Multivariate Gaussian Random Fields Research Articles

Statistical Modeling of Right-Censored Spatial Data Using Gaussian Random Fields

Consider a fixed number of clustered areas identified by their geographical coordinates that are monitored for the occurrences of an event such as a pandemic, epidemic, or migration. Data collected on units at all areas include covariates and environmental factors. We apply a probit transformation to the time to event and embed an isotropic spatial correlation function into our models for better modeling as compared to existing methodologies that use frailty or copula. Composite likelihood technique is employed for the construction of a multivariate Gaussian random field that preserves the spatial correlation function. The data are analyzed using counting process and geostatistical formulation that led to a class of weighted pairwise semiparametric estimating functions. The estimators of model parameters are shown to be consistent and asymptotically normally distributed under infill-type asymptotic spatial statistics. Detailed small sample numerical studies that are in agreement with theoretical results are provided. The foregoing procedures are applied to the leukemia survival data in Northeast England. A comparison to existing methodologies provides improvement.

A Bayesian model for multivariate discrete data using spatial and expert information with application to inferring building attributes

When modeling sparsely observed multivariate data, strong prior information elicited from experts can be used to bolster predictive accuracy and counteract sampling bias. Similarly, modeling autocorrelation in space can help make use of co-occurrence patterns present in many types of spatial data. To make use of both expert prior information and spatial structure, we propose a novel graphical model for a spatial Bayesian network developed specifically to address challenges in inferring the attributes of buildings from geographically sparse observational data. This model is implemented as the sum of a spatial multivariate Gaussian random field and a tabular conditional probability function in real-valued space prior to projection onto the probability simplex. This modeling form is especially suitable for the usage of prior information in the form of sets of atomic rules obtained from experts. To perform inference with missing data, we implement a Markov chain Monte Carlo scheme composed of alternating steps of Gibbs sampling of missing entries and Hamiltonian Monte Carlo for model parameters. A case study in building attribution is presented to highlight the advantages and limitations of this approach.

Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus

The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings. We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.

Asymptotically equivalent prediction in multivariate geostatistics

Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geostatistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typically multivariate, call for a deeper study of cokriging. In particular, we deal with the problem of misspecified cokriging prediction within the framework of fixed domain asymptotics. Specifically, we provide conditions for equivalence of measures associated with multivariate Gaussian random fields, with index set in a compact set of a d-dimensional Euclidean space. Such conditions have been elusive for over about 50 years of spatial statistics. We then focus on the multivariate Matérn and Generalized Wendland classes of matrix valued covariance functions, that have been very popular for having parameters that are crucial to spatial interpolation, and that control the mean square differentiability of the associated Gaussian process. We provide sufficient conditions, for equivalence of Gaussian measures, relying on the covariance parameters of these two classes. This enables to identify the parameters that are crucial to asymptotically equivalent interpolation in multivariate geostatistics. Our findings are then illustrated through simulation studies.

Uncertainty modelling and multiscale simulation of woven composite twisted structure

The woven composite twisted structure under cantilever load is studied using experimental and multiscale numerical methods. The uncertain geometry parameters inside the twisted specimen are identified using Micro-CT scanning technology, and quantified using multivariate Gaussian random field theory. A high-fidelity meso-Representative Volume Cell (RVC) database after data clustering is established using these uncertainties, in which the warp and weft yarns are no longer regular orthogonality, the yarn cross-sectional area and the yarns gap exhibit variability. The statistical uncertainty quantification and propagation are further integrated into the coupled Finite Element-Fast Fourier Transformation (FE-FFT) concurrent multiscale method to reveal the damage and failure mechanisms of the woven composite twisted structure. The numerical results are in good agreement with the experiment results. This study can be helpful of designing the woven composite blade structures.

