Generalized Spatial Regression with Differential Regularization

We aim at analyzing geostatistical and areal data observed over irregularly shaped spatial domains and having a distribution within the exponential family. We propose a generalized additive model that allows to account for spatially-varying covariate information. The model is fitted by maximizing a penalized log-likelihood function, with a roughness penalty term that involves a differential quantity of the spatial field, computed over the domain of interest. Efficient estimation of the spatial field is achieved resorting to the finite element method, which provides a basis for piecewise polynomial surfaces. The proposed model is illustrated by an application to the study of criminality in the city of Portland, Oregon, USA

IGS: an IsoGeometric approach for Smoothing on surfaces

We propose an Isogeometric approach for smoothing on surfaces, namely estimating a function starting from noisy and discrete measurements. More precisely, we aim at estimating functions lying on a surface represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional. The latter is equivalent to solve a 4th-order Partial Differential Equation (PDE). In this context, we use Isogeometric Analysis (IGA) for the numerical approximation of such surface PDE, leading to an IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a surface. Indeed, IGA facilitates encapsulating the exact geometrical representation of the surface in the analysis and also allows the use of at least globally $C^1-$continuous NURBS basis functions for which the 4th-order PDE can be solved using the standard Galerkin method. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet of the SOAR space shuttle

Integrated Depths for Partially Observed Functional Data

Partially observed functional data are frequently encountered in applications and are the object of an increasing interest by the literature. We here address the problem of measuring the centrality of a datum in a partially observed functional sample. We propose an integrated functional depth for partially observed functional data, dealing with the very challenging case where partial observability can occur systematically on any observation of the functional dataset. In particular, differently from many techniques for partially observed functional data, we do not request that some functional datum is fully observed, nor we require that a common domain exist, where all of the functional data are recorded. Because of this, our proposal can also be used in those frequent situations where reconstructions methods and other techniques for partially observed functional data are inapplicable. By means of simulation studies, we demonstrate the very good performances of the proposed depth on finite samples. Our proposal enables the use of benchmark methods based on depths, originally introduced for fully observed data, in the case of partially observed functional data. This includes the functional boxplot, the outliergram and the depth versus depth classifiers. We illustrate our proposal on two case studies, the first concerning a problem of outlier detection in German electricity supply functions, the second regarding a classification problem with data obtained from medical imaging. for this article are available online

Functional Data Analysis of Amplitude and Phase Variation

The abundance of functional observations in scientific endeavors has led to a significant development in tools for functional data analysis (FDA). This kind of data comes with several challenges: infinite-dimensionality of function spaces, observation noise, and so on. However, there is another interesting phenomena that creates problems in FDA. The functional data often comes with lateral displacements/deformations in curves, a phenomenon which is different from the height or amplitude variability and is termed phase variation. The presence of phase variability artificially often inflates data variance, blurs underlying data structures, and distorts principal components. While the separation and/or removal of phase from amplitude data is desirable, this is a difficult problem. In particular, a commonly used alignment procedure, based on minimizing the $\mathbb{L}^2$ norm between functions, does not provide satisfactory results. In this paper we motivate the importance of dealing with the phase variability and summarize several current ideas for separating phase and amplitude components. These approaches differ in the following: (1) the definition and mathematical representation of phase variability, (2) the objective functions that are used in functional data alignment, and (3) the algorithmic tools for solving estimation/optimization problems. We use simple examples to illustrate various approaches and to provide useful contrast between them.Comment: Published at http://dx.doi.org/10.1214/15-STS524 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province

We propose a method for the analysis of functional data with complex dependencies, such as spatially dependent curves or time dependent surfaces, over highly textured domains. The models are based on the idea of regression with partial differential regularizations. In particular, we consider here two roughness penalties that account separately for the regularity of the field in space and in time. Among the various modelling features, the proposed method is able to deal with spatial domains featuring peninsulas, islands and other complex geometries. Space-time varying covariate information is included in the model via a semi-parametric framework. The proposed method is compared via simulation studies to other spatiotemporal techniques and it is applied to the analysis of the annual production of waste in the towns of Venice province