Model Of Coregionalization Research Articles

Isotropic Variogram Matrix Functions on Spheres

This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ 2, log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization.

Scale-dependent correlations between soil properties and environmental factors across the Loess Plateau of China

Traditional statistical analysis of the correlations between spatially distributed variables takes no account of their regionalised nature. Factorial kriging analysis (FKA) was developed and widely used to overcome this problem. In our study, we applied FKA to investigate scale-dependent correlations between selected soil properties and environmental factors across the Loess Plateau of China. Surface soil samples were collected from 382 sampling sites throughout the region, and soil organic carbon (SOC), soil total nitrogen (STN), soil total phosphorus (STP), soil total potassium (STK), soil pH, bulk density (BD), and clay and silt contents were determined. Five environmental factors (elevation, precipitation, temperature, land-use type, and soil type) were also included in the FKA to identify influential processes. A linear model of co-regionalisation, including a nugget effect and two spherical structures (effective ranges of 200 and 400 km), was fitted to the experimental auto- and cross-variograms of the variables. Scale-dependent correlations were calculated for nugget-effect scale (<30–50 km), short-range scale with a range of 200 km, and long-range scale with a range of 400 km. Principal component analysis was conducted to clearly illustrate the correlations at each spatial scale. The scale-dependent correlations were different from the general correlations and varied at different scales. Generally, SOC and STN were strongly correlated at the nugget-effect scale and the long-range scale, but not at the short-range scale. Precipitation and clay content showed close correlations with STP at the nugget-effect scale and long-range scale. The STK was weakly correlated with the other variables at each spatial scale, and closely correlated with soil type at the long-range scale. Soil pH was closely correlated with BD, soil type, and elevation at the nugget-effect, short, and long spatial scales, respectively. Close correlations were found between BD and land-use type at each spatial scale. Land use and soil type were considered to be the important factors controlling spatial variation of soil properties at the short-range scale, while at the long-range scale the likely factors were identified as precipitation, temperature, and elevation. Our study provided an insight into the spatial-dependent correlations between soil properties and environmental factors from a regional perspective.

Mapping the three-dimensional variation of soil salinity in a rice-paddy soil

Soil salinity is widespread in a variety of environments, and land managers need to map its severity and extent both laterally and vertically. In this research we explore the inversion of apparent electrical conductivity (ECa) measured with an EM38 using a linear model and Tikhonov regularization to model electrical conductivity (σ) profiles in a saline paddy field in the Yangtze delta of China. The modelled σ matched closely the directly measured bulk electrical conductivity (σb) in the topsoil within our calibration field. Discrepancies were greatest between 0.4 and 0.8m, below which they converged again, and were judged small enough to map soil salinity. Equivalent ECa data, recorded in an adjacent field, was similarly inverted with the modelled σ analysed geostatistically. In this regard, the σ data at 10 depths were treated as 10 correlated variates, and experimental auto-and cross-variograms were computed by the method of moments from them. A linear model of coregionalization fitted well, and it was used to cokrige σ on 5m×5m blocks on a fine grid. The kriging errors, computed as the square roots of the cokriging variances, were typically about 5% of the kriged estimates. Estimates of σ were then converted into the universal standard of soil salinity measurement (i.e. electrical conductivity of a saturated soil paste extract — ECe). The results indicate that an irregularly shaped patch of strongly saline topsoil (i.e. 8–12dSm−1) and subsoil salinity (i.e. >16dSm−1) at the southern end of the field was consistent with a yield reduction of some 33%; and as compared with the weakly saline conditions evident at the northern end of the field (e.g. topsoil ECe 2–4dSm−1) where yield was much larger. We conclude that the approach has merit and might be useful in providing a baseline set of data and a method that can used to monitor and evaluate the management of salinity.

Automatic Variogram Modeling by Iterative Least Squares: Univariate and Multivariate Cases

In this paper, we propose a new methodology to automatically find a model that fits on an experimental variogram. Starting with a linear combination of some basic authorized structures (for instance, spherical and exponential), a numerical algorithm is used to compute the parameters, which minimize a distance between the model and the experimental variogram. The initial values are automatically chosen and the algorithm is iterative. After this first step, parameters with a negligible influence are discarded from the model and the more parsimonious model is estimated by using the numerical algorithm again. This process is iterated until no more parameters can be discarded. A procedure based on a profiled cost function is also developed in order to use the numerical algorithm for multivariate data sets (possibly with a lot of variables) modeled in the scope of a linear model of coregionalization. The efficiency of the method is illustrated on several examples (including variogram maps) and on two multivariate cases.

