105 research outputs found
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 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
Analyzing data in complicated 3D domains: Smoothing, semiparametric regression, and functional principal component analysis
In this work, we introduce a family of methods for the analysis of data observed at locations scattered in three-dimensional (3D) domains, with possibly complicated shapes. The proposed family of methods includes smoothing, regression, and functional principal component analysis for functional signals defined over (possibly nonconvex) 3D domains, appropriately complying with the nontrivial shape of the domain. This constitutes an important advance with respect to the literature, because the available methods to analyze data observed in 3D domains rely on Euclidean distances, which are inappropriate when the shape of the domain influences the phenomenon under study. The common building block of the proposed methods is a nonparametric regression model with differential regularization. We derive the asymptotic properties of the methods and show, through simulation studies, that they are superior to the available alternatives for the analysis of data in 3D domains, even when considering domains with simple shapes. We finally illustrate an application to a neurosciences study, with neuroimaging signals from functional magnetic resonance imaging, measuring neural activity in the gray matter, a nonconvex volume with a highly complicated structure
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 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
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