arXiv subjects

Marc G. Genton

Publications and source records attributed to Marc G. Genton.

At least 37 records · Page 2Linked to original sources

arXiv2024

Conjugacy properties of multivariate unified skew-elliptical distributions

The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess important conjugacy properties. When used as priors for the coefficients vector in probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although this result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such a gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that this family is conjugate to the likelihood induced by multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.

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arXiv2024

Efficient stochastic generators with spherical harmonic transformation for high-resolution global climate simulations from CESM2-LENS2

Earth system models (ESMs) are fundamental for understanding Earth's complex climate system. However, the computational demands and storage requirements of ESM simulations limit their utility. For the newly published CESM2-LENS2 data, which suffer from this issue, we propose a novel stochastic generator (SG) as a practical complement to the CESM2, capable of rapidly producing emulations closely mirroring training simulations. Our SG leverages the spherical harmonic transformation (SHT) to shift from spatial to spectral domains, enabling efficient low-rank approximations that significantly reduce computational and storage costs. By accounting for axial symmetry and retaining distinct ranks for land and ocean regions, our SG captures intricate non-stationary spatial dependencies. Additionally, a modified Tukey g-and-h (TGH) transformation accommodates non-Gaussianity in high-temporal-resolution data. We apply the proposed SG to generate emulations for surface temperature simulations from the CESM2-LENS2 data across various scales, marking the first attempt of reproducing daily data. These emulations are then meticulously validated against training simulations. This work offers a promising complementary pathway for efficient climate modeling and analysis while overcoming computational and storage limitations.

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arXiv2024

Parallel Approximations for High-Dimensional Multivariate Normal Probability Computation in Confidence Region Detection Applications

Addressing the statistical challenge of computing the multivariate normal (MVN) probability in high dimensions holds significant potential for enhancing various applications. One common way to compute high-dimensional MVN probabilities is the Separation-of-Variables (SOV) algorithm. This algorithm is known for its high computational complexity of O(n^3) and space complexity of O(n^2), mainly due to a Cholesky factorization operation for an n X n covariance matrix, where $n$ represents the dimensionality of the MVN problem. This work proposes a high-performance computing framework that allows scaling the SOV algorithm and, subsequently, the confidence region detection algorithm. The framework leverages parallel linear algebra algorithms with a task-based programming model to achieve performance scalability in computing process probabilities, especially on large-scale systems. In addition, we enhance our implementation by incorporating Tile Low-Rank (TLR) approximation techniques to reduce algorithmic complexity without compromising the necessary accuracy. To evaluate the performance and accuracy of our framework, we conduct assessments using simulated data and a wind speed dataset. Our proposed implementation effectively handles high-dimensional multivariate normal (MVN) probability computations on shared and distributed-memory systems using finite precision arithmetics and TLR approximation computation. Performance results show a significant speedup of up to 20X in solving the MVN problem using TLR approximation compared to the reference dense solution without sacrificing the application's accuracy. The qualitative results on synthetic and real datasets demonstrate how we maintain high accuracy in detecting confidence regions even when relying on TLR approximation to perform the underlying linear algebra operations.

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arXiv2024

GPU-Accelerated Vecchia Approximations of Gaussian Processes for Geospatial Data using Batched Matrix Computations