Prediction of particle pollution through spatio-temporal multivariate geostatistical analysis: spatial special issue

Vehicular traffic, industrial activity and street dust are important sources of atmospheric particles, which cause pollution and serious health problems, including respiratory illness. Hence, techniques for analyzing and modeling the spatio-temporal behavior of particulate matter (PM), in the recent statistical literature, represent an essential support for environmental and human health protection. In this paper, air pollution from particles with diameters smaller than 10 \({\rm \mu}\)m and related meteorological variables, such as temperature and wind speed, measured during November 2009 in the south of Apulian region (Lecce, Brindisi, and Taranto districts) are studied. A thorough multivariate geostatistical analysis is proposed, where different tools for testing the symmetry assumption of the spatio-temporal linear coregionalization model (ST-LCM) are considered, as well as a recent fitting procedure of the ST-LCM, based on the simultaneous diagonalization of symmetric real-valued matrix variograms, is adopted and two non-separable classes of variogram models, the product–sum and Gneiting classes, are fitted to the basic components. The most significant aspects of this study are (a) the quantitative assessment of the assumption of symmetry of the ST-LCM, (b) the use of different non-separable spatio-temporal models for fitting the basic components of a ST-LCM and, more importantly, (c) the application of the spatio-temporal multivariate geostatistical analysis to predict particle pollution in one of the most polluted geographical area. Prediction maps for particle pollution levels with the corresponding validation results are given.

Multiscale analysis of heavy metal contents in soils: Spatial variability in the Duero river basin (Spain)

Source identification of heavy metals in soil is not straightforward since several inputs of either an anthropogenic or natural origin contribute to their total content. Here we explore the spatial variation and covariation of seven heavy metals (Cd, Cr, Ni, Pb, Zn, Cu and Hg) in the agricultural soils of the Duero river basin (one of the largest in Spain) where both anthropogenic activities (mainly agriculture and industry) and natural factors may be responsible for their total concentration. Factorial kriging and principal components analysis were used on a data set that comprised 721 soil samples. We found that the concentrations of heavy metals in the analysed samples do not exceed the limits set by Spanish legislation excepting mercury that presents high values in a limited number of samples (maximum 1041μg/kg). The linear model of coregionalization–the basic model for factorial kriging analysis–was composed of two structures (representing two scales of variation) with ranges of 20km (local scale) and 130km (regional scale). Six of seven elements (Cd, Cr, Ni, Pb, Zn and Cu) were found to be strongly correlated regardless of the spatial scale considered. In contrast, correlations of Hg with other elements were small at the local spatial scale but augmented substantially at the regional scale. We conclude that agricultural practices in the Duero basin have not altered yet the natural content for Cd, Cr, Ni, Pb, Zn and Cu. On the other hand, Hg inputs from human origin, most probably related to airborne emission and deposition from industrial plants, are observable at the local spatial scale. Finally, no human-induced correlations among heavy metals were detected at the regional spatial scale. Based on the results of this study and in accordance with the results obtained in the nearby Ebro river basin (Rodríguez et al., 2008) we conclude that anthropogenic heavy metals in soil are visible only at local spatial scales. In contrast, natural factors maximize their influence on the distribution of heavy metals when considering larger spatial scales.

Multivariate Gaussian Process Emulators With Nonseparable Covariance Structures

The Gaussian process regression model is a popular type of “emulator” used as a fast surrogate for computationally expensive simulators (deterministic computer models). For simulators with multivariate output, common practice is to specify a separable covariance structure for the Gaussian process. Though computationally convenient, this can be too restrictive, leading to poor performance of the emulator, particularly when the different simulator outputs represent different physical quantities. Also, treating the simulator outputs as independent can lead to inappropriate representations of joint uncertainty. We develop nonseparable covariance structures for Gaussian process emulators, based on the linear model of coregionalization and convolution methods. Using two case studies, we compare the performance of these covariance structures both with standard separable covariance structures and with emulators that assume independence between the outputs. In each case study, we find that only emulators with nonseparable covariances structures have sufficient flexibility both to give good predictions and to represent joint uncertainty about the simulator outputs appropriately. This article has supplementary material online.

