A universal kriging approach for spatial functional data

We consider the problem of predicting the spatial field of particle-size curves (PSCs) from a sample observed at a finite set of locations within an alluvial aquifer near the city of Tübingen, Germany. We interpret PSCs as cumulative distribution functions and their derivatives as probability density functions. We thus (a) embed the available data into an infinite-dimensional Hilbert Space of compositional functions endowed with the Aitchison geometry and (b) develop new geostatistical methods for the analysis of spatially dependent functional compositional data. This approach enables one to provide predictions at unsampled locations for these types of data, which are commonly available in hydrogeological applications, together with a quantification of the associated uncertainty. The proposed functional compositional kriging (FCK) predictor is tested on a one-dimensional application relying on a set of 60 PSCs collected along a 5-m deep borehole at the test site. The quality of FCK predictions of PSCs is evaluated through leave-one-out cross-validation on the available data, smoothed by means of Bernstein Polynomials. A comparison of estimates of hydraulic conductivity obtained via our FCK approach against those rendered by classical kriging of effective particle diameters (i.e., quantiles of the PSCs) is provided. Unlike traditional approaches, our method fully exploits the functional form of PSCs and enables one to project the complete information content embedded in the PSC to unsampled locations in the system.

Section: Introductionmentioning

confidence: 99%

A kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers

Menafoglio

Guadagnini

Secchi

2014

“…Applications of functional data analysis can be found in various scientific areas, including climatological and environmental ones (see e.g. [24], [6], [2], [13]). However, to our knowledge, little reference is made to heteroskedasticity in the functional data literature.…”

Section: Introductionmentioning

confidence: 99%

Modelling collocation uncertainty of 3D atmospheric profiles

Ignaccolo

Franco‐Villoria

Fassò

2014

Atmospheric thermodynamic data are gathered by high technology remote instruments such as radiosondes, giving rise to profiles that are usually modelled as functions depending only on height. The radiosonde balloons, however, drift away in the atmosphere resulting in not necessarily vertical but threedimensional (3D) trajectories. To model this kind of functional data, we introduce a "point based" formulation of an heteroskedastic functional regression model that includes a trivariate smooth function and results to be an extension of a previously introduced unidimensional model. Functional coefficients of both the conditional mean and variance are estimated by reformulating the model as a standard generalized additive model and subsequently as a mixed model. This reformulation leads to a double mixed model whose parameters are fitted by using an iterative algorithm that allows to adjust for heteroskedasticity. The proposed modelling approach is applied to describe collocation mismatch when we deal with couples of balloons launched at two

“…However, in many applied cases, the assumption of a constant mean function is clearly not realistic. To address this problem, there have been a number of contributions dealing with this situation (see Caballero et al 2013;Menafoglio et al 2013;Ignaccolo et al 2014;Reyes et al 2015). In all these cases, the stationarity assumption is relaxed.…”

Section: Introductionmentioning

confidence: 99%

“…In all these cases, the stationarity assumption is relaxed. Caballero et al (2013) propose a new predictor by extending the classical universal kriging predictor for univariate data to the context of functional data. Menafoglio et al (2013) establish a kriging theory for random fields in any separable Hilbert space, allowing for the analysis of a broad range of object data, such as curves, surfaces or images.…”

Section: Introductionmentioning

confidence: 99%

Advances in spatial functional statistics

Mateu

Romano

2016

Self Cite

This is the editorial letter for the special issue dedicated to Spatial Functional Statistics, motivated by the joint VII International Workshop on Spatio-temporal Modelling (METMAVII) and the 2014 meeting of the research group for Statistical Applications to Environmental Problems (GRASPA14), which took place in Turin (Italy) from 10 to 12 September 2014. This special issue summarises and discusses peer-reviewed contributions related to the analysis of functional data showing complex characteristics such as spatial dependence structures. The selection of papers comprises both new methodological proposals and a wide range of applications. In particular, we cover a wide range of statistical aspects, comprising prediction of functional data with spatial dependence, optimal sampling designs using functional covariates, nonparametric clustering methods for dependent functional data, and depth measures for spatially dependent functional data.