Multivariate isotropic random fields on spheres: Nonparametric Bayesian modeling and Lp fast approximations

We study multivariate Gaussian random fields defined over d-dimensional spheres. First, we provide a nonparametric Bayesian framework for modeling and inference on matrix-valued covariance functions. We determine the support (under the topology of uniform convergence) of the proposed random matrices, which cover the whole class of matrix-valued geodesically isotropic covariance functions on spheres. We provide a thorough inspection of the properties of the proposed model in terms of (a) first moments, (b) posterior distributions, and (c) Lipschitz continuities. We then provide an approximation method for multivariate fields on the sphere for which measures of Lp accuracy are established. Our findings are supported through simulation studies that show the rate of convergence when truncating a spectral expansion of a multivariate random field at a finite order. To illustrate the modeling framework developed in this paper, we consider a bivariate spatial data set of two 2019 NCEP/NCAR Flux Reanalyses.

물성치의 공간분포를 고려한 빙 시험편의 확률론적 강도평가

As the Arctic sea ice decreases due to various reasons such as global warming, the demand for ships and offshore structures operating in the Arctic region is steadily increasing. In the case of sea ice, the anisotropy is caused by the uncertainty inside the material. For most of the research, nevertheless, estimating the ice load has been treated deterministically. With regard to this, in this paper, a four-point bending strength analysis of an ice specimen was attempted using a stochastic finite element method. First, spatial distribution of the material properties used in the yield criterion was assumed to be a multivariate Gaussian random field. After that, a direct method, which is a sort of stochastic finite element method, and a sensitivity method using the sensitivity of response for random variables were proposed for calculating the probabilistic distribution of ice specimen strength. A parametric study was conducted with different mean vectors and correlation lengths for each material property used in the above procedure. The calculation time was about ten seconds for the direct method and about three minutes for the sensitivity methods. As the cohesion and correlation length increased, the mean value of the critical load and the standard deviation increased. On the contrary, they decreased as the friction angle increased. Also, in all cases, the direct and sensitivity methods yielded very similar results.

Generalized Sparse Precision Matrix Selection for Fitting Multivariate Gaussian Random Fields to Large Data Sets

We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed by Davanloo et al. (2015) for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance matrix and for the estimated parameters of the underlying spatial correlation function are established. Numerical tests using simulated and real datasets validate our theoretical findings. Data segmentation is used to handle large data sets.

A Matérn-Based Multivariate Gaussian Random Process for a Consistent Model of the Horizontal Wind Components and Related Variables

Abstract The integration of physical relationships into stochastic models is of major interest, for example, in data assimilation. Here, a multivariate Gaussian random field formulation is introduced that represents the differential relations of the two-dimensional wind field and related variables such as the streamfunction, velocity potential, vorticity, and divergence. The covariance model is based on a flexible bivariate Matérn covariance function for the streamfunction and velocity potential. It allows for different variances in the potentials, nonzero correlations between them, anisotropy, and a flexible smoothness parameter. The joint covariance function of the related variables is derived analytically. Further, it is shown that a consistent model with nonzero correlations between the potentials and positive definite covariance function is possible. The statistical model is fitted to forecasts of the horizontal wind fields of a mesoscale numerical weather prediction system. Parameter uncertainty is assessed by a parametric bootstrap method. The estimates reveal only physically negligible correlations between the potentials.

Composite Likelihood Inference for Multivariate Gaussian Random Fields

In the recent years, there has been a growing interest in proposing covariance models for multivariate Gaussian random fields. Some of these covariance models are very flexible and can capture both the marginal and the cross-spatial dependence of the components of the associated multivariate Gaussian random field. However, effective estimation methods for these models are somehow unexplored. Maximum likelihood is certainly a useful tool, but it is impractical in all the circumstances where the number of observations is very large. In this work, we consider two possible approaches based on composite likelihood for multivariate covariance model estimation. We illustrate, through simulation experiments, that our methods offer a good balance between statistical efficiency and computational complexity. Asymptotic properties of the proposed estimators are assessed under increasing domain asymptotics. Finally, we apply the method for the analysis of a bivariate dataset on chlorophyll concentration and sea surface temperature in the Chilean coast.

