Bayesian Nonstationary and Nonparametric Covariance Estimation for Large Spatial Data (with Discussion)

Kidd, Brian; Katzfuß, Matthias

doi:10.1214/21-ba1273

Cited by 9 publications

(3 citation statements)

References 36 publications

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“…Special cases of our general CVecchia idea have already been successfully employed in several applications (which were started later but completed earlier than the present paper): Katzfuss et al (2022) used the idea to approximate anisotropic GPs for computer-model emulation in high input dimension; Messier and Katzfuss (2021) approximated spatio-temporal land-use regression for ground-level nitrogen dioxide; and in the context of nonparametric inference (Kidd and Katzfuss, 2021), ideas related to CVecchia were used with sample correlations instead of parametric correlations.…”

Section: Discussionmentioning

confidence: 99%

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Myeongjong¹,

Katzfuß²

2021

Preprint

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Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The Euclidean-and correlation-based Vecchia approximations are equivalent for strictly decreasing isotropic covariances, but the correlation-based approach has two advantages for more complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.

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Section: Discussionmentioning

confidence: 99%

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Myeongjong¹,

Katzfuß²

2021

Preprint

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“…A different approach is taken in Kirsner and Sansó (2020), that build a multi‐resolution model with spatially‐varying resolution. Finally, more recently, a nonparametric and non‐stationary approach that looks at inferring the sparse elements of the Cholesky factor of an inverse covariance matrix has been presented in Kidd and Katzfuss (2022).…”

Section: Introductionmentioning

confidence: 99%

Multivariate nearest‐neighbors Gaussian processes with random covariance matrices

Grenier,

Sansó,

Matthews

2024

Environmetrics

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We propose a non‐stationary spatial model based on a normal‐inverse‐Wishart framework, conditioning on a set of nearest‐neighbors. The model, called nearest‐neighbor Gaussian process with random covariance matrices is developed for both univariate and multivariate spatial settings and allows for fully flexible covariance structures that impose no stationarity or isotropic restrictions. In addition, the model can handle duplicate observations and missing data. We consider an approach based on integrating out the spatial random effects that allows fast inference for the model parameters. We also consider a full hierarchical approach that leverages the sparse structures induced by the model to perform fast Monte Carlo computations. Strong computational efficiency is achieved by leveraging the adaptive localized structure of the model that allows for a high level of parallelization. We illustrate the performance of the model with univariate and bivariate simulations, as well as with observations from two stationary satellites consisting of albedo measurements.

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“…The resulting sparse non-linear transport maps can be seen as a nonparametric and non-Gaussian generalization of Vecchia approximations (e.g., Vecchia, 1988;Stein et al, 2004;Datta et al, 2016;Katzfuss and Guinness, 2021;Schäfer et al, 2021a), which implicitly utilize linear transport map given by a sparse inverse Cholesky factor. Kidd and Katzfuss (2022) proposed a Bayesian non-parametric inference on the Cholesky factor.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Nonparametric Generative Modeling of Large Multivariate Non-Gaussian Spatial Fields

Wiemann,

Katzfuss

2023

JABES

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Multivariate spatial fields are of interest in many applications, including climate model emulation. Not only can the marginal spatial fields be subject to nonstationarity, but the dependence structure among the marginal fields and between the fields might also differ substantially. Extending a recently proposed Bayesian approach to describe the distribution of a nonstationary univariate spatial field using a triangular transport map, we cast the inference problem for a multivariate spatial field for a small number of replicates into a series of independent Gaussian process (GP) regression tasks with Gaussian errors. Due to the potential nonlinearity in the conditional means, the joint distribution modeled can be non-Gaussian. The resulting nonparametric Bayesian methodology scales well to high-dimensional spatial fields. It is especially useful when only a few training samples are available, because it employs regularization priors and quantifies uncertainty. Inference is conducted in an empirical Bayes setting by a highly scalable stochastic gradient approach. The implementation benefits from minibatching and could be accelerated with parallel computing. We illustrate the extended transport-map model by studying hydrological variables from non-Gaussian climatemodel output.

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Bayesian Nonstationary and Nonparametric Covariance Estimation for Large Spatial Data (with Discussion)

Cited by 9 publications

References 36 publications

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Multivariate nearest‐neighbors Gaussian processes with random covariance matrices

Bayesian Nonparametric Generative Modeling of Large Multivariate Non-Gaussian Spatial Fields

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