Spatial prediction of seabed sediment texture classes by cokriging from a legacy database of point observations

This paper illustrates the potential for statistical mapping of seabed sediment texture classes. It reports the analysis of legacy data on the composition of seabed sediment samples from the UK Continental Shelf with respect to three particle size classes (sand, mud, gravel). After appropriate transformation for compositional variables the spatial variation of the sediment particle size classes was modelled geostatistically using robust variogram estimators to produce a validated linear model of coregionalization. This was then used to predict the composition of seabed sediments at the nodes of a fine grid. The predictions were back-transformed to the original scales of measurement by a Monte Carlo integration over the prediction distribution on the transformed scale. This approach allowed the probability to be computed for each class in a classification of seabed sediment texture, at each node on the grid. The probability of each class, and derived information such as the class of maximum probability could therefore be mapped. Predictions were validated at a set of 2000 randomly sampled locations. The class of maximum probability corresponded to the observed class with a frequency of 0.7, and the uncertainty of this prediction was shown to depend on the absolute probability of the class of maximum probability. Other tests showed that this geostatistical approach gives reliable predictions with meaningful uncertainty measures. This provides a basis for rapid mapping of seabed sediment texture to classes with sound quantification of the uncertainty. Remapping to revised class definitions can also be done rapidly, which will be of particular value in habitat mapping where the seabed geology is an important factor in biotope modelling.

Multi-task learning for pKa prediction

Many compound properties depend directly on the dissociation constants of its acidic and basic groups. Significant effort has been invested in computational models to predict these constants. For linear regression models, compounds are often divided into chemically motivated classes, with a separate model for each class. However, sometimes too few measurements are available for a class to build a reasonable model, e.g., when investigating a new compound series. If data for related classes are available, we show that multi-task learning can be used to improve predictions by utilizing data from these other classes. We investigate performance of linear Gaussian process regression models (single task, pooling, and multi-task models) in the low sample size regime, using a published data set (n = 698, mostly monoprotic, in aqueous solution) divided beforehand into 15 classes. A multi-task regression model using the intrinsic model of co-regionalization and incomplete Cholesky decomposition performed best in 85% of all experiments. The presented approach can be applied to estimate other molecular properties where few measurements are available.

Using Simultaneous Diagonalization to Identify a Space–Time Linear Coregionalization Model

Although there are multiple methods for modeling matrix covariance functions and matrix variograms in the geostatistical literature, the linear coregionalization model is still widely used. In particular it is easy to check to ensure whether the matrix covariance function is positive definite or that the matrix variogram is conditionally negative definite. One of the difficulties in using a linear coregionalization model is in determining the number of basic structures and the corresponding covariance functions or variograms. In this paper, a new procedure is given for identifying the basic structures of the space–time linear coregionalization model and modeling the matrix variogram. This procedure is based on the near simultaneous diagonalization of the sample matrix variograms computed for a set of spatiotemporal lags. A case study using a multivariate spatiotemporal data set provided by the Environmental Protection Agency of Lombardy, Italy, illustrates how nearly simultaneous diagonalization of the empirical matrix variograms simplifies modeling of the matrix variograms. The new methodology is compared with a previous one by analyzing various indices and statistics.

Reducing the number of orthogonal factors in linear coregionalization modeling

The simulation of vector random fields whose spatial correlation structure is represented by a linear coregionalization model can be performed by decomposing the vector components into spatially orthogonal factors and by simulating each factor separately. However, when the number of basic nested structures is large, so is the number of factors, making simulation computationally demanding.This paper proposes a methodology to construct linear coregionalization models with as many nested structures as desired, together with as few orthogonal factors as possible. The construction rests on the decomposition of the model coregionalization matrices into pairwise commuting matrices, followed by a factorization by principal component analysis. The proposed approach is illustrated through a case study in mineral resources evaluation and compared to the traditional fitting procedure, obtaining an equally good fit of the direct and cross variograms but with significantly less factors.

STOX1 overexpression in mice induces severe preeclampsia-like symptoms prevented by aspirin at low doses

Soil salinity is widespread in a variety of environments, and land managers need to map its severity and extent both laterally and vertically. In this research we explore the inversion of apparent electrical conductivity (ECa) measured with an EM38 using a linear model and Tikhonov regularization to model electrical conductivity (σ) profiles in a saline paddy field in the Yangtze delta of China. The modelled σ matched closely the directly measured bulk electrical conductivity (σb) in the topsoil within our calibration field. Discrepancies were greatest between 0.4 and 0.8 m, below which they converged again, and were judged small enough to map soil salinity. Equivalent ECa data, recorded in an adjacent field, was similarly inverted with the modelled σ analysed geostatistically. In this regard, the σ data at 10 depths were treated as 10 correlated variates, and experimental auto-and cross-variograms were computed by the method of moments from them. A linear model of coregionalization fitted well, and it was used to cokrige σ on 5 m × 5 m blocks on a fine grid. The kriging errors, computed as the square roots of the cokriging variances, were typically about 5% of the kriged estimates. Estimates of σ were then converted into the universal standard of soil salinity measurement (i.e. electrical conductivity of a saturated soil paste extract — ECe). The results indicate that an irregularly shaped patch of strongly saline topsoil (i.e. 8–12 dS m− 1) and subsoil salinity (i.e. > 16 dS m− 1) at the southern end of the field was consistent with a yield reduction of some 33%; and as compared with the weakly saline conditions evident at the northern end of the field (e.g. topsoil ECe 2–4 dS m− 1) where yield was much larger. We conclude that the approach has merit and might be useful in providing a baseline set of data and a method that can used to monitor and evaluate the management of salinity.

