Smoothed Full-Scale Approximation of Gaussian Process Models for Computation of Large Spatial Datasets

Zhang, Bohai; Sang, Huiyan; Huang, Jianhua Z.

doi:10.5705/ss.202017.0008

Cited by 10 publications

(19 citation statements)

References 38 publications

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“…where E(y i |y 1 ) is the predictive process with knots S 1 evaluated at S i . The recently proposed smoothed FSA (Zhang, Sang and Huang, 2019), which is billed as a generalization of the Vecchia approach, can also be viewed as a special case of general Vecchia, for which the conditioning vectors include some nearby blocks in addition to the knot vector y 1 .…”

Section: Full-scale Approximation (Fsa)mentioning

confidence: 99%

A General Framework for Vecchia Approximations of Gaussian Processes

Katzfuß¹,

Guinness²

2021

Statist. Sci.

170

162

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Gaussian processes (GPs) are commonly used as models for functions, time series, and spatial fields, but they are computationally infeasible for large datasets. Focusing on the typical setting of modeling data as a GP plus an additive noise term, we propose a generalization of the Vecchia (J. Roy. Statist. Soc. Ser. B 50 (1988) 297-312) approach as a framework for GP approximations. We show that our general Vecchia approach contains many popular existing GP approximations as special cases, allowing for comparisons among the different methods within a unified framework. Representing the models by directed acyclic graphs, we determine the sparsity of the matrices necessary for inference, which leads to new insights regarding the computational properties. Based on these results, we propose a novel sparse general Vecchia approximation, which ensures computational feasibility for large spatial datasets but can lead to considerable improvements in approximation accuracy over Vecchia's original approach. We provide several theoretical results and conduct numerical comparisons. We conclude with guidelines for the use of Vecchia approximations in spatial statistics.

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Section: Full-scale Approximation (Fsa)mentioning

confidence: 99%

A General Framework for Vecchia Approximations of Gaussian Processes

Katzfuß¹,

Guinness²

2021

Statist. Sci.

170

162

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“…We do compare to other methods as well. In Zhang et al (2018) paper, they compare the Smoothed Full-Scale Approximation (SFSA), Full-Scale Approximation using a block modulating function (FSAB) (Sang and Huang, 2012), NNGP, and a local Gaussian process method with adaptive local designs (LaGP) (Gramacy and Apley, 2015). Their results…”

Section: Discussionmentioning

confidence: 99%

“…As an illustration, we analyze the ozone dataset used in Cressie and Johannesson (2008), which has become a benchmark dataset in the spatial statistics literature (Zhang et al, 2018). This dataset consists of n = 173, 405 values of total column ozone (TCO) in Dobson units (see Figure…”

Section: Ozone Data Application: Data Descriptionmentioning

confidence: 99%

“…In Section 4, we present a simulation study, and compare our approach to two different state-of-the art methods in spatial statistics referred to as the Nearest Neighbor Gaussian Process (NNGP; Datta et al, 2016b) and the general Vecchia approximation (Katzfuss and Guinness, 2017). In Section 5, we implement our model using the benchmark ozone dataset analyzed in (Cressie and Johannesson, 2008) and (Zhang et al, 2018).…”

Section: Introductionmentioning

confidence: 99%

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Bayesian Inference for Big Spatial Data Using Non-stationary Spectral Simulation

Yang¹,

Bradley²

2020

Preprint

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It is increasingly understood that the assumption of stationarity is unrealistic for many spatial processes. In this article, we combine dimension expansion with a spectral method to model big non-stationary spatial fields in a computationally efficient manner. Specifically, we use Mejía and Rodríguez-Iturbe (1974)'s spectral simulation approach to simulate a spatial process with a covariogram at locations that have an expanded dimension. We introduce Bayesian hierarchical modelling to dimension expansion, which originally has only been modeled using a method of moments approach. In particular, we simulate from the posterior distribution using a collapsed Gibbs sampler. Our method is both full rank and non-stationary, and can be applied to big spatial data because it does not involve storing and inverting large covariance matrices. Additionally, we have fewer parameters than many other non-stationary spatial models. We demonstrate the wide applicability of our approach using a simulation study, and an application using ozone data obtained from the National Aeronautics and Space Administration (NASA).

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“…Our contribution lies in providing a practical approach for implementing classes of so called "full scale approximation" models that have been explored in [9] and, more recently, in [1]. More specifically, we avoid more expensive iterative algorithms such as Markov chain Monte Carlo (MCMC), such as in [1], and formulate conjugate Bayesian models by modeling the response itself as a scalable "sparse plus low rank Gaussian process" (SLGP) to carry out inference (including estimation and prediction with uncertainty quantification) using exact distribution theory. This is especially beneficial for massive data sets of the magnitude we consider here.…”

Section: Introductionmentioning

confidence: 99%

Conjugate Nearest Neighbor Gaussian Process Models for Efficient Statistical Interpolation of Large Spatial Data

Shirota¹,

Finley²,

Cook³

et al. 2019

Preprint

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A key challenge in spatial statistics is the analysis for massive spatially-referenced data sets. Such analyses often proceed from Gaussian process specifications that can produce rich and robust inference, but involve dense covariance matrices that lack computationally exploitable structures. The matrix computations required for fitting such models involve floating point operations in cubic order of the number of spatial locations and dynamic memory storage in quadratic order. Recent developments in spatial statistics offer a variety of massively scalable approaches. Bayesian inference and hierarchical models, in particular, have gained popularity due to their richness and flexibility in accommodating spatial processes. Our current contribution is to provide computationally efficient exact algorithms for spatial interpolation of massive data sets using scalable spatial processes. We combine lowrank Gaussian processes with efficient sparse approximations. Following recent work by [1], we model the low-rank process using a Gaussian predictive process (GPP) and the residual process as a sparsity-inducing nearest-neighbor Gaussian process (NNGP). A key contribution here is to implement these models using exact conjugate Bayesian modeling to avoid expensive iterative algorithms. Through the simulation studies, we evaluate performance of the proposed approach and the robustness of our models, especially for long range prediction. We implement our approaches for remotely sensed light detection and ranging (LiDAR) data collected over the US Forest Service Tanana Inventory Unit (TIU) in a remote portion of Interior Alaska.

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Smoothed Full-Scale Approximation of Gaussian Process Models for Computation of Large Spatial Datasets

Cited by 10 publications

References 38 publications

A General Framework for Vecchia Approximations of Gaussian Processes

A General Framework for Vecchia Approximations of Gaussian Processes

Bayesian Inference for Big Spatial Data Using Non-stationary Spectral Simulation

Conjugate Nearest Neighbor Gaussian Process Models for Efficient Statistical Interpolation of Large Spatial Data

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