arXiv subjects

Alejandro Cholaquidis

Publications and source records attributed to Alejandro Cholaquidis.

At least 37 records · Page 2Linked to original sources

arXiv2021

Level sets of depth measures in abstract spaces

The lens depth of a point has been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points X and Y, with the same radius d(X, Y). We study the consistency in Hausdorff and measure distance, of the level sets of the empirical lens depth, based on an iid sample on a general metric space. We also prove that the boundary of the empirical level sets are consistent estimators of their population counterparts, and analyze two real-life examples

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arXiv2020

On a general definition of the functional linear model

A general formulation of the linear model with functional (random) explanatory variable $X = X(t), t \in T$ , and scalar response Y is proposed. It includes the standard functional linear model, based on the inner product in the space $L^2[0,1]$, as a particular case. It also includes all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite or countable collections of marginal variables X(t_j), with $t_j\in T$ or a linear combination of a finite number of linear projections of X. This general formulation can be interpreted in terms of the RKHS space generated by the covariance function of the process X(t). Some consistency results are proved. A few experimental results are given in order to show the practical interest of considering, in a unified framework, linear models based on a finite number of marginals $X(t_j)$ of the process $X(t)$.

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arXiv2020

Weighted lens depth: Some applications to supervised classification

Starting with Tukey's pioneering work in the 1970's, the notion of depth in statistics has been widely extended especially in the last decade. These extensions include high dimensional data, functional data, and manifold-valued data. In particular, in the learning paradigm, the depth-depth method has become a useful technique. In this paper we extend the notion of lens depth to the case of data in metric spaces, and prove its main properties, with particular emphasis on the case of Riemannian manifolds, where we extend the concept of lens depth in such a way that it takes into account non-convex structures on the data distribution. Next we illustrate our results with some simulation results and also in some interesting real datasets, including pattern recognition in phylogenetic trees using the depth--depth approach.

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arXiv2020

Set Estimation Under Biconvexity Restrictions

A set in the Euclidean plane is said to be biconvex if, for some angle $θ\in[0,π/2)$, all its sections along straight lines with inclination angles $θ$ and $θ+π/2$ 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 $S$ from a random sample of points drawn on $S$. By analogy with the classical convex case, one would like to define the "biconvex hull" of the sample points as a natural estimator for $S$. However, as previously pointed out by several authors, the notion of "hull" for a given set $A$ (understood as the "minimal" set including $A$ 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 $S$ 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 $S$, the biconvexity angle $θ$ is also given.

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arXiv2019

On semi-supervised learning

Semi-supervised learning deals with the problem of how, if possible, to take advantage of a huge amount of unclassified data, to perform a classification in situations when, typically, there is little labeled data. Even though this is not always possible (it depends on how useful, for inferring the labels, it would be to know the distribution of the unlabeled data), several algorithm have been proposed recently. %but in general they are not proved to outperform A new algorithm is proposed, that under almost necessary conditions, %and it is proved that it attains asymptotically the performance of the best theoretical rule as the amount of unlabeled data tends to infinity. The set of necessary assumptions, although reasonable, show that semi-supervised classification only works for very well conditioned problems. The focus is on understanding when and why semi-supervised learning works when the size of the initial training sample remains fixed and the asymptotic is on the size of the unlabeled data. The performance of the algorithm is assessed in the well known "Isolet" real-data of phonemes, where a strong dependence on the choice of the initial training sample is shown.

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arXiv2019

On boundary detection

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.

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arXiv2018

A combined strategy for multivariate density estimation

Non-linear aggregation strategies have recently been proposed in response to the problem of how to combine, in a non-linear way, estimators of the regression function (see for instance \cite{biau:16}), classification rules (see \cite{ch:16}), among others. Although there are several linear strategies to aggregate density estimators, most of them are hard to compute (even in moderate dimensions). Our approach aims to overcome this problem by estimating the density at a point $x$ using not just sample points close to $x$ but in a neighborhood of the (estimated) level set $f(x)$. We show, both theoretically and through a simulation study, that the mean squared error of our proposal is smaller than that of the aggregated densities. A Central Limit Theorem is also proven.

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arXiv2018

Level sets and drift estimation for reflected Brownian motion with drift

We consider the estimation of the drift and the level sets of the stationary distri- bution of a Brownian motion with drift, reflected in the boundary of a compact set $S\subset R^d$ , departing from the observation of a trajectory of this process. We obtain the uniform consistency and rates of convergence for the proposed kernel based estimators. This problem has relevant applications in ecology, in estimating the home-range and the core-area of an animal based on tracking data. Recently, the problem of estimat- ing the domain of a reflected Brownian motion was considered in Cholaquidis, et al. (2016), when the stationary distribution is uniform and the estimation of the core-area, defined as a level set of the stationary distribution, is meaningless. As a by-product of our results, we obtain an estimation of the drift function. In order to prove our re- sults, some new theoretical properties of the reflected Brownian motion with drift are obtained, under fairly general assumptions. These properties allow us to perform the estimation for flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.