Hybrid fitting of a hydrosystem model: Long‐term insight into the Beauce aquifer functioning (France)

This study aims at analyzing the water budget of the unconfined Beauce aquifer (8000 km2) over a 35 year period, by modeling the hydrological functioning and quantifying exchanged water fluxes inside the system. A distributed process‐based model (DPBM) is implemented to model the surface, the unsaturated zone and the aquifer subsystems. Based on an extensive literature review on multiparameter optimization and inverse problem, a pragmatic hybrid fitting method that couples manual and automatic calibration is developed. Three data subsets are used for calibration (10 year), validation (10 year) and test (35 year). The global piezometric head root‐mean‐square error is around 2.5 m for the three subsets and is rather uniformly spatially distributed over 78 piezometers. The sensitivity of the simulation to the different steps of the calibration process is investigated. The transmissivity field permits the fitting of the low‐frequency signal for long‐term filtering of the recharge signal, whereas the storage coefficient filters the signal with a higher frequency. For long‐term insight into aquifer system functioning, the priority is thus to first fit the transmissivity field and to assess the distributed aquifer recharge accurately. The fitted DPBM, coupled with a linear model of coregionalization, is then used to quantify the hydrosystem water mass balance between 1974 and 2009, indicating that there is yet no trend of water resources decrease neither due to climate nor to human activities.

Use of EMI, gamma-ray emission and GPS height as multi-sensor data for soil characterisation

Geo-electrical sensors are often used as auxiliary variables with sparse direct measurements to estimate soil properties. Using a single sensor is not ideal in some circumstances. For example, sandy, sandy gravelly, sandy salt-affected and clayey soils are poorly identified using an EMI or gamma-ray sensor singularly. The complementary use of these sensors should improve interpretation in landscapes containing these soils. Analysis of multi-sensor data is however problematic. Several methods have been developed to integrate multi-sensor data but there is currently no unequivocally accepted methodology. The objectives of this work were: 1) to define a combined approach of geostatistics and sensor data fusion to integrate field data from electromagnetic induction (EMI) measured with EM38 and EM31, gamma (γ)-ray and RTK GPS sensors for delineating areas of homogeneous soil; 2) to show the potential of gamma radiometric sensor by estimating a relationship for crop available soil potassium (K) from the γ-ray signal. The geophysical survey was carried out on an 80-ha cropping field in Corrigin, Western Australia. Seventy-seven soil samples were collected at the nodes of a 100×100m-mesh grid and analysed for different properties. The EM38 and EM31 data were strongly correlated with each other and so were γ-radiometric counts from thorium (Th), uranium (U) and all elements (TC). The multi-sensor data were split into 4 subgroups, based on their similarities: 1) EMI data; 2) γ-radiometric counts from potassium (emitted from all forms of K including readily plant available, non-exchangeable and structural K); 3) γ-radiometric counts from Th, U and TC and 4) RTK GPS height. Each group of data was separately analysed using geostatistical techniques. The soil samples and geophysical data were jointly interpolated using multi collocated cokriging. The EMI data showed anisotropy and an anisotropic Linear Model of Coregionalization was fitted before cokriging. The EM31 and EM38 maps looked quite similar. The maps of γ-U, Th and TC were also similar, suggesting that they reflected the same soil properties, but were somewhat different from the γ-K maps. High values of EMI coincided with both low γ-radiometric values at the valley bottom, due to moist sandy salinity-prone soil of varying depth to texture contrast, and high γ-radiometric values at the elevated areas of the field due to emission from finer textured soil. High γ-radiometric values coincided also with low values of EMI over gravelly sands. Only the use of a multi-sensor platform could discriminate soils that gave similar outputs to one sensor. The first two principal components of the geophysical data were used to partition the field into homogeneous areas.In order to test the utility of geophysical survey for K recommendations, the spatial association between the maps of the estimates of plant available soil K content and γ-K counts was demonstrated by using different agreement coefficients and a regression model with correlated errors was estimated between the two variables.

