At Stromboli, coastline populations are exposed to near-field tsunami generated by flank collapses. Landslides on the Sciara del Fuoco can generate waves arriving in the village in less than a few minutes. Due to the hazard posed on Stromboli by tsunami, we have focused our research on answering three key questions in terms of mitigation: (1) How can a tsunami be detected and characterized in real-time? (2) What will be the likely event scenario and on-island impact of the resulting tsunami? (3) When and where will the tsunami arrive, and what is the evacuation capacity? To this end, we have convolved a method to assess escape times from tsunami-exposed coastal areas, with wave travel times and inundation assessments output from numerical simulations. Here we review the local situation in terms of hazard, risk and mitigation measures, and assess progress to date in preparing for tsunami on Stromboli to explore a blueprint for management actions at any near-field tsunami-exposed community.

The volcanic island of Stromboli (southern Tyrrhenian Sea, Italy) is renowned for its persistent, periodic, low-intensity explosive activity, whose spectacular manifestations attract tens of thousands of tourists every year. However, sporadic more intense major explosive and effusive eruptions and paroxysms pose serious threats to the island. In addition to direct hazards, granular slides of volcanic debris and pyroclastic avalanches, which can rapidly reach the sea and potentially generate tsunamis, are often associated with such unpredictable eruptive activity. Due to the very fast propagation of the tsunami around the island and the consequent short tsunami warning time (ranging from less than a minute to only a few minutes), mitigation efforts and evacuation from the Strombolian coast must be carefully planned. In this paper, we describe a new GIS-assisted procedure that allows us to combine the outputs of an ensemble of 156 pre-computed landslide-generated tsunami hazard scenarios (with variable landslide volume, position, and density), statistical exposure data (i.e. the number of inhabitants and tourists), and digital geographic information to obtain a quantitative (scenario-based) risk analysis. By means of the analysis of the road network and coastal morphology, we develop a model with routes and times to reach a safe area from every pixel in the inundated area and an appraisal of the time needed to escape versus the wave arrival time. This allows us to evaluate and quantify the effectiveness of potential risk mitigation by means of evacuation. The creation of an impact score linking the predicted inundation extent and the tsunami warning signals is intended, in the long term, to be used to predict the intensity of future tsunamis and to adapt evacuation plans accordingly. The model, here applied to Stromboli, is general and can be applied to other volcanic islands. Evacuating an island hosting several thousand tourists every summer with very little warning time underlines the absolute necessity for such mitigation efforts, aimed at informing hazard planners and managers and all other stakeholders.

We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M \subset \mathbb{R}^D$ with (possibly) non-empty boundary $\partial M$. The model reunites and extends the most prevalent $\mathcal{C}^2$-type set estimation models: manifolds without boundary, and full-dimensional domains. We consider both the estimation of the manifold $M$ itself and that of its boundary $\partial M$ if non-empty. Given $n$ samples, the minimax rates are of order $O\bigl((\log n/n)^{2/d}\bigr)$ if $\partial M = \emptyset$ and $O\bigl((\log n/n)^{2/(d+1)}\bigr)$ if $\partial M \neq \emptyset$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $O\bigl((\log n/n)^{2/(d+1)}\bigr)$-close to $\partial M$ for reconstructing it.

We study the problem of estimating the surface area of the boundary partial derivative S of a sufficiently smooth set S subset of R-d when the available information is only a finite subset X-n subset of S. We propose two estimators. The first makes use of the Devroye-Wise support estimator and is based on Crofton's formula, which, roughly speaking, states that the (d - 1)-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye-Wise support estimators. The second surface area estimator makes use of the alpha-convex hull of X-n, which is denoted by C alpha (X-n). More precisely, it is the (d- 1)-dimensional surface area of C-alpha (X-n), as denoted by vertical bar C-alpha(X-n)vertical bar(d-1), which is proven to converge to the (d - 1)-dimensional surface area of partial derivative S. Moreover, vertical bar C-alpha(X-n)vertical bar(d-1) can be computed using Crofton's formula. Our results depend on the Hausdorff distance between S and X-n for the Devroye-Wise estimator, and the Hausdorff distance between partial derivative S and partial derivative C-alpha (X-n) for the second estimator.

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 esti-mator and the α-convex hull of the data. As a by-product, our results also cover the convex case, for any dimension.