Efficient Simulation of Stationary Multivariate Gaussian Random Fields with Given Cross-Covariance

The present paper introduces a new approach to simulate any stationary multivariate Gaussian random field whose cross-covariances are predefined continuous and integrable functions. Such a field is given by convolution of a vector of univariate random fields and a functional matrix which is derived by Cholesky decomposition of the Fourier transform of the predefined cross-covariance matrix. In contrast to common methods, no restrictive model for the cross-covariance is needed. It is stationary and can also be reduced to the isotropic case. The computational effort is very low since fast Fourier transform can be used for simulation. As will be shown the algorithm is computationally faster than a recently published spectral turning bands model. The applicability is demonstrated using a common numerical example with varied spatial correlation structure. The model was developed to support simulation algorithms for mineral microstructures in geoscience.

Covariance tapering for multivariate Gaussian random fields estimation

In recent literature there has been a growing interest in the construction of covariance models for multivariate Gaussian random fields. However, effective estimation methods for these models are somehow unexplored. The maximum likelihood method has attractive features, but when we deal with large data sets this solution becomes impractical, so computationally efficient solutions have to be devised. In this paper we explore the use of the covariance tapering method for the estimation of multivariate covariance models. In particular, through a simulation study, we compare the use of simple separable tapers with more flexible multivariate tapers recently proposed in the literature and we discuss the asymptotic properties of the method under increasing domain asymptotics.

Correct Classification Rates in Multi-Category Discriminant Analysis of Spatial Gaussian Data

This paper discusses the problem of classifying a multivariate Gaussian random field observation into one of the several categories specified by different parametric mean models. Investigation is conducted on the classifier based on plug-in Bayes classification rule (PBCR) formed by replacing unknown parameters in Bayes classification rule (BCR) with category parameters estimators. This is the extension of the previous one from the two category cases to the multi-category case. The novel closed-form expressions for the Bayes classification probability and actual correct classification rate associated with PBCR are derived. These correct classification rates are suggested as performance measures for the classifications procedure. An empirical study has been carried out to analyze the dependence of derived classification rates on category parameters.

Equivalence of Gaussian measures of multivariate random fields

Problems related to weather forecast, forest attributes estimation and prediction, disease propagation, among others, are commonly approximated in the framework of multivariate Gaussian random field modeling. This paper deals with the equivalence condition of two zero-mean Gaussian infinite-dimensional vector measures defined on the finite product of separable Hilbert spaces. In particular, sufficient conditions are provided. The results derived are applied to obtain the equivalence of Gaussian measures associated with two stationary zero-mean Gaussian vector random fields. Classical problems related to, for example, asymptotic properties of maximum likelihood vector Gaussian random field parameter estimators from tapered multivariate covariance functions, often arising in Multivariate Geostatistics, can be solved as direct application of the results derived.

Error rates in classification of multivariate Gaussian random field observation

We consider the problem of supervised classifying the multivariate Gaussian random field (GRF) single observation into one of two populations in case of given training sample. The populations are specified by different regression mean models and by common factorized covariance function. For completely specified populations, we derive a formula for Bayes error rate. In the case of unknown regression parameters and feature covariance matrix, the plug-in Bayes discriminant function based on ML estimators of parameters is used for classification. We derive the actual error rate and the asymptotic expansion of the expected error rate associated with plug-in Bayes discriminant function. These results are multivariate generalizations of previous ones. Numerical analysis of the derived formulas is implemented for the bivariate GRF observations at locations belonging to the two-dimensional lattice with unit spacing.