The U-WEDGE Transformation Method for Multivariate Geostatistical Simulation

To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. Spatial decorrelation methods are usually based on the diagonalisation of the variance/covariance and semivariogram matrices of the set of attributes for a chosen family of lag spacings. These matrices are symmetric and there are several efficient methods for the approximate joint diagonalisation of a family of symmetric matrices. One of these is the uniformly weighted exhaustive diagonalisation with Gauss iterations (U-WEDGE) method. In contrast to the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed, U-WEDGE can be applied directly to the set of experimental semivariogram matrices without having to place restrictions on the number of structures in the linear model of coregionalisation, thus removing one of the restrictions placed on the subsequent modelling of the spatial structure of the factors. We use an iron-ore data set to illustrate the method and present a comparison between the simulated attributes obtained from U-WEDGE and MAF with the full co-simulation of the attributes.

Spatial prediction of soil organic carbon from data on large and variable spatial supports. II. Mapping temporal change

We calculate statistical predictions of changes in soil organic carbon (SOC), and attendant uncertainty from areal data across a region of France. The data consist of measurements of SOC from farms across the region collected in two time periods: 1995–1999, and 2000–2004. To protect the anonymity of farms that contributed, the data were summarised by commune; we were only able to use the average value, sample variance and number of observations from each commune. We consider how we can use data of this form to map temporal changes in SOC. We account for the dependence between data from the two surveys through a linear model of coregionalization. Cross‐validation shows that by using the linear model of coregionalization to model inter‐survey dependence, we obtain better estimates of SOC changes and better uncertainty assessments. We compare maps produced using the approaches showing the estimated SOC changes and probabilities of SOC decrease between the times of the two surveys. Copyright © 2012 John Wiley & Sons, Ltd.

3D stochastic joint inversion of gravity and magnetic data

Abstract A novel stochastic joint inversion method based on cokriging is applied to estimate density and magnetic susceptibility distributions from gravity and total magnetic field data. The method fully integrates the physical relations between density–gravity, on one hand, and magnetic susceptibility–total magnetic field, on the other hand. As a consequence, when the data are considered noise-free, the responses from the inverted density and susceptibility data exactly reproduce the observed data. The required density and magnetic susceptibility auto- and cross covariance are assumed to follow a linear model of coregionalization (LCM). The parameters of the LCM are estimated from v–v plot fitting of the gravity and total magnetic experimental covariances. The model is tested on two synthetic cases and one real data set, the Perseverance mine (Quebec, Canada). Joint inversions are compared to separate inversions. The joint inversions better recover the known models in the synthetic cases. With the real data set, better definition and location of the mineralized lenses are achieved by joint inversion.

Co-simulating Total and Soluble Copper Grades in an Oxide Ore Deposit

In oxide copper deposits, the acid soluble copper represents the fraction of total copper recoverable by heap leaching. Two difficulties often complicate the joint modeling and simulation of total and soluble copper grades: the inequality constraint linking both grade variables and the sampling design for soluble copper grade, which may be preferential and cause biases in sample statistics. A methodology is presented in order to accurately estimate the total and soluble copper grade bivariate distribution, based on an explicit modeling of the conditional distributions of soluble copper grade. Co-simulation is then realized by converting the copper grades into Gaussian random fields, through stepwise conditional transformation, and by fitting a coregionalization model while accounting for the preferential sampling design. The proposed approach is illustrated through an application to an ore deposit located in northern Chile.

A Remark on the Use of a Weight Matrix in the Linear Model of Coregionalization

The linear model of coregionalization (LMC) is generally fit to multivariate geostatistical data by minimizing a least-squares criterion. It is commonly believed that weighting the criterion by inverse variances will reduce the influence of those variables with large variance. We point out that this need not be so, and that in some cases the weights will have no effect whatsoever on the estimated sill matrices. When there is an effect, it is due not to a reduction of these variables’ influence, but rather due to a lack of invariance of the minimization problem; moreover, sometimes the influence may actually increase. The correct way to reduce influence is to fit the LMC after standardizing the variables to have unit variance.

Enhanced coregionalization analysis for simulating vector Gaussian random fields

This paper deals with the simulation of a stationary vector Gaussian random field whose spatial correlation structure is given by a linear model of coregionalization. Traditionally, simulation is performed by decomposing the vector random field into a set of independent vector random fields with coregionalization models that contain a single nested structure, and a factorization of these fields into principal components. A variation of this approach is proposed, by considering the minimum/maximum autocorrelation factors associated with groups of two nested structures. This variation reduces the total number of independent factors by one-half, thus considerably decreases memory requirements and CPU time for simulation, without any loss of accuracy for reproducing the linear model of coregionalization, regardless of how many nested structures are contained in this model. The proposed approach is implemented in a set of computer programs and illustrated through a synthetic example and a case study in mineral resources evaluation.