Eduardo García-Portugués

University Carlos III de Madrid | UC3M · Department of Statistics

We solve the non-discounted, finite-horizon optimal stopping problem of a Gauss–Markov bridge by using a time-space transformation approach. The associated optimal stopping boundary is proved to be Lipschitz continuous on any closed interval that excludes the horizon, and it is characterized by the unique solution of an integral equation. A Picard...

A kernel density estimator for data on the polysphere $\mathbb{S}^{d_1}\times\cdots\times\mathbb{S}^{d_r}$, with $r,d_1,\ldots,d_r\geq 1$, is presented in this paper. We derive the main asymptotic properties of the estimator, including mean square error, normality, and optimal bandwidths. We address the kernel theory of the estimator beyond the von...

We provide a class of diffusion processes for continuous time-varying multivariate angular data with explicit transition probability densities, enabling exact likelihood inference. The presented diffusions are time-reversible and can be constructed for any pre-specified stationary distribution on the torus, including highly-multimodal mixtures. We...

We introduce a test of uniformity for (hyper)spherical data motivated by the stereographic projection. The closed-form expression of the test statistic and its null asymptotic distribution are derived using Gegenbauer polynomials. The power against rotationally symmetric local alternatives is provided, and simulations illustrate the non-null asympt...

A goodness-of-fit test for the Functional Linear Model with Scalar Response (FLMSR) with responses Missing at Random (MAR) is proposed in this paper. The test statistic relies on a marked empirical process indexed by the projected functional covariate and its distribution under the null hypothesis is calibrated using a wild bootstrap procedure. The...

The degree to which unimodal circular data are concentrated around the mean direction can be quantified using the mean resultant length, a measure known under many alternative names, such as the phase locking value or the Kuramoto order parameter. For maximal concentration, achieved when all of the data take the same value, the mean resultant lengt...

Two new omnibus tests of uniformity for data on the hypersphere are proposed. The new test statistics exploit closed-form expressions for orthogonal polynomials, feature tuning parameters, and are related to a “smooth maximum” function and the Poisson kernel. We obtain exact moments of the test statistics under uniformity and rotationally symmetric...

Principal Component Analysis (PCA) is a well-known linear dimension-reduction technique designed for Euclidean data. In a wide spectrum of applied fields, however, it is common to observe multivariate circular data (also known as toroidal data), rendering spurious the use of PCA on it due to the periodicity of its support. This paper introduces Tor...

This paper presents a comprehensive investigation of existing feature extraction tools for symbolic music and contrasts their performance to determine the set of features that best characterizes the musical style of a given music score. In this regard, we propose a novel feature extraction tool, named musif, and evaluate its efficacy on various rep...

Skeletal representations (s-reps) have been successfully adopted to parsimoniously parametrize the shape of three-dimensional objects and have been particularly employed in analyzing hippocampus shape variation. Within this context, we provide a fully nonparametric dimension-reduction tool based on kernel smoothing for determining the main source o...

We construct a goodness-of-fit test for the Functional Linear Model with Scalar Response (FLMSR) with responses Missing At Random (MAR). For that, we extend an existing testing procedure for the case where all responses have been observed to the case where the responses are MAR. The testing procedure gives rise to a statistic based on a marked empi...

Two new omnibus tests of uniformity for data on the hypersphere are proposed. The new test statistics leverage closed-form expressions for orthogonal polynomials, feature tuning parameters, and are related to a "smooth maximum" function and the Poisson kernel. We obtain exact moments of the test statistics under uniformity and rotationally symmetri...

A sizable amount of goodness-of-fit tests involving functional data have appeared in the last decade. We provide a relatively compact revision of most of these contributions, within the independent and identically distributed framework, by reviewing goodness-of-fit tests for distribution and regression models with functional predictor and either sc...

Principal Component Analysis (PCA) is a well-known linear dimension-reduction technique designed for Euclidean data. In a wide spectrum of applied fields, however, it is common to observe multivariate circular data (also known as toroidal data), rendering spurious the use of PCA on it due to the periodicity of its support. This paper introduces Tor...

We solve the non-discounted, finite-horizon optimal stopping problem of a Gauss-Markov bridge by using a time-space transformation approach. The associated optimal stopping boundary is proved to be Lipschitz continuous and differentiable anywhere away from the horizon, and it is characterized by the unique solution of an integral equation. A Picard...

