A class of multi-resolution approximations for large spatial datasets

Katzfuß, Matthias; Gong, Wenlong

doi:10.5705/ss.202018.0285

Cited by 17 publications

(18 citation statements)

References 30 publications

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“…Scalability of the MRA is ensured in that for increasing resolution, the number of basis functions increases while the support of each function (i.e., the part of the spatial domain in which it is nonzero) decreases. Decreasing support (and increasing sparsity of the covariance matrices of the corresponding weights) is achieved either by increasingly severe tapering of the covariance function (MRA-taper; Katzfuss and Gong 2017) or by recursively partitioning the spatial domain (MRAblock; Katzfuss, 2017). This can lead to (nearly) exact approximations with quasilinear computational complexity.…”

Section: Multiresolution Approximationsmentioning

confidence: 99%

A Case Study Competition Among Methods for Analyzing Large Spatial Data

Heaton

Datta

Finley

et al. 2018

JABES

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349

277

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The Gaussian process is an indispensable tool for spatial data analysts. The onset of the “big data” era, however, has lead to the traditional Gaussian process being computationally infeasible for modern spatial data. As such, various alternatives to the full Gaussian process that are more amenable to handling big spatial data have been proposed. These modern methods often exploit low-rank structures and/or multi-core and multi-threaded computing environments to facilitate computation. This study provides, first, an introductory overview of several methods for analyzing large spatial data. Second, this study describes the results of a predictive competition among the described methods as implemented by different groups with strong expertise in the methodology. Specifically, each research group was provided with two training datasets (one simulated and one observed) along with a set of prediction locations. Each group then wrote their own implementation of their method to produce predictions at the given location and each was subsequently run on a common computing environment. The methods were then compared in terms of various predictive diagnostics. Supplementary materials regarding implementation details of the methods and code are available for this article online. Electronic Supplementary Material Supplementary materials for this article are available at 10.1007/s13253-018-00348-w.

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Section: Multiresolution Approximationsmentioning

confidence: 99%

A Case Study Competition Among Methods for Analyzing Large Spatial Data

Heaton

Datta

Finley

et al. 2018

JABES

Self Cite

349

277

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“…Near-linear computational complexity can be achieved by applying these mechanisms hierarchically on multiple scales. Examples of hierarchical sparse approximations include wavelet methods (e.g., [7]), the multi-resolution approximation [29,30], and (implicitly) some versions of the Vecchia approximation [31]. Hierarchical application of low-rank approximations leads to hierarchical matrices [22,24,23,8,1,25,9,10,57], which are an algebraic abstraction of the fast multipole method [19].…”

Section: Introductionmentioning

confidence: 99%

Sparse Cholesky Factorization by Kullback--Leibler Minimization

Schäfer¹,

Katzfuß²,

Owhadi³

2021

SIAM J. Sci. Comput.

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We propose to compute a sparse approximate inverse Cholesky factor L of a dense covariance matrix Θ by minimizing the Kullback-Leibler divergence between the Gaussian distributions N (0, Θ) and N (0, L − L −1 ), subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of Θ obtained from pairwise evaluation of Green's functions of elliptic boundary-value problems at points {x i } 1≤i≤N ⊂ R d , we propose an elimination ordering and sparsity pattern that allows us to compute -approximate inverse Cholesky factors of such Θ in computational complexity O(N log(N/ ) d ) in space and O(N log(N/ ) 2d ) in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussianprocess regression in linear (in N ) space complexity. Motivated by the optimality properties of our methods, we propose methods for applying it to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, sacrificing neither accuracy nor computational complexity.

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“…Here, we provide an efficient method for analyzing global Level-2 satellite data based on the multi-resolution approximation (M -RA) for spatial processes first introduced in Katzfuss (2017) and Katzfuss and Gong (2020), which uses a large number of basis functions to capture spatial variation at all scales while being highly computationally efficient. In contrast to related multi-resolution wavelet models (e.g., the MRVA), the M -RA is directly applicable to irregularly spaced observations, allows proper probabilistic inference, and the basis functions can adjust flexibly to spatially-varying dependence structure.…”

Section: Introductionmentioning

confidence: 99%

Nonstationary Spatial Modeling of Massive Global Satellite Data

Huang

Blake

Katzfuß

et al. 2021

Preprint

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Earth-observing satellite instruments obtain a massive number of observations every day. For example, tens of millions of sea surface temperature (SST) observations on a global scale are collected daily by the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument. Despite their size, such datasets are incomplete and noisy, necessitating spatial statistical inference to obtain complete, high-resolution fields with quantified uncertainties. Such inference is challenging due to the high computational cost, the nonstationary behavior of environmental processes on a global scale, and land barriers affecting the dependence of SST. In this work, we develop a multi-resolution approximation (M -RA) of a Gaussian process (GP) whose nonstationary, global covariance function is obtained using local fits. The M -RA requires domain partitioning, which can be set up application-specifically. In the SST case, we partition the domain purposefully to account for and weaken dependence across land barriers. Our M -RA implementation is tailored to distributed-memory computation in high-performance-computing environments. We analyze a MODIS SST dataset consisting of more than 43 million observations, to our knowledge the largest dataset ever analyzed using a probabilistic GP model. We show that our nonstationary model based on local fits provides substantially improved predictive performance relative to a stationary approach.

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A class of multi-resolution approximations for large spatial datasets

Cited by 17 publications

References 30 publications

A Case Study Competition Among Methods for Analyzing Large Spatial Data

A Case Study Competition Among Methods for Analyzing Large Spatial Data

Sparse Cholesky Factorization by Kullback--Leibler Minimization

Nonstationary Spatial Modeling of Massive Global Satellite Data

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