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arXiv2018

Robust Fusion Methods for Structured Big Data

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.

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arXiv2017

Semi-supervised learning

Semi-supervised learning deals with the problem of how, if possible, to take advantage of a huge amount of not classified data, to perform classification, in situations when, typically, the labelled data are few. Even though this is not always possible (it depends on how useful is to know the distribution of the unlabelled data in the inference of the labels), several algorithm have been proposed recently. A new algorithm is proposed, that under almost neccesary conditions, attains asymptotically the performance of the best theoretical rule, when the size of unlabeled data tends to infinity. The set of necessary assumptions, although reasonables, show that semi-parametric classification only works for very well conditioned problems.

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arXiv2017

Detection of low dimensionality and data denoising via set estimation techniques

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.

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arXiv2017

Robust Fusion Methods for Big Data

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.

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arXiv2016

On the classification problem for Poisson Point Processes

We study the binary classification problem for Poisson point processes, which are allowed to take values in a general metric space. The problem is tackled in two different ways: estimating nonparametricaly the intensity functions of the processes (and then plugged into a deterministic formula which expresses the regression function in terms of the intensities), and performing the classical $k$ nearest neighbor rule by introducing a suitable distance between patterns of points. In the first approach we prove the consistency of the estimated intensity so that the rule turns out to be also consistent. For the $k$-NN classifier, we prove that the regression function fulfils the so called "Besicovitch condition", usually required for the consistency of the classical classification rules. The theoretical findings are illustrated on simulated data, where in one case the $k$-NN rule outperforms the first approach.

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arXiv2016

A Generalization of the maximal-spacings in several dimensions and a convexity test

The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (1983) for data uniformly distributed on the unit cube. Later on, Janson (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.

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arXiv2015

Set estimation from reflected Brownian motion

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 an animal based on tracking data. There are a variety of studies on the habitat of animals that employ the notion of home range. This paper offers theoretical foundations for a new methodology that, under fairly unrestrictive shape assumptions, allows one to find flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.

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arXiv2015

On visual distances for spectrum-type functional data

A functional distance ${\mathbb H}$, based on the Hausdorff metric between the function hypographs, is proposed for the space ${\mathcal E}$ of non-negative real upper semicontinuous functions on a compact interval. The main goal of the paper is to show that the space $({\mathcal E},{\mathbb H})$ is particularly suitable in some statistical problems with functional data which involve functions with very wiggly graphs and narrow, sharp peaks. A typical example is given by spectrograms, either obtained by magnetic resonance or by mass spectrometry. On the theoretical side, we show that $({\mathcal E},{\mathbb H})$ is a complete, separable locally compact space and that the ${\mathbb H}$-convergence of a sequence of functions implies the convergence of the respective maximum values of these functions. The probabilistic and statistical implications of these results are discussed in particular, regarding the consistency of $k$-NN classifiers for supervised classification problems with functional data in ${\mathbb H}$. On the practical side, we provide the results of a small simulation study and check also the performance of our method in two real data problems of supervised classification involving mass spectra.

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arXiv2015

A nonlinear 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 $M$ 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 $M$ classifiers. The results of a small simulation are reported both, for high dimensional and functional data, and a real data example is analyzed.

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arXiv2014

A geometrically motivated parametric model in manifold estimation,

The general aim of manifold estimation is reconstructing, by statistical methods, an $m$-dimensional compact manifold $S$ on ${\mathbb R}^d$ (with $m\leq d$) or estimating some relevant quantities related to the geometric properties of $S$. We will assume that the sample data are given by the distances to the $(d-1)$-dimensional manifold $S$ from points randomly chosen on a band surrounding $S$, with $d=2$ and $d=3$. The point in this paper is to show that, if $S$ belongs to a wide class of compact sets (which we call \it sets with polynomial volume\rm), the proposed statistical model leads to a relatively simple parametric formulation. In this setup, standard methodologies (method of moments, maximum likelihood) can be used to estimate some interesting geometric parameters, including curvatures and Euler characteristic. We will particularly focus on the estimation of the $(d-1)$-dimensional boundary measure (in Minkowski's sense) of $S$. It turns out, however, that the estimation problem is not straightforward since the standard estimators show a remarkably pathological behavior: while they are consistent and asymptotically normal, their expectations are infinite. The theoretical and practical consequences of this fact are discussed in some detail.

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