Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions

Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since, in general, convolution does not provide closed-form integration, we discuss efficient computation. We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze a trivariate pollution dataset from California.     

Development and Evaluation of Geostatistical Methods for Non-Euclidean-Based Spatial Covariance Matrices

Mathematical Geosciences - Customary and routine practice of geostatistical modeling assumes that inter-point distances are a Euclidean metric (i.e., as the crow flies) when characterizing spatial...     

Correction to: Development and Evaluation of Geostatistical Methods for Non-Euclidean-Based Spatial Covariance Matrices

Mathematical Geosciences - The original version of this article unfortunately contained a mistake in equation 9.     

Compositional Kriging: A Spatial Interpolation Method for Compositional Data

Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable.     

Geostatistics for Compositional Data: An Overview

Mathematical Geosciences - This paper presents an overview of results for the geostatistical analysis of collocated multivariate data sets, whose variables form a composition, where the components...     

The Characterization Problem for Isotropic Covariance Functions

Isotropic covariance functions are successfully used to model spatial continuity in a multitude of scientific disciplines. Nevertheless, a satisfactory characterization of the class of permissible isotropic covariance models has been missing. The intention of this note is to review, complete, and extend the existing literature on the problem. As it turns out, a famous conjecture of Schoenberg (1938) holds true: any measurable, isotropic covariance function on ^d (d 2) admits a decomposition as the sum of a pure nugget effect and a continuous covariance function. Moreover, any measurable, isotropic covariance function defined on a ball in ^d can be extended to an isotropic covariance function defined on the entire space ^d .     

Correlation Analysis for Compositional Data

Compositional data need a special treatment prior to correlation analysis. In this paper we argue why standard transformations for compositional data are not suitable for computing correlations, and why the use of raw or log-transformed data is neither meaningful. As a solution, a procedure based on balances is outlined, leading to sensible correlation measures. The construction of the balances is demonstrated using a real data example from geochemistry. It is shown that the considered correlation measures are invariant with respect to the choice of the binary partitions forming the balances. Robust counterparts to the classical, non-robust correlation measures are introduced and applied. By using appropriate graphical representations, it is shown how the resulting correlation coefficients can be interpreted.     

Geostatistical Data Integration Model for Contamination Assessment

Soil contamination assessments can be improved with new methods aimed at the accurate estimation of the volume and extension of contaminated soil to be remediated. Geostatistical models that use secondary information to characterize soil contamination are incorporated into a new integration model to provide accurate three-dimensional maps. The proposed integration model is based on a stochastic inversion approach and uses sequential indicator simulation. A two-dimensional reference image representing the areal extension of the contamination is combined with local measurements of contamination in the vertical direction, to render a three-dimensional contamination map. To demonstrate how well the integration model performs, the case study presented focuses on geophysical data and how it can be integrated with soil contamination measurements to improve the characterization of a contaminated site. The results show that the model reproduces successfully the reference image thus providing an accurate three-dimensional contamination map.     

Covariance-Based Variable Selection for Compositional Data

Omitting variables in compositional data analysis may lead to a substantial change in results from that of multivariate statistical analysis. In particular, this is the case for principal component analysis and the compositional biplot, where both the interpretation of loadings and scores of the remaining subcomposition are affected. A stepwise procedure is introduced that allows for a reduction of the original composition to a subcomposition by avoiding a substantial change of the information, like those carried by the compositional biplot. The subcomposition is easier to handle and interpret. Numerical results give evidence of the usefulness of the procedure.     

BLU Estimators and Compositional Data

One of the principal objections to the logratio approach for the statistical analysis of compositional data has been the absence of unbiasedness and minimum variance properties of some estimators: they seem not to be BLU estimator. Using a geometric approach, we introduce the concept of metric variance and of a compositional unbiased estimator, and we show that the closed geometric mean is a c-BLU estimator (compositional best linear unbiased estimator with respect to the geometry of the simplex) of the center of the distribution of a random composition. Thus, it satisfies analogous properties to the arithmetic mean as a BLU estimator of the expected value in real space. The geometric approach used gives real meaning to the concepts of measure of central tendency and measure of dispersion and opens up a new way of understanding the statistical analysis of compositional data.     