We study the barrier that gives the optimal time to exercise an American option written on a time-dependent Ornstein-Uhlenbeck process, a diffusion often adopted by practitioners to model commodity prices and interest rates. By framing the optimal exercise of the American option as a problem of optimal stopping and relying on probabilistic argument...

Exact null distributions of goodness-of-fit test statistics are generally challenging to obtain in tractable forms. Practitioners are therefore usually obliged to rely on asymptotic null distributions or Monte Carlo methods, either in the form of a lookup table or carried out on demand, to apply a goodness-of-fit test. There exist simple and useful...

A particularly challenging context for dimensionality reduction is multivariate circular data, i.e., data supported on a torus. Such kind of data appears, e.g., in the analysis of various phenomena in ecology and astronomy, as well as in molecular structures. This paper introduces Scaled Torus Principal Component Analysis (ST-PCA), a novel approach...

Exact null distributions of goodness-of-fit test statistics are generally challenging to obtain in tractable forms. Practitioners are therefore usually obliged to rely on asymptotic null distributions or Monte Carlo methods, either in the form of a lookup table or carried out on demand, to apply a goodness-of-fit test. Stephens (1970) provided rema...

Markov bridges may be useful models in finance to describe situations in which information on the underlying processes is known in advance. However, within the framework of optimal stopping problems, Markov bridges are inherently challenging processes as they are time-inhomogeneous and account for explosive drifts. Consequently, few results are kno...

A particularly challenging context for dimensionality reduction is multivariate circular data, i.e., data supported on a torus. Such kind of data appears, e.g., in the analysis of various phenomena in ecology and astronomy, as well as in molecular structures. This paper introduces Scaled Torus Principal Component Analysis (ST-PCA), a novel approach...

We consider one of the most classical problems in multivariate statistics, namely the problem of testing isotropy, or equivalently, the problem of testing uniformity on the unit hypersphere $\mathcal{S}^{p-1}$ of $\mathbb{R}^p$. Rather than restricting to tests that can detect specific types of alternatives only, we consider the broad class of Sobo...

Implementation of several goodness-of-fit tests for functional data. Currently, mostly related with the functional linear model with functional/scalar response and functional/scalar predictor. The package allows for the replication of the data applications considered in García-Portugués, Álvarez-Liébana, Álvarez-Pérez and González-Manteiga (2021)

Testing uniformity of a sample supported on the hypersphere is one of the first steps when analysing multivariate data for which only the directions (and not the magnitudes) are of interest. In this work, a projection-based Cramér–von Mises test of uniformity on the hypersphere is introduced. This test can be regarded as an extension of the well-kn...

A sizable amount of goodness-of-fit tests involving functional data have appeared in the last decade. We provide a relatively compact revision of most of these contributions, within the independent and identically distributed framework, by reviewing goodness-of-fit tests for distribution and regression models with functional predictor and either sc...

We introduce nonparametric tests of independence for bivariate circular data based on trigonometric moments. Our contributions lie in (i) proposing nonparametric tests that are locally and asymptotically optimal against bivariate cosine von Mises alternatives and (ii) extending these tests, via the empirical characteristic function, to obtain consi...

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere, and their extensions. Typically, such data can be repre...

Testing uniformity of a sample supported on the hypersphere is one of the first steps when analysing multivariate data for which only the directions (and not the magnitudes) are of interest. In this work, a projection-based Cram\'er-von Mises test of uniformity on the hypersphere is introduced. This test can be regarded as an extension of the well-...

Functional linear models are one of the most fundamental tools to assess the relation between two random variables of a functional or scalar nature. This contribution proposes a goodness-of-fit test for the functional linear model with functional response that neatly adapts to functional/scalar responses/predictors. In particular, the new goodness-...

We propose a projection-based class of uniformity tests on the hypersphere using statistics that integrate, along all possible directions, the weighted quadratic discrepancy between the empirical cumulative distribution function of the projected data and the projected uniform distribution. Simple expressions for several test statistics are obtained...

The Functional Linear Model with Functional Response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this paper, we propose a novel goodness‐of‐fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cramer‐von Mises norm over...

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by mo...

Functional linear models are one of the most fundamental tools to assess the relation between two random variables of a functional or scalar nature. This contribution proposes a goodness-of-fit test for the functional linear model with functional response that neatly adapts to functional/scalar responses/predictors. In particular, the new goodness-...

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be repres...

Software companion for the paper “A goodness-of-fit test for the functional linear model with functional response” (García-Portugués, Álvarez-Liébana, Álvarez-Pérez and González-Manteiga, 2019). It implements the proposed estimators and goodness-of-fit tests for the functional linear model with scalar response. It also allows to replicate the data...