Gaussian processes (GPs) are commonly used for geospatial analysis, but they suffer from high computational complexity when dealing with massive data. For instance, the log-likelihood function required in estimating the statistical model parameters for geospatial data is a computationally intensive procedure that involves computing the inverse of a covariance matrix with size n X n, where n represents the number of geographical locations. As a result, in the literature, studies have shifted towards approximation methods to handle larger values of n effectively while maintaining high accuracy. These methods encompass a range of techniques, including low-rank and sparse approximations. Vecchia approximation is one of the most promising methods to speed up evaluating the log-likelihood function. This study presents a parallel implementation of the Vecchia approximation, utilizing batched matrix computations on contemporary GPUs. The proposed implementation relies on batched linear algebra routines to efficiently execute individual conditional distributions in the Vecchia algorithm. We rely on the KBLAS linear algebra library to perform batched linear algebra operations, reducing the time to solution compared to the state-of-the-art parallel implementation of the likelihood estimation operation in the ExaGeoStat software by up to 700X, 833X, 1380X on 32GB GV100, 80GB A100, and 80GB H100 GPUs, respectively. We also successfully manage larger problem sizes on a single NVIDIA GPU, accommodating up to 1M locations with 80GB A100 and H100 GPUs while maintaining the necessary application accuracy. We further assess the accuracy performance of the implemented algorithm, identifying the optimal settings for the Vecchia approximation algorithm to preserve accuracy on two real geospatial datasets: soil moisture data in the Mississippi Basin area and wind speed data in the Middle East.

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arXiv2024

Uniform-over-dimension convergence with application to location tests for high-dimensional data

Asymptotic methods for hypothesis testing in high-dimensional data usually require the dimension of the observations to increase to infinity, often with an additional condition on its rate of increase compared to the sample size. On the other hand, multivariate asymptotic methods are valid for fixed dimension only, and their practical implementations in hypothesis testing methodology typically require the sample size to be large compared to the dimension for yielding desirable results. However, in practical scenarios, it is usually not possible to determine whether the dimension of the data at hand conform to the conditions required for the validity of the high-dimensional asymptotic methods, or whether the sample size is large enough compared to the dimension of the data. In this work, a theory of asymptotic convergence is proposed, which holds uniformly over the dimension of the random vectors. This theory attempts to unify the asymptotic results for fixed-dimensional multivariate data and high-dimensional data, and accounts for the effect of the dimension of the data on the performance of the hypothesis testing procedures. The methodology developed based on this asymptotic theory can be applied to data of any dimension. An application of this theory is demonstrated in the two-sample test for the equality of locations. The test statistic proposed is unscaled by the sample covariance, similar to usual tests for high-dimensional data. Using simulated examples, it is demonstrated that the proposed test exhibits better performance compared to several popular tests in the literature for high-dimensional data. Further, it is demonstrated in simulated models that the proposed unscaled test performs better than the usual scaled two-sample tests for multivariate data, including the Hotelling's $T^2$ test for multivariate Gaussian data.

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arXiv2024

On the Impact of Spatial Covariance Matrix Ordering on Tile Low-Rank Estimation of Matérn Parameters

Spatial statistical modeling and prediction involve generating and manipulating an n*n symmetric positive definite covariance matrix, where n denotes the number of spatial locations. However, when n is large, processing this covariance matrix using traditional methods becomes prohibitive. Thus, coupling parallel processing with approximation can be an elegant solution to this challenge by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. Each processing unit can process a single tile, allowing better performance. The approximation can also be performed at the tile level for better compression and faster execution. The Tile Low-Rank (TLR) approximation, a tile-based approximation algorithm, has recently been used in spatial statistics applications. However, the quality of TLR algorithms mainly relies on ordering the matrix elements. This order can impact the compression quality and, therefore, the efficiency of the underlying linear solvers, which highly depends on the individual ranks of each tile. Thus, herein, we aim to investigate the accuracy and performance of some existing ordering algorithms that are used to order the geospatial locations before generating the spatial covariance matrix. Furthermore, we highlight the pros and cons of each ordering algorithm in the context of spatial statistics applications and give hints to practitioners on how to choose the ordering algorithm carefully. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations.

