The SAR Model for Very Large Datasets: A Reduced Rank Approach

Burden, Sandy; Cressie, Noel; Steel, David G

doi:10.3390/econometrics3020317

Cited by 23 publications

(17 citation statements)

References 40 publications

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“…Notes 1 Assumptions of a known spatial correlation matrix and known number of basis functions, L, are common in the literature about reduced rank spatial modeling (e.g., Cressie and Johannesson 2008;Hughes and Haran 2013;Burden, Cressie, and Steel 2015). 2 Use of this property is needed because the Nystr€ om extension, which we use in an eigenfunction approximation using the Nystr€ om extension section, is only for positive semidefinite matrix, while MCM is by definition an indefinite matrix.…”

Section: Acknowledgementmentioning

confidence: 99%

“…A number of computationally efficient approximations have been developed in these study areas. They include likelihood approximations (e.g., Stein, Chi, and Welty ; Griffith ; LeSage and Pace ; Arbia ), low rank approximations (e.g., Cressie and Johannesson ; Hughes and Haran ; Burden, Cressie, and Steel ), spatial process approximations (e.g., Banerjee et al ; Datta et al ), and Gaussian Markov random field‐based approximations (Lindgren, Rue, and Lindström ) (see Sun, Li, and Genton for review).…”

Section: Introductionmentioning

confidence: 99%

“… Assumptions of a known spatial correlation matrix and known number of basis functions, L , are common in the literature about reduced rank spatial modeling (e.g., Cressie and Johannesson ; Hughes and Haran ; Burden, Cressie, and Steel ). …”

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confidence: 99%

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Eigenvector Spatial Filtering for Large Data Sets: Fixed and Random Effects Approaches

Murakami

Griffith

2018

Geographical Analysis

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Eigenvector spatial filtering (ESF) is a spatial modeling approach, which has been applied in urban and regional studies, ecological studies, and so on. However, it is computationally demanding, and may not be suitable for large data modeling. The objective of this study is developing fast ESF and random effects ESF (RE‐ESF), which are capable of handling very large samples. To achieve it, we accelerate eigen‐decomposition and parameter estimation, which make ESF and RE‐ESF slow. The former is accelerated by utilizing the Nyström extension, whereas the latter is by small matrix tricks. The resulting fast ESF and fast RE‐ESF are compared with nonapproximated ESF and RE‐ESF in Monte Carlo simulation experiments. The result shows that, while ESF and RE‐ESF are slow for several thousand samples, fast ESF and RE‐ESF require only several seconds for the samples. It is also suggested that the proposed approaches effectively remove positive spatial dependence in the residuals with very small approximation errors when the number of eigenvectors considered is 200 or more. Note that these approaches cannot deal with negative spatial dependence. The proposed approaches are implemented in an R package “spmoran.”

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Section: Acknowledgementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Eigenvector Spatial Filtering for Large Data Sets: Fixed and Random Effects Approaches

Murakami

Griffith

2018

Geographical Analysis

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“…A search of the remote sensing literature reveals the use of conventional rather than spatial linear regression techniques to analyze remotely sensed data, although these data tend to contain very high levels of positive spatial autocorrelation, and some remote sensing and other scientists recognize the usefulness of employing spatial autoregressive techniques (e.g., Wimberly et al 2009, Burden et al 2015. The absence of spatial autoregressive modeling in part arises from severe numerical complications.…”

Section: Introductionmentioning

confidence: 98%

Approximation of Gaussian spatial autoregressive models for massive regular square tessellation data

Griffith

2015

International Journal of Geographical Information Science

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Most of the literature to date proposes approximations to the determinant of a positive definite n × n spatial covariance matrix (the Jacobian term) for Gaussian spatial autoregressive models that fail to support the analysis of massive georeferenced data sets. This paper briefly surveys this literature, recalls and refines much simpler Jacobian approximations, presents selected eigenvalue estimation techniques, summarizes validation results (for estimated eigenvalues, Jacobian approximations, and estimation of a spatial autocorrelation parameter), and illustrates the estimation of the spatial autocorrelation parameter in a spatial autoregressive model specification for cases as large as n = 37,214,101. The principal contribution of this paper is to the implementation of spatial autoregressive model specifications for any size of georeferenced data set. Its specific additions to the literature include (1) new, more efficient estimation algorithms; (2) an approximation of the Jacobian term for remotely sensed data forming incomplete rectangular regions; (3) issues of inference; and (4) timing results.

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“…This propagator matrix is referred to as the Moran's I (MI) propagator matrix because of a connection to the MI statistic from Moran (1950). This motivation is similar to the specification of the MI basis functions used in Griffith (2000Griffith ( , 2002Griffith ( , 2004, Tiefelsdorf and Griffith (2007), Hughes and Haran (2013), Porter et al (2013), and Burden et al (2015). The dynamic properties associated with the MI propagator matrix still need development.…”

Section: Introductionmentioning

confidence: 99%

Spatio‐temporal models for big multinomial data using the conditional multivariate logit‐beta distribution

Bradley

Wikle

Holan

2019

Journal Time Series Analysis

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We introduce a Bayesian approach for analyzing high‐dimensional multinomial data that are referenced over space and time. In particular, the proportions associated with multinomial data are assumed to have a logit link to a latent spatio‐temporal mixed effects model. This strategy allows for covariances that are nonstationarity in both space and time, asymmetric, and parsimonious. We also introduce the use of the conditional multivariate logit‐beta distribution into the dependent multinomial data setting, which leads to conjugate full‐conditional distributions for use in a collapsed Gibbs sampler. We refer to this model as the multinomial spatio‐temporal mixed effects model (MN‐STM). Additionally, we provide methodological developments including: the derivation of the associated full‐conditional distributions, a relationship with a latent Gaussian process model, and the stability of the non‐stationary vector autoregressive model. We illustrate the MN‐STM through simulations and through a demonstration with public‐use quarterly workforce indicators data from the longitudinal employer household dynamics program of the US Census Bureau.

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The SAR Model for Very Large Datasets: A Reduced Rank Approach

Cited by 23 publications

References 40 publications

Eigenvector Spatial Filtering for Large Data Sets: Fixed and Random Effects Approaches

Eigenvector Spatial Filtering for Large Data Sets: Fixed and Random Effects Approaches

Approximation of Gaussian spatial autoregressive models for massive regular square tessellation data

Spatio‐temporal models for big multinomial data using the conditional multivariate logit‐beta distribution

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