Estimation of the Volume of a Set under the Rolling Ball Condition

ABSTRACT 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 achieves a minimax lower bound that we derive. Some numerical experiments corroborate our theoretical findings. Supplementary materials for this article are available online.