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arXiv2023

Visualization and Assessment of Copula Symmetry

Visualization and assessment of copula structures are crucial for accurately understanding and modeling the dependencies in multivariate data analysis. In this paper, we introduce an innovative method that employs functional boxplots and rank-based testing procedures to evaluate copula symmetry. This approach is specifically designed to assess key characteristics such as reflection symmetry, radial symmetry, and joint symmetry. We first construct test functions for each specific property and then investigate the asymptotic properties of their empirical estimators. We demonstrate that the functional boxplot of these sample test functions serves as an informative visualization tool of a given copula structure, effectively measuring the departure from zero of the test function. Furthermore, we introduce a nonparametric testing procedure to assess the significance of deviations from symmetry, ensuring the accuracy and reliability of our visualization method. Through extensive simulation studies involving various copula models, we demonstrate the effectiveness of our testing approach. Finally, we apply our visualization and testing techniques to two real-world datasets: a nutritional habits survey with five variables and wind speed data from three locations in Saudi Arabia.

stat.ME↗

arXiv2023

Portability and Scalability Evaluation of Large-Scale Statistical Modeling and Prediction Software through HPC-Ready Containers

HPC-based applications often have complex workflows with many software dependencies that hinder their portability on contemporary HPC architectures. In addition, these applications often require extraordinary efforts to deploy and execute at performance potential on new HPC systems, while the users expert in these applications generally have less expertise in HPC and related technologies. This paper provides a dynamic solution that facilitates containerization for transferring HPC software onto diverse parallel systems. The study relies on the HPC Workflow as a Service (HPCWaaS) paradigm proposed by the EuroHPC eFlows4HPC project. It offers to deploy workflows through containers tailored for any of a number of specific HPC systems. Traditional container image creation tools rely on OS system packages compiled for generic architecture families (x86\_64, amd64, ppc64, ...) and specific MPI or GPU runtime library versions. The containerization solution proposed in this paper leverages HPC Builders such as Spack or Easybuild and multi-platform builders such as buildx to create a service for automating the creation of container images for the software specific to each hardware architecture, aiming to sustain the overall performance of the software. We assess the efficiency of our proposed solution for porting the geostatistics ExaGeoStat software on various parallel systems while preserving the computational performance. The results show that the performance of the generated images is comparable with the native execution of the software on the same architectures. On the distributed-memory system, the containerized version can scale up to 256 nodes without impacting performance.

cs.DC↗

arXiv2023

Multivariate Unified Skew-t Distributions And Their Properties

The unified skew-t (SUT) is a flexible parametric multivariate distribution that accounts for skewness and heavy tails in the data. A few of its properties can be found scattered in the literature or in a parameterization that does not follow the original one for unified skew-normal (SUN) distributions, yet a systematic study is lacking. In this work, explicit properties of the multivariate SUT distribution are presented, such as its stochastic representations, moments, SUN-scale mixture representation, linear transformation, additivity, marginal distribution, canonical form, quadratic form, conditional distribution, change of latent dimensions, Mardia measures of multivariate skewness and kurtosis, and non-identifiability issue. These results are given in a parametrization that reduces to the original SUN distribution as a sub-model, hence facilitating the use of the SUT for applications. Several models based on the SUT distribution are provided for illustration.

stat.ME↗

arXiv2023

A Multivariate Skew-Normal-Tukey-h Distribution

We introduce a new family of multivariate distributions by taking the component-wise Tukey-h transformation of a random vector following a skew-normal distribution. The proposed distribution is named the skew-normal-Tukey-h distribution and is an extension of the skew-normal distribution for handling heavy-tailed data. We compare this proposed distribution to the skew-t distribution, which is another extension of the skew-normal distribution for modeling tail-thickness, and demonstrate that when there are substantial differences in marginal kurtosis, the proposed distribution is more appropriate. Moreover, we derive many appealing stochastic properties of the proposed distribution and provide a methodology for the estimation of the parameters in which the computational requirement increases linearly with the dimension. Using simulations, as well as a wine and a wind speed data application, we illustrate how to draw inferences based on the multivariate skew-normal-Tukey-h distribution.

stat.ME↗

arXiv2023

Which Parameterization of the Matérn Covariance Function?