Given a sample of a random variable supported by a smooth compact manifold $M\subset \mathbb{R}^d$, we propose a test to decide whether the boundary of $M$ 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 $k_n$-nearest neighbors. We prove that the level of the test can be estimated, that, with probability one, its power is one for $n$ large enough, and that there exists a consistent decision rule. Heuristics for choosing a convenient value for the $k_n$ parameter and identifying observations close to the boundary are also given. We provide a simulation study of the test.

We address one of the important problems in Big Data, namely how to combine estimators from different subsamples by robust fusion procedures, when we are unable to deal with the whole sample.

Let $\mathcal{X}=\{X_1,\ldots X_n\}\subset \mathbb{R}^d$ be a random sample of observations drawn with a probability distribution supported on $S$ satisfying that both $S$ and $\overline{S^c}$ are $r_0$-convex ($r_0>0$). In this paper we propose an estimator of the medial axis of $S$ based on the $\lambda$-medial axis and the $r$-convex hull. Its convergence rate is derived. An heuristic to tune the parameters of the estimator is given and a small simulation study is performed.

We address one of the important problems in Big Data, namely how to combine estimators from different subsamples by robust fusion procedures, when we are unable to deal with the whole sample. We propose a general framework based on the classic idea of ‘divide and conquer’. In particular we address in some detail the case of a multivariate location and scatter matrix, the covariance operator for functional data, and clustering problems.

We address one of the important problems in Big Data, namely how to combine estimators from different subsamples by robust fusion procedures, when we are unable to deal with the whole sample. We propose a general framework based on the classic idea of `divide and conquer'. In particular we address in some detail the case of a multivariate location and scatter matrix, the covariance operator for functional data, and clustering problems.

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (Probab. Theory Related Fields 64(4), 411–424, 1983), for data uniformly distributed on the unit cube. Later on, Janson (Ann. Prob. 15, 274–280, 1987) extended the results to data uniformly distributed on any bounded set, and obtained a very fine result, namely, he derived the asymptotic distribution of different maximal-spacings notions. These results have been very useful in many statistical applications. We extend Janson's results to the case where the data are generated from a Hölder continuous density that is bounded from below and whose support is bounded. As an application, we develop a convexity test for the support of a distribution.

The notion of multivariate spacings was introduced and studied by Deheuvels, P. (1983) for data uniformly distributed on the unit cube. Later on, Janson, S. (1987) extended the results to bounded sets, and obtained a very fine result, namely, he derived the exact asymptotic distribution of the maximal spacing. These results have been very useful in many statistical applications. We extend Janson's result to the case where the data are generated from a positive, bounded support Lipchitz continuous density function, and develop a convexity test for the support of a distribution.

We address one of the important problems in Big Data, namely how to combine estimators from different subsamples by robust fusion procedures, when we are unable to deal with the whole sample.

This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set $S \subset {\mathbb R}^d$. The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of $S$, of geometric or topological character. The available information is just a random sample of points drawn on $S$. The term "to identify" means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is $S$ full dimensional? Is $S$ "close to a lower dimensional set" $\mathcal{M}$? If so, can we estimate $\mathcal{M}$ or some functionals of $\mathcal{M}$ (in particular, the Minkowski content of $\mathcal{M}$)? As an important auxiliary tool in the answers of these questions, a denoising procedure is proposed in order to partially remove the noise in the original data. The theoretical results are complemented with some simulations and graphical illustrations.

This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set S ⊂ R d. The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of S, of geometric or topological character. The available information is just a random sample of points drawn on S. The term to identify means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: 1. Is S full dimensional? 2. If S is full dimensional, is it close to a lower dimensional set M? 3. If S is close to a lower dimensional M , can we a) estimate M? b) estimate some functionals defined on M (in particular, the Minkowski content of M)? The theoretical results are complemented with some simulations and graphical illustrations .

In this paper we introduce a new estimator for the support of a multivariate density. It is defined as a union of convex hulls of observations contained in balls of fixed radius. We study the asymptotic behavior of this "local convex hull" for the estimation of the support and its boundary. When the support is smooth enough, the proposed estimator is proved to be, eventually almost surely, homeomorphic to the support. Numerical simulations on simulated data illustrate the performance of our estimator.