Advances in Principal Balances for Compositional Data

Compositional data analysis requires selecting an orthonormal basis with which to work on coordinates. In most cases this selection is based on a data driven criterion. Principal component analysis provides bases that are, in general, functions of all the original parts, each with a different weight hindering their interpretation. For interpretative purposes, it would be better to have each basis component as a ratio or balance of the geometric means of two groups of parts, leaving irrelevant parts with a zero weight. This is the role of principal balances, defined as a sequence of orthonormal balances which successively maximize the explained variance in a data set. The new algorithm to compute principal balances requires an exhaustive search along all the possible sets of orthonormal balances. To reduce computational time, the sets of possible partitions for up to 15 parts are stored. Two other suboptimal, but feasible, algorithms are also introduced: (i) a new search for balances following a constrained principal component approach and (ii) the hierarchical cluster analysis of variables. The latter is a new approach based on the relation between the variation matrix and the Aitchison distance. The properties and performance of these three algorithms are illustrated using a typical data set of geochemical compositions and a simulation exercise.     

Isometric Logratio Transformations for Compositional Data Analysis

Geometry in the simplex has been developed in the last 15 years mainly based on the contributions due to J. Aitchison. The main goal was to develop analytical tools for the statistical analysis of compositional data. Our present aim is to get a further insight into some aspects of this geometry in order to clarify the way for more complex statistical approaches. This is done by way of orthonormal bases, which allow for a straightforward handling of geometric elements in the simplex. The transformation into real coordinates preserves all metric properties and is thus called isometric logratio transformation (ilr). An important result is the decomposition of the simplex, as a vector space, into orthogonal subspaces associated with nonoverlapping subcompositions. This gives the key to join compositions with different parts into a single composition by using a balancing element. The relationship between ilr transformations and the centered-logratio (clr) and additive-logratio (alr) transformations is also studied. Exponential growth or decay of mass is used to illustrate compositional linear processes, parallelism and orthogonality in the simplex.     

A Graph Clustering Approach to Localization for Adaptive Covariance Tuning in Data Assimilation Based on State-Observation Mapping

Mathematical Geosciences - An original graph clustering approach for the efficient localization of error covariances is proposed within an ensemble-variational data assimilation framework. Here,...     

A Parametric Approach for Dealing with Compositional Rounded Zeros

In this work, a parametric approach for replacing data below the detection limit, also known as rounded zeros, in compositional data sets is proposed. Compositional rounded zeros correspond to small proportions of some whole that cannot be reliably detected by the analytical instruments under given operating conditions. This kind of zeros appear frequently in the data collection process in geosciences. They must be treated in an adequate way before some multivariate analysis can be applied. Our procedure results from a modification of the Expectation-Maximization (EM) algorithm and is based on the additive log-ratio transformation. Its coherence with the nature of compositional data and with basic operations in the simplex sample space is checked. Using real data sets, we find that this approach improves other parametric and non-parametric techniques for compositional rounded zeros.     

Discriminant Analysis for Compositional Data Incorporating Cell-Wise Uncertainties

Mathematical Geosciences - In the geosciences it is still uncommon to include measurement uncertainties into statistical methods such as discriminant analysis, but, especially for trace elements,...     

Michael Greenacre: Compositional Data Analysis in Practice

Mathematical Geosciences -     

Outlier Detection for Compositional Data Using Robust Methods

Outlier detection based on the Mahalanobis distance (MD) requires an appropriate transformation in case of compositional data. For the family of logratio transformations (additive, centered and isometric logratio transformation) it is shown that the MDs based on classical estimates are invariant to these transformations, and that the MDs based on affine equivariant estimators of location and covariance are the same for additive and isometric logratio transformation. Moreover, for 3-dimensional compositions the data structure can be visualized by contour lines. In higher dimension the MDs of closed and opened data give an impression of the multivariate data behavior.