The Matérn family of covariance functions is currently the most popularly used model in spatial statistics, geostatistics, and machine learning to specify the correlation between two geographical locations based on spatial distance. Compared to existing covariance functions, the Matérn family has more flexibility in data fitting because it allows the control of the field smoothness through a dedicated parameter. Moreover, it generalizes other popular covariance functions. However, fitting the smoothness parameter is computationally challenging since it complicates the optimization process. As a result, some practitioners set the smoothness parameter at an arbitrary value to reduce the optimization convergence time. In the literature, studies have used various parameterizations of the Matérn covariance function, assuming they are equivalent. This work aims at studying the effectiveness of different parameterizations under various settings. We demonstrate the feasibility of inferring all parameters simultaneously and quantifying their uncertainties on large-scale data using the ExaGeoStat parallel software. We also highlight the importance of the smoothness parameter by analyzing the Fisher information of the statistical parameters. We show that the various parameterizations have different properties and differ from several perspectives. In particular, we study the three most popular parameterizations in terms of parameter estimation accuracy, modeling accuracy and efficiency, prediction efficiency, uncertainty quantification, and asymptotic properties. We further demonstrate their differing performances under nugget effects and approximated covariance. Lastly, we give recommendations for parameterization selection based on our experimental results.

stat.ME↗

arXiv2023

On the non-identifiability of unified skew-normal distributions

In this note, we investigate the non-identifiability of the multivariate unified skew-normal distribution under permutation of its latent variables. We show that the non-identifiability issue also holds with other parametrizations and extends to the family of unified skew-elliptical distributions and more generally to selection distibutions. We provide several suggestions to make the unified skew-normal model identifiable and describe various sub-models that are identifiable.

math.ST↗

arXiv2023

Efficient Large-scale Nonstationary Spatial Covariance Function Estimation Using Convolutional Neural Networks

Spatial processes observed in various fields, such as climate and environmental science, often occur on a large scale and demonstrate spatial nonstationarity. Fitting a Gaussian process with a nonstationary Matérn covariance is challenging. Previous studies in the literature have tackled this challenge by employing spatial partitioning techniques to estimate the parameters that vary spatially in the covariance function. The selection of partitions is an important consideration, but it is often subjective and lacks a data-driven approach. To address this issue, in this study, we utilize the power of Convolutional Neural Networks (ConvNets) to derive subregions from the nonstationary data. We employ a selection mechanism to identify subregions that exhibit similar behavior to stationary fields. In order to distinguish between stationary and nonstationary random fields, we conducted training on ConvNet using various simulated data. These simulations are generated from Gaussian processes with Matérn covariance models under a wide range of parameter settings, ensuring adequate representation of both stationary and nonstationary spatial data. We assess the performance of the proposed method with synthetic and real datasets at a large scale. The results revealed enhanced accuracy in parameter estimations when relying on ConvNet-based partition compared to traditional user-defined approaches.

stat.ML↗

arXiv2023

Robust Two-Layer Partition Clustering of Sparse Multivariate Functional Data

A novel elastic time distance for sparse multivariate functional data is proposed and used to develop a robust distance-based two-layer partition clustering method. With this proposed distance, the new approach not only can detect correct clusters for sparse multivariate functional data under outlier settings but also can detect those outliers that do not belong to any clusters. Classical distance-based clustering methods such as density-based spatial clustering of applications with noise (DBSCAN), agglomerative hierarchical clustering, and $K$-medoids are extended to the sparse multivariate functional case based on the newly-proposed distance. Numerical experiments on simulated data highlight that the performance of the proposed algorithm is superior to the performances of existing model-based and extended distance-based methods. The effectiveness of the proposed approach is demonstrated using Northwest Pacific cyclone tracks data as an example.

stat.ME↗

arXiv2023

Test and Visualization of Covariance Properties for Multivariate Spatio-Temporal Random Fields

