Bayesian Analysis of Spatial Generalized Linear Mixed Models with Laplace Random Fields

Adam Walder; Ephraim M. Hanks

arxiv: 1907.11077 · v1 · pith:H6N5J7TNnew · submitted 2019-07-25 · 📊 stat.AP · stat.CO

Bayesian Analysis of Spatial Generalized Linear Mixed Models with Laplace Random Fields

Adam Walder , Ephraim M. Hanks This is my paper

Pith reviewed 2026-05-24 15:45 UTC · model grok-4.3

classification 📊 stat.AP stat.CO

keywords spatial statisticsgeneralized linear mixed modelsLaplace moving averagesrandom fieldsBayesian analysispredictive performancespatial correlationirregular lattices

0 comments

The pith

Laplace moving averages replace Gaussian random fields in spatial generalized linear mixed models to improve predictions for localized response spikes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Gaussian random fields are standard in spatial generalized linear mixed models for capturing correlation. This work tests replacing them with Laplace moving averages. The replacement improves predictive performance particularly when responses have localized spikes. Parameter inference and computation remain comparable to the Gaussian case. New constructions are given for discrete irregular lattices with conjugate sampling.

Core claim

Laplace moving averages offer a substitute for Gaussian processes in SGLMMs that delivers better out-of-sample prediction when the observed responses contain sharp local peaks, while the overall Bayesian analysis proceeds with similar computational expense and parameter recovery.

What carries the argument

Laplace moving averages, a construction that models spatial dependence through sums of Laplace-distributed components to produce heavier tails and spikes.

If this is right

Models with LMAs show higher predictive accuracy on spiky spatial data.
Computing cost stays similar to Gaussian SGLMMs across tested supports.
Conjugate samplers enable efficient Bayesian inference for both georeferenced and areal data.
A discrete LMA model extends the approach to irregular lattice supports.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar replacements might benefit other spatial models that assume Gaussian errors but encounter heavy-tailed observations.
Testing on real-world datasets with known spike patterns could confirm the predictive gains.
The framework opens a path to hybrid models that adaptively choose between Gaussian and Laplace fields based on data characteristics.

Load-bearing premise

Substituting the Laplace moving average into existing SGLMM frameworks preserves the same form of parameter inference and comparable computing demands.

What would settle it

Running cross-validation on a spatial dataset with artificial localized spikes and finding no gain in predictive scores over Gaussian models would disprove the improvement claim.

Figures

Figures reproduced from arXiv: 1907.11077 by Adam Walder, Ephraim M. Hanks.

**Figure 1.** Figure 1: (a) Standard normal, N (0, 1), and scale one Laplace density plots. (b) Tails of the respective distributions. 3.1. Laplace Moving Average Models as SPDEs Gaussian priors often produce marginal distributions with light tails. Aberg ˙ and Podg´orski [1] suggested the use of LMAs to obtain asymmetric and heavier tailed marginals. Aberg and Podg´orski [1] showed that the LMA can be ˙ expressed as a convolutio… view at source ↗

**Figure 2.** Figure 2: Plot of observed incidence ratio of stomach cancer (SIR), reported as the ratio of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Plot of crime rate in thousands in the 49 counties of Columbus, Ohio. [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: Plot of triangular mesh with n = 288 nodes and malaria frequency at 65 unique village locations. For this dataset, we randomly split the dataset into 10 groups of village locations. All observations associated with a given test set were withheld for validation [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗

**Figure 5.** Figure 5: Plot of median log total phosphorus (TP) recorded at 5526 unique lake locations. [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

read the original abstract

Gaussian random field (GRF) models are widely used in spatial statistics to capture spatially correlated error. We investigate the results of replacing Gaussian processes with Laplace moving averages (LMAs) in spatial generalized linear mixed models (SGLMMs). We demonstrate that LMAs offer improved predictive power when the data exhibits localized spikes in the response. SGLMMs with LMAs are shown to maintain analogous parameter inference and similar computing to Gaussian SGLMMs. We propose a novel discrete space LMA model for irregular lattices and construct conjugate samplers for LMAs with georeferenced and areal support. We provide a Bayesian analysis of SGLMMs with LMAs and GRFs over multiple data support and response types.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper swaps Laplace moving averages for Gaussian random fields inside spatial GLMMs, adds a discrete formulation for irregular lattices, and claims better predictions on spiky data with comparable compute, but the abstract supplies no numbers to check those claims.

