A Class-Kriging predictor for Functional Compositions with Application to Particle-Size Curves in Heterogeneous Aquifers

This work addresses the problem of characterizing the spatial field of soil particle-size distributions within a heterogeneous aquifer system. The medium is conceptualized as a composite system, characterized by spatially varying soil textural properties associated with diverse geomaterials. The heterogeneity of the system is modeled through an original hierarchical model for particle-size distributions that are here interpreted as points in the Bayes space of functional compositions. This theoretical framework allows performing spatial prediction of functional compositions through a functional compositional Class-Kriging predictor. To tackle the problem of lack of information arising when the spatial arrangement of soil types is unobserved, a novel clustering method is proposed, allowing to infer a grouping structure from sampled particle-size distributions. The proposed methodology enables one to project the complete information content embedded in the set of heterogeneous particle-size distributions to unsampled locations in the system. These developments are tested on a field application relying on a set of particle-size data observed within an alluvial aquifer in the Neckar river valley, in Germany

Kriging prediction for manifold-valued random fields

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many cases, data are spatially distributed but it is not trivial to take into account spatial dependence in the analysis because of the non linear geometry of the manifold. This work proposes a solution to the problem of spatial prediction for manifold valued data, with a particular focus on the case of positive definite symmetric matrices. Under the hypothesis that the dispersion of the observations on the manifold is not too large, data can be projected on a suitably chosen tangent space, where an additive model can be used to describe the relationship between response variable and covariates. Thus, we generalize classical kriging prediction, dealing with the spatial dependence in this tangent space, where well established Euclidean methods can be used. The proposed kriging prediction is applied to the matrix field of covariances between temperature and precipitation in Quebec, Canada.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jmva.2015.12.00

Object Oriented Geostatistical Simulation of Functional Compositions via Dimensionality Reduction in Bayes spaces

We address the problem of geostatistical simulation of spatial complex data, with emphasis on functional compositions (FCs). We pursue an object oriented geostatistical approach and interpret FCs as random points in a Bayes Hilbert space. This enables us to deal with data dimensionality and constraints by relying on a solid geometric basis, and to develop a simulation strategy consisting of: (i) optimal dimensionality reduction of the problem through a simplicial principal component analysis, and (ii) geostatistical simulation of random realizations of FCs via an approximate multivariate problem.We illustrate our methodology on a dataset of natural soil particle-size densities collected in an alluvial aquifer

A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space

We address the problem of predicting spatially dependent functional data belonging to a Hilbert space, with a Functional Data Analysis approach. Having defined new global measures of spatial variability for functional random processes, we derive a Universal Kriging predictor for functional data. Consistently with the new established theoretical results, we develop a two-step procedure for predicting georeferenced functional data: first model selection and estimation of the spatial mean (drift), then Universal Kriging prediction on the basis of the identified model. The proposed methodology is applied to daily mean temperatures curves recorded in the Maritimes Provinces of Canada

An object-oriented approach to the analysis of spatial complex data over stream-network domains

We address the problem of spatial prediction for Hilbert data, when their spatial domain of observation is a river network. The reticular nature of the domain requires to use geostatistical methods based on the concept of Stream Distance, which captures the spatial connectivity of the points in the river induced by the network branching. Within the framework of Object Oriented Spatial Statistics (O2S2), where the data are considered as points of an appropriate (functional) embedding space, we develop a class of functional moving average models based on the Stream Distance. Both the geometry of the data and that of the spatial domain are thus taken into account. A consistent definition of covariance structure is developed, and associated estimators are studied. Through the analysis of the summer water temperature profiles in the Middle Fork River (Idaho, USA), our methodology proved to be effective, both in terms of covariance structure characterization and forecasting performance

Stochastic simulation of soil particle-size curves in heterogeneous aquifer systems through a Bayes space approach

We address the problem of stochastic simulation of soil particle-size curves (PSCs) in heterogeneous aquifer systems. Unlike traditional approaches that focus solely on a few selected features of PSCs (e.g., selected quantiles), our approach considers the entire particle-size curves and can optionally include conditioning on available data. We rely on our prior work to model PSCs as cumulative distribution functions and interpret their density functions as functional compositions. We thus approximate the latter through an expansion over an appropriate basis of functions. This enables us to (a) effectively deal with the data dimensionality and constraints and (b) to develop a simulation method for PSCs based upon a suitable and well defined projection procedure. The new theoretical framework allows representing and reproducing the complete information content embedded in PSC data. As a first field application, we demonstrate the quality of unconditional and conditional simulations obtained with our methodology by considering a set of particle-size curves collected within a shallow alluvial aquifer in the Neckar river valley, Germany

Simplicial principal component analysis for density functions in Bayes spaces

Probability density functions are frequently used to characterize the distributional properties of large-scale database systems. As functional compositions, densities primarily carry relative information. As such, standard methods of functional data analysis (FDA) are not appropriate for their statistical processing. The specific features of density functions are accounted for in Bayes spaces, which result from the generalization to the infinite dimensional setting of the Aitchison geometry for compositional data. The aim is to build up a concise methodology for functional principal component analysis of densities. A simplicial functional principal component analysis (SFPCA) is proposed, based on the geometry of the Bayes space B2 of functional compositions. SFPCA is performed by exploiting the centred log-ratio transform, an isometric isomorphism between B2 and L2 which enables one to resort to standard FDA tools. The advantages of the proposed approach with respect to existing techniques are demonstrated using simulated data and a real-world example of population pyramids in Upper Austria