The prevalence of multivariate space-time data collected from monitoring networks and satellites, or generated from numerical models, has brought much attention to multivariate spatio-temporal statistical models, where the covariance function plays a key role in modeling, inference, and prediction. For multivariate space-time data, understanding the spatio-temporal variability, within and across variables, is essential in employing a realistic covariance model. Meanwhile, the complexity of generic covariances often makes model fitting very challenging, and simplified covariance structures, including symmetry and separability, can reduce the model complexity and facilitate the inference procedure. However, a careful examination of these properties is needed in real applications. In the work presented here, we formally define these properties for multivariate spatio-temporal random fields and use functional data analysis techniques to visualize them, hence providing intuitive interpretations. We then propose a rigorous rank-based testing procedure to conclude whether the simplified properties of covariance are suitable for the underlying multivariate space-time data. The good performance of our method is illustrated through synthetic data, for which we know the true structure. We also investigate the covariance of bivariate wind speed, a key variable in renewable energy, over a coastal and an inland area in Saudi Arabia. The Supplementary Material is available online, including the R code for our developed methods.

stat.ME↗

arXiv2023

Goodness-of-fit tests for multivariate skewed distributions based on the characteristic function

We employ a general Monte Carlo method to test composite hypotheses of goodness-of-fit for several popular multivariate models that can accommodate both asymmetry and heavy tails. Specifically, we consider weighted L2-type tests based on a discrepancy measure involving the distance between empirical characteristic functions and thus avoid the need for employing corresponding population quantities which may be unknown or complicated to work with. The only requirements of our tests are that we should be able to draw samples from the distribution under test and possess a reasonable method of estimation of the unknown distributional parameters. Monte Carlo studies are conducted to investigate the performance of the test criteria in finite samples for several families of skewed distributions. Real-data examples are also included to illustrate our method.

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arXiv2022

Global Depths for Irregularly Observed Multivariate Functional Data

Two frameworks for multivariate functional depth based on multivariate depths are introduced in this paper. The first framework is multivariate functional integrated depth, and the second framework involves multivariate functional extremal depth, which is an extension of the extremal depth for univariate functional data. In each framework, global and local multivariate functional depths are proposed. The properties of population multivariate functional depths and consistency of finite sample depths to their population versions are established. In addition, finite sample depths under irregularly observed time grids are estimated. As a by-product, the simplified sparse functional boxplot and simplified intensity sparse functional boxplot are proposed for visualization without data reconstruction. A simulation study demonstrates the advantages of global multivariate functional depths over local multivariate functional depths in outlier detection and running time for big functional data. An application of our frameworks to cyclone tracks data demonstrates the excellent performance of our global multivariate functional depths.

stat.ME↗

arXiv2022

The Second Competition on Spatial Statistics for Large Datasets

In the last few decades, the size of spatial and spatio-temporal datasets in many research areas has rapidly increased with the development of data collection technologies. As a result, classical statistical methods in spatial statistics are facing computational challenges. For example, the kriging predictor in geostatistics becomes prohibitive on traditional hardware architectures for large datasets as it requires high computing power and memory footprint when dealing with large dense matrix operations. Over the years, various approximation methods have been proposed to address such computational issues, however, the community lacks a holistic process to assess their approximation efficiency. To provide a fair assessment, in 2021, we organized the first competition on spatial statistics for large datasets, generated by our {\em ExaGeoStat} software, and asked participants to report the results of estimation and prediction. Thanks to its widely acknowledged success and at the request of many participants, we organized the second competition in 2022 focusing on predictions for more complex spatial and spatio-temporal processes, including univariate nonstationary spatial processes, univariate stationary space-time processes, and bivariate stationary spatial processes. In this paper, we describe in detail the data generation procedure and make the valuable datasets publicly available for a wider adoption. Then, we review the submitted methods from fourteen teams worldwide, analyze the competition outcomes, and assess the performance of each team.

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