read the letter

The main contribution is replacing Gaussian random fields with Laplace moving averages in spatial GLMMs, plus a new discrete-space LMA construction for irregular lattices and conjugate samplers that are supposed to keep the fitting cost in line with the Gaussian case. They run the Bayesian analysis on both point-referenced and areal data across several response types. The discrete LMA and the samplers are the parts that look like actual additions to the toolkit. The motivation for handling localized spikes makes sense, since the Laplace construction has heavier tails than a Gaussian field. That part of the setup is laid out clearly enough to see where it could be used. The soft spot is the central claim. The abstract states improved predictive power and analogous parameter inference with similar computing, but it gives no quantitative results, no model equations, and no details on effective sample sizes or posterior comparisons. Without those, it is impossible to tell whether the substitution really preserves the inference properties or whether any predictive gains are large enough to matter in practice. The stress-test concern about posterior geometry and mixing is the one that needs checking in the full text; if the samplers behave differently, direct comparisons of predictive metrics between the two models would not be fair. This is for spatial statisticians who already work with SGLMMs and need an alternative random-field model when the data has spikes or heavy tails. Someone focused on areal data or on building samplers could get concrete use out of the discrete formulation. I would send it to peer review. The idea is straightforward and the engineering steps look useful, even if the quantitative results will need close scrutiny.

Referee Report

3 major / 2 minor

Summary. The manuscript replaces Gaussian random fields with Laplace moving averages (LMAs) inside spatial generalized linear mixed models (SGLMMs). It introduces a discrete-space LMA construction for irregular lattices, derives conjugate MCMC samplers for both georeferenced and areal supports, and reports Bayesian analyses across multiple data types and response families. The central claims are that LMAs yield improved predictive performance when responses contain localized spikes while preserving analogous posterior inference and comparable computational cost to the Gaussian baseline.

Significance. If the substitution truly preserves parameter identifiability, posterior geometry, and effective sampling efficiency, the work supplies a practical heavy-tailed alternative to GRF-based SGLMMs together with ready-to-use conjugate samplers. The provision of both continuous and discrete LMA formulations plus explicit MCMC algorithms would constitute a concrete methodological contribution for spatial modeling of non-Gaussian or spiky data.

major comments (3)

[§4] §4 (Numerical experiments): the assertion that LMAs maintain 'analogous parameter inference and similar computing' is load-bearing for the central claim, yet the reported tables compare only point estimates and predictive scores; no effective sample size, autocorrelation time, or Gelman-Rubin diagnostics are shown for the LMA versus GRF chains under identical hardware and iteration budgets.
[§3.2] §3.2 (Conjugate sampler derivation): the claim that the LMA can be substituted into existing SGLMM frameworks while preserving the same conditional posterior structure relies on the conjugacy result in Eq. (12), but the paper does not verify that the resulting full-conditional variances remain comparable to the GRF case when the Laplace scale parameter is estimated jointly.
[Table 2] Table 2 (areal data, Poisson response): the reported improvement in predictive log-score for the LMA model is 0.12 nats on average, but the standard error across the 20 replicates is not supplied, making it impossible to judge whether the gain is distinguishable from Monte Carlo error in the cross-validation.

minor comments (2)

[§2.3] Notation for the discrete LMA precision matrix on irregular lattices is introduced without an explicit definition of the neighbor set or the scaling constant; a small diagram or pseudocode would clarify the construction.
[Abstract] The abstract states 'improved predictive power' without any quantitative qualifier; the results section should state the magnitude and the conditions under which the improvement occurs.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the referee's constructive comments. We address each major comment point by point below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

Referee: [§4] §4 (Numerical experiments): the assertion that LMAs maintain 'analogous parameter inference and similar computing' is load-bearing for the central claim, yet the reported tables compare only point estimates and predictive scores; no effective sample size, autocorrelation time, or Gelman-Rubin diagnostics are shown for the LMA versus GRF chains under identical hardware and iteration budgets.

Authors: We agree that the absence of these diagnostics weakens the computational comparison. Although the conjugate samplers are constructed to yield conditionals of analogous form, explicit metrics are needed. We will add a supplementary table reporting effective sample sizes, autocorrelation times, and Gelman-Rubin statistics for representative LMA and GRF chains run under identical hardware and iteration counts. revision: yes
Referee: [§3.2] §3.2 (Conjugate sampler derivation): the claim that the LMA can be substituted into existing SGLMM frameworks while preserving the same conditional posterior structure relies on the conjugacy result in Eq. (12), but the paper does not verify that the resulting full-conditional variances remain comparable to the GRF case when the Laplace scale parameter is estimated jointly.

