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Geostatistical analysis of spatially dependent functional data: Universal Kriging in a Hilbert space
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Statistical analysis of complex and spatially dependent data: A review of Object Oriented Spatial Statistics
Full text linkWe review recent advances in Object Oriented Spatial Statistics, a system of ideas, algorithms and methods that allows the analysis of high dimensional and complex data when their spatial dependence is an important issue. At the intersection of different disciplines – including mathematics, statistics, computer science and engineering – Object Oriented Spatial Statistics provides the right perspective to address key problems in varied contexts, from Earth and life sciences to urban planning. We illustrate a few paradigmatic methods applied to problems of prediction, classification and smoothing, giving emphasis to the key ideas Object Oriented Spatial Statistics relies upon
A Kriging Approach based on Aitchison Geometry for the Characterization of Particle-Size Curves in Heterogeneous Aquifers
Full text linkWe 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 T\"{u}bingen, Germany.
We interpret particle-size curves 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
particle-size curves 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 particle-size curves and enables one to project the complete information content embedded in the PSC to unsampled locations in the system
A Class-Kriging predictor for Functional Compositions with Application to Particle-Size Curves in Heterogeneous Aquifers
Full text linkThis 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
Universal kriging of functional data: trace-variography vs cross-variography? Application to forecasting in unconventional shales
Full text linkIn this paper we investigate the practical and methodological use of Universal Kriging of functional data to predict unconventional shale gas production in undrilled locations from known production data. In Universal Kriging of functional data, two approaches are considered: (1) estimation by means of Cokriging of functional components (Universal Cokriging, UCok), requiring cross-variography and (2) estimation by means of trace-variography (Universal Trace-Kriging, UTrK), which avoids cross-variogram modeling. While theoretically, under known variogram structures, such approaches may be quite equivalent, their practical application implies different estimation procedures and modeling efforts. We investigate these differences from the methodological viewpoint and by means of a real field application in the Barnett shale play. An extensive Monte Carlo study inspired from such real field application is employed to support our conclusions
Kriging prediction for functional compositional data and application to particle-size curves
No full textWe present a new geometric approach to krige functional compositional
data which embraces the viewpoints of both Functional and Compositional Data Analysis. Our theoretical framework enables one to characterize and predict random fields valued in the Hilbert space of functional compositions endowed with the Aitchison geometry. We show the application of the methodology to a field case scenario dealing with particle-size data collected within a heterogeneous aquifer near Tubingen, Germany. We consider particle-size densities, interpreted as functional compositional data, and perform kriging of these curves to obtain a complete characterization of the soil textural properties within the aquife
A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space
Full text linkWe 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
Full text linkWe 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
Prediction of non-stationary functional data: Universal Kriging in a Hilbert space
No full textIn an increasing number of studies, collected data are curves; when functional data are spatially dependent, the problem of prediction assumes a key role. In this work we deal with spatially distributed functional data proposing an extension of some geostatistical tools to non-stationary functional random fields, with a Functional Data Analysis approach. An extension of the Universal Kriging method to elements of a Hilbert space is proposed, in a coherent frame of definitions and assumptions.
Consistently with these new theoretical results, a method for prediction of non-stationary spatial dependent functional data is proposed and then developed in three steps: model selection for the drift term, decomposition of the original process into a deterministic term (the drift) and a residual stochastic process, Universal Kriging prediction.
The proposed procedure is applied to daily mean temperatures curves observed in 35 meteorological stations located in Canada's Maritimes Provinces
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