A cluster mass permutation test with contextual enhancement for fMRI activation detection

Gaussian random field (GRF)-based methods are commonly used for statistical inference and to control the family-wise error rate (FWE) in neuroimaging. They require that the error fields are reasonable lattice approximations to an underlying continuous multivariate Gaussian random field and have differentiable and invertible spatial autocorrelation function. Permutation test estimates the distribution of the test statistic from the data and adjusts automatically for the FWE. Here we present a new analysis procedure, the cluster mass permutation test with contextual enhancement (CMPCE), and compare it to GRF. In CMPCE, the data are first pre-whitened to remove temporal autocorrelations. The FWE rates, the cluster detection probability and delineation accuracy of CMPCE and GRF were compared using measured null data and null data containing simulated activations. We also applied both methods to an fMRI experiment where tactile somatosensory stimulation into the right hand was used. When analyzing the FWE using null data, both CMPCE and GRF gave significantly higher FWEs (CMPCE up to 0.12, GRF up to 0.18) than the nominal significance level 0.05, indicating that the pre-whitening, motion correction or high-pass filtering partially failed. In the simulated activation data, CMPCE gave less falsely classified voxels for the same cluster detection probability level than GRF. The maximal cluster detection probability was on the other hand higher in the GRF-based method. Both methods gave qualitatively similar results in the tactile fMRI data. CMPCE seems to be a promising fMRI analysis method, especially if high delineation accuracy is required.

The Effect of Spatial Autocorrelation on the Error Rates of the Linear Discriminant Functions

The problem of classification of the realization of the intrinsically stationary multivariate Gaussian random field into one of two populations with different means and factorized covariance matrices is considered. Unknown means and the common covariance matrix of the feature vector components are estimated from the spatially correlated training samples assuming spatial correlations to be known. Two plug‐in linear discriminant functions (DF) are considered. The first linear DF uses the maximum likelihood (ML) estimators of means and the bias‐adjusted ML estimator of covariance. The second one uses usual sample means and bias‐adjusted sample covariance. The first‐order asymptotic expansions with respect to the inverses of training sample sizes of the expected error rate associated with two plug‐in DF's are presented. The numerical results obtained allow us to compare the performance of the suggested DF's. The numerical calculations are done for the exponential spatial correlation function.

On Estimation of the Mean and Covariance Parameter for Gaussian Random Fields

In the paper we consider the problem of estimation for a multivariate Gaussian random field whose vector mean depends linearly of a multidimensional parameter θ. A Bayesian estimator is derived and its distributions are analyzed under the assumption that the covariance function is completely known. Further, an invariant structure is introduced and the optimal invariant terminal decision function obtained. Finally, assuming that the covariance function is known up to a parameter σ 2 we estimate the parameters θ and σ 2 and prove some of their properties. The results of Ibarrola and Vélez (1990), (1992), for multidimensional stochastic processes are extended to the the present context and similar results are obtained.

Simulation of Multivariate Gaussian Fields Conditioned by Realizations of the Fields and Their Derivatives

This paper investigates conditional simulation technique of multivariate Gaussian random fields by stochastic interpolation technique. For the first time in the literature a situation is studied when the random fields are conditioned not only by a set of realizations of the fields, but also by a set of realizations of their derivatives. The kriging estimate of multivariate Gaussian field is proposed, which takes into account both the random field as well as its derivative. Special conditions are imposed on the kriging estimate to determine the kriging weights. Basic formulation for simulation of conditioned multivariate random fields is established. As a particular case of uncorrelated components of multivariate field without realizations of the derivative of the random field, the present formulation includes that of univariate field given by Hoshiya. Examples of a univariate field and a three component field are elucidated and some numerical results are discussed. It is concluded that the information on the derivatives may significantly alter the results of the conditional simulation.

Spatial classification using fuzzy membership models

In the usual statistical approach to spatial classification, it is assumed that each pixel belongs to precisely one of a small number of known groups. This framework is extended to include mixed-pixel data; then, only a proportion of each pixel belongs to each group. Two models based on multivariate Gaussian random fields are proposed to model this fuzzy membership process. The problems of predicting the group membership and estimating the parameters are discussed. Some simulations are presented to study the properties of this approach, and an example is given using Landsat remote-sensing data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>