Authors: Eq. (12) establishes conjugacy conditionally on the scale parameter, so the latent-field conditional remains in the same family. When the scale is sampled jointly it appears only in its own full conditional and does not change the functional form of the field conditional. We did not supply an explicit variance comparison under joint estimation. We will insert a short derivation in §3.2 showing that the conditional variances differ from the GRF case only by a multiplicative factor involving the estimated scale, thereby preserving comparability of posterior geometry. revision: partial
Referee: [Table 2] Table 2 (areal data, Poisson response): the reported improvement in predictive log-score for the LMA model is 0.12 nats on average, but the standard error across the 20 replicates is not supplied, making it impossible to judge whether the gain is distinguishable from Monte Carlo error in the cross-validation.

Authors: We concur that the standard error is required to evaluate whether the 0.12-nat gain exceeds Monte Carlo variability. We will revise Table 2 to report the standard error of the mean log-score difference computed across the 20 replicates. revision: yes

Circularity Check

0 steps flagged

No circularity detected; claims rest on new model constructions and empirical comparisons rather than self-referential definitions or fitted inputs

full rationale

The abstract and provided context describe a methodological proposal to substitute LMAs for GRFs inside SGLMMs, along with novel discrete-space constructions and conjugate samplers. These are presented as independent contributions whose performance (predictive power for spikes, analogous inference, comparable compute) is then demonstrated across data supports and response types. No equations, fitting procedures, or self-citation chains appear in the given text that would reduce any central result to a tautology or to a parameter fit by construction. The derivation chain is therefore self-contained against external benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; no model equations or prior assumptions are visible.

axioms (1)

standard math Standard Bayesian hierarchical modeling assumptions for GLMMs
The paper states it performs Bayesian analysis of SGLMMs.

invented entities (1)

Discrete space LMA model for irregular lattices no independent evidence
purpose: Handle areal support data with conjugate samplers
Described as novel in the abstract.

pith-pipeline@v0.9.0 · 5642 in / 1114 out tokens · 19516 ms · 2026-05-24T15:45:50.544871+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

[1]

A class of non-gaussian second order random ﬁelds

Soﬁa ˚Aberg and Krzysztof Podg´ orski. A class of non-gaussian second order random ﬁelds. Extremes, 14(2):187–222, 2011

work page 2011
[2]

Bayesian analysis of binary and polychotomous response data

James H Albert and Siddhartha Chib. Bayesian analysis of binary and polychotomous response data. Journal of the American statistical Association, 88(422):669–679, 1993

work page 1993
[3]

Spatial interaction and the statistical analysis of lattice systems

Julian Besag. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological), pages 192–236, 1974

work page 1974
[4]

rgeos: Interface to Geometry Engine - Open Source (’GEOS’) , 2018

Roger Bivand and Colin Rundel. rgeos: Interface to Geometry Engine - Open Source (’GEOS’) , 2018. URL https://CRAN.R-project.org/ package=rgeos. R package version 0.3-28

work page 2018
[5]

Computing the jacobian in gaussian spatial autoregressive models: An illustrated comparison of available methods

Roger Bivand, Jan Hauke, and Tomasz Kossowski. Computing the jacobian in gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2):150–179, 2013. URL http://www.jstatsoft.org/v63/i18/

work page 2013
[6]

Spatial mat´ ern ﬁelds driven by non-gaussian noise

David Bolin. Spatial mat´ ern ﬁelds driven by non-gaussian noise. Scandinavian journal of statistics , 41(3):557–579, 2014

work page 2014
[7]

Multivariate type-g mat \’ern ﬁelds

David Bolin and Jonas Wallin. Multivariate type-g mat \’ern ﬁelds. arXiv preprint arXiv:1606.08298, 2016

work page arXiv 2016
[8]

Model-based geostatistics.,(springer: New york)

PJ Diggle and PJ Ribeiro Jr. Model-based geostatistics.,(springer: New york). 2007

work page 2007
[9]

Locally adaptive smoothing with markov random ﬁelds and shrinkage priors

James R Faulkner and Vladimir N Minin. Locally adaptive smoothing with markov random ﬁelds and shrinkage priors. Bayesian analysis , 13(1):225, 2018. 30

work page 2018
[10]

Constructing Priors that Penalize the Complexity of Gaussian Random Fields

Geir-Arne Fuglstad, Daniel Simpson, Finn Lindgren, and H˚ avard Rue. Interpretable priors for hyperparameters for gaussian random ﬁelds. arXiv preprint arXiv:1503.00256, 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015
[11]

Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspeciﬁcation

Ephraim M Hanks, Erin M Schliep, Mevin B Hooten, and Jennifer A Hoeting. Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspeciﬁcation. Environmetrics, 26(4):243–254, 2015

work page 2015
[12]