The Functional Linear Model with Functional Response (FLMFR) is one of the most fundamental models to asses the relation between two functional random variables. In this paper, we propose a novel goodness-of-fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cram\'er-von Mises norm ove...

This chapter shows how toroidal diffusions are convenient methodological tools for modelling protein evolution in a probabilistic framework. The chapter addresses the construction of ergodic diffusions with stationary distributions equal to well-known directional distributions, which can be regarded as toroidal analogues of the Ornstein-Uhlenbeck p...

When modeling directional data, that is, unit-norm multivariate vectors, a first natural question is to ask whether the directions are uniformly distributed or, on the contrary, whether there exist modes of variation significantly different from uniformity. We review in this article a reasonably exhaustive collection of uniformity tests for assessi...

Testing procedures for assessing specific parametric model forms, or for checking the plausibility of simplifying assumptions, play a central role in the mathematical treatment of the uncertain. No certain answers are obtained by testing methods, but at least the uncertainty of these answers is properly quantified. This is the case for tests design...

Testing procedures for assessing specific parametric model forms, or for checking the plausibility of simplifying assumptions, play a central role in the mathematical treatment of the uncertain. No certain answers are obtained by testing methods, but at least the uncertainty of these answers is properly quantified. This is the case for tests design...

We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the Ornstein-Uhlenbeck process. Their likelihood fun...

We illustrate the advantages of distance weighted discrimination for classification and feature extraction in a High Dimension Low Sample Size (HDLSS) situation. The HDLSS context is a gender classification problem of face images in which the dimension of the data is several orders of magnitude larger than the sample size. We compare distance weigh...

Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach and tackle the situations where the location of the symmetry axis is either specified or unspecified. For each problem, we define two te...

Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach and tackle problems where the location of the symmetry axis is either specified or unspecified. For each problem, we define two tests an...

We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the Ornstein-Uhlenbeck process. Their likelihood fun...

We consider marked empirical processes, indexed by a randomly projected functional covariate, to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rate...

We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rates...

Recently described stochastic models of protein evolution have demonstrated that the inclusion of structural information in addition to amino acid sequences leads to a more reliable estimation of evolutionary parameters. We present a generative, evolutionary model of protein structure and sequence that is valid on a local length scale. The model co...

Wildfires represent a threat to natural resources, causing a huge economic and environmental damage, so an effective management of wildfires is required in order to avoid devastating effects. Preventive policies recommend fuel management practices at landscape level, but these measurements will only be successful if strategically placed in order to...

This paper presents a goodness-of-fit test for parametric regression models with scalar response and directional predictor, that is, vectors in a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by...

A central limit theorem for the integrated squared error of the directional-linear kernel density estimator is established. The result enables the construction and analysis of two testing procedures based on the squared loss: a nonparametric independence test for directional and linear random variables and a goodness-of-fit test for parametric fami...

New bandwidth selectors for kernel density estimation with directional data are presented in this work. These selectors are based on asymptotic and exact error expressions for the kernel density estimator combined with mixtures of von Mises distributions. The performance of the proposed selectors is investigated in a simulation study and compared w...

A nonparametric test for assessing the independence between a directional random variable (circular or spherical, as particular cases) and a linear one is proposed in this paper. The statistic is based on the squared distance between nonparametric kernel density estimates and its calibration is done by a permutation approach. The size and power cha...

A nonparametric kernel density estimator for directional-linear data is introduced. The proposal is based on a product kernel accounting for the different nature of both (directional and linear) components of the random vector. Expressions for bias, variance and mean integrated square error MISE are derived, jointly with an asymptotic normality res...

Functional data have been the subject of many research works over the last years. Functional regression is one of the most discussed issues. Specifically, significant advances have been made for functional linear regression models with scalar response. Let $(\mathcal{H},<\cdot,\cdot>)$ be a separable Hilbert space. We focus on the model $Y=<\Theta,...

The study of environmental problems usually requires the description of variables with different nature and the assessment of relations between them. In this work, an algorithm for flexible estimation of the joint density for a circular-linear variable is proposed. The method is applied for exploring the relation between wind direction and SO2 conc...

In this work, a goodness-of-fit test for the null hypothesis of a functional linear model with scalar response is proposed. The test is based on a generalization to the functional framework of a previous one, designed for the goodness--of--fit of regression models with multivariate covariates using random projections. The test statistic is easy to...