Geostatistical Simulation of Cross-Correlated Variables: a Case Study through Cerro Matoso Nickel-Laterite Deposit

Geostatistical methods have been increasingly used as powerful techniques for predicting spatial attributes and modelling the uncertainty of predictions in un-sampled locations, especially through multi-element deposits. Independent Gaussian simulation constructs precise outputs over each variable, in most cases by simulating using the multi-Gaussian assumption. However, this approach does not consider the underlying correlations between the variables. Spatial uncertainty can also be quantified by co-simulation, where the relationship of the co-regionalized variables is accounted for and the spatial relationships between variables are reproduced. In this study, we apply the two aforementioned approaches (independent simulation and co-simulation) for modelling two correlated elements (Fe & MgO) at Cerro Matoso S.A. Nickel laterite deposit located in Colombia. Results show that co-simulation provides a reasonable outcome in regards to the correlation coefficient parameter and relative error as expected.Nazarbayev University, School of Mining and Geoscience

High-Dimensional Bayesian Geostatistics

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. This article offers a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as "priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has $\sim n$ floating point operations (flops), where $n$ the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings

Nonparametric Multi-shape Modeling with Uncertainty Quantification

The modeling and uncertainty quantification of closed curves is an important problem in the field of shape analysis, and can have significant ramifications for subsequent statistical tasks. Many of these tasks involve collections of closed curves, which often exhibit structural similarities at multiple levels. Modeling multiple closed curves in a way that efficiently incorporates such between-curve dependence remains a challenging problem. In this work, we propose and investigate a multiple-output (a.k.a. multi-output), multi-dimensional Gaussian process modeling framework. We illustrate the proposed methodological advances, and demonstrate the utility of meaningful uncertainty quantification, on several curve and shape-related tasks. This model-based approach not only addresses the problem of inference on closed curves (and their shapes) with kernel constructions, but also opens doors to nonparametric modeling of multi-level dependence for functional objects in general.Comment: 66 pages, 20 figure

Spatial Statistical Data Fusion on Java-enabled Machines in Ubiquitous Sensor Networks

Wireless Sensor Networks (WSN) consist of small, cheap devices that have a combination of sensing, computing and communication capabilities. They must be able to communicate and process data efficiently using minimum amount of energy and cover an area of interest with the minimum number of sensors. This thesis proposes the use of techniques that were designed for Geostatistics and applies them to WSN field. Kriging and Cokriging interpolation that can be considered as Information Fusion algorithms were tested to prove the feasibility of the methods to increase coverage. To reduce energy consumption, a compression method that models correlations based on variograms was developed. A second challenge is to establish the communication to the external networks and to react to unexpected events. A demonstrator that uses commercial Java-enabled devices was implemented. It is able to perform remote monitoring, send SMS alarms and deploy remote updates

A hybrid approach for joint simulation of geometallurgical variables with inequality constraint

Geometallurgical variables have a significant impact on downstream activities of mining projects. Reliable 3D spatial modelling of these variables plays an important role in mine planning and mineral processing, in which it can improve the overall viability of the mining projects. This interdisciplinary paradigm involves geology, geostatistics, mineral processing and metallurgy that creates a need for enhanced techniques of modelling. In some circumstances, the geometallurgical responses demonstrate a decent intrinsic correlation that motivates one to use co-estimation or co-simulation approaches rather than independent estimation or simulation. The latter approach allows us to reproduce that dependency characteristic in the final model. In this paper, two problems have been addressed, one is concerning the inequality constraint that might exist among geometallurgical variables, and the second is dealing with difficulty in variogram analysis. To alleviate the first problem, the variables can be converted to new variables free of inequality constraint. The second problem can also be solved by taking into account the minimum/maximum autocorrelation factors (MAF) transformation technique which allows defining a hybrid approach of joint simulation rather than conventional method of co-simulation. A case study was carried out for the total and acid soluble copper grades obtained from an oxide copper deposit. Firstly, these two geometallurgical variables are transferred to the new variables without inequality constraint and then MAF analysis is used for joint simulation and modelling. After back transformation of the results, they are compared with traditional approaches of co-simulation, for which they showed that the MAF methodology is able to reproduce the spatial correlation between the variables without loss of generality while the inequality constraint is honored. The results are then post processed to support probabilistic domaining of geometallurgical zones

On Simplified Bayesian Modeling for Massive Geostatistical Datasets: Conjugacy and Beyond

With continued advances in Geographic Information Systems and related computational technologies, researchers in diverse fields like forestry, environmental health, climate sciences etc. have growing interests in analyzing large scale data sets measured at a substantial number of geographic locations. Geostatistical models used to capture the space varying relationships in such data are often accompanied by onerous computations which prohibit the analysis of large scale spatial data sets. Less burdensome alternatives proposed recently for analyzing massive spatial datasets often lead to inaccurate inference or require slow sampling process. Bayesian inference, while attractive for accommodating uncertainties through their hierarchical structures, can become computationally onerous for modeling massive spatial data sets because of their reliance on iterative estimation algorithms. My dissertation research aims at developing computationally scalable Bayesian geostatistical models that provide valid inference through highly accelerated sampling process. We also study the asymptotic properties of estimators in spatial analysis.In Chapter 2 and 3, we develop conjugate Bayesian frameworks for analyzing univariate and multivariate spatial data. We propose a conjugate latent Nearest-Neighbor Gaussian Process (NNGP) model in Chapter 2, which uses analytically tractable posterior distributions to obtain posterior inferences, including the large dimensional latent process. In Chapter 3, we focus on building conjugate Bayesian frameworks for analyzing multivariate spatial data. We utilize Matrix-Normal Inverse-Wishart(MNIW) prior to propose conjugate Bayesian frameworks and algorithms that can incorporate a family of scalable spatial modeling methodologies.In Chapter 4, we pursue general Bayesian modeling methodologies beyond a conjugate Bayesian hierarchical modeling. We build scalable versions of a hierarchical linear model of coregionalization (LMC) and spatial factor models, and propose a highly accelerated block update MCMC algorithm. Using the proposed Bayesian LMC model, we extend scalable modeling strategies for a single process into multivariate process cases. All proposed frameworks are tested on simulated data and fit to real data sets with observed locations numbering in the millions. Our contribution is to offer practicing scientists and spatial analysts practical and flexible scalable hierarchical models for analyzing massive spatial data sets.In Chapter 5, we investigate the asymptotic properties of the estimators in spatial analysis. We formally establish results on the identifiability and consistency of the nugget in spatial models based upon the Gaussian process within the framework of in-fill asymptotics, i.e. the sample size increases within a sampling domain that is bounded. We establish the identifiability of parameters in the Matern covariance function and the consistency of their maximum likelihood estimators in the presence of discontinuities due to the nugget

Bayesian Anal

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. This article offers a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as "priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has ~ | floating point operations (flops), where | the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings.R01 ES027027/ES/NIEHS NIH HHS/United StatesR01 OH010093/OH/NIOSH CDC HHS/United StatesRC1 GM092400/GM/NIGMS NIH HHS/United States2018-01-30T00:00:00Z29391920PMC5790125vault:2616