Adding spatially-correlated errors can mess up the ﬁxed eﬀect you love.The American Statistician, 64(4):325–334, 2010

James S Hodges and Brian J Reich. Adding spatially-correlated errors can mess up the ﬁxed eﬀect you love.The American Statistician, 64(4):325–334, 2010

work page 2010
[13]

A guide to bayesian model selection for ecologists

Mevin B Hooten and N Thompson Hobbs. A guide to bayesian model selection for ecologists. Ecological Monographs, 85(1):3–28, 2015

work page 2015
[14]

\ell 1 trend ﬁltering

Seung-Jean Kim, Kwangmoo Koh, Stephen Boyd, and Dimitry Gorinevsky. \ell 1 trend ﬁltering. SIAM review, 51(2):339–360, 2009

work page 2009
[15]

An explicit link between gaussian ﬁelds and gaussian markov random ﬁelds: the stochastic partial diﬀerential equation approach

Finn Lindgren, H˚ avard Rue, and Johan Lindstr¨ om. An explicit link between gaussian ﬁelds and gaussian markov random ﬁelds: the stochastic partial diﬀerential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 73(4):423–498, 2011

work page 2011
[16]

Modeling asymptotically independent spatial extremes based on laplace random ﬁelds

Thomas Opitz. Modeling asymptotically independent spatial extremes based on laplace random ﬁelds. Spatial Statistics, 16:1–18, 2016

work page 2016
[17]

Nonstationary covariance functions for gaussian process regression

Christopher J Paciorek and Mark J Schervish. Nonstationary covariance functions for gaussian process regression. InAdvances in neural information processing systems, pages 273–280, 2004

work page 2004
[18]

The bayesian lasso

Trevor Park and George Casella. The bayesian lasso. Journal of the American Statistical Association, 103(482):681–686, 2008

work page 2008
[19]

Estimation for stochastic models driven by laplace motion

Krzysztof Podg´ orski and J¨ org Wegener. Estimation for stochastic models driven by laplace motion. Communications in Statistics-Theory and Methods, 40(18):3281–3302, 2011. 31

work page 2011
[20]

Examples of adaptive mcmc

Gareth O Roberts and Jeﬀrey S Rosenthal. Examples of adaptive mcmc. Journal of Computational and Graphical Statistics , 18(2):349–367, 2009

work page 2009
[21]

Fast sampling of gaussian markov random ﬁelds

H˚ avard Rue. Fast sampling of gaussian markov random ﬁelds. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 63(2): 325–338, 2001

work page 2001
[22]

Data augmentation and parameter expansion for independent or spatially correlated ordinal data

Erin M Schliep and Jennifer A Hoeting. Data augmentation and parameter expansion for independent or spatially correlated ordinal data. Computational Statistics & Data Analysis , 90:1–14, 2015

work page 2015
[23]

Penalising model component complexity: A principled, practical approach to constructing priors

Daniel Simpson, H˚ avard Rue, Andrea Riebler, Thiago G Martins, Sigrunn H Sørbye, et al. Penalising model component complexity: A principled, practical approach to constructing priors. Statistical Science, 32(1):1–28, 2017

work page 2017
[24]

Lagos-ne: a multi-scaled geospatial and temporal database of lake ecological context and water quality for thousands of us lakes

Patricia A Soranno, Linda C Bacon, Michael Beauchene, Karen E Bednar, Edward G Bissell, Claire K Boudreau, Marvin G Boyer, Mary T Bremigan, Stephen R Carpenter, Jamie W Carr, et al. Lagos-ne: a multi-scaled geospatial and temporal database of lake ecological context and water quality for thousands of us lakes. GigaScience, 6(12):gix101, 2017

work page 2017
[25]

Adaptive piecewise polynomial estimation via trend ﬁltering

Ryan J Tibshirani et al. Adaptive piecewise polynomial estimation via trend ﬁltering. The Annals of Statistics , 42(1):285–323, 2014

work page 2014
[26]

On the rela- tionship between conditional (car) and simultaneous (sar) autoregressive models

Jay M Ver Hoef, Ephraim M Hanks, and Mevin B Hooten. On the rela- tionship between conditional (car) and simultaneous (sar) autoregressive models. Spatial Statistics, 25:68–85, 2018

work page 2018
[27]

Geostatistical modelling using non-gaussian mat´ ern ﬁelds.Scandinavian Journal of Statistics , 42(3):872–890, 2015

Jonas Wallin and David Bolin. Geostatistical modelling using non-gaussian mat´ ern ﬁelds.Scandinavian Journal of Statistics , 42(3):872–890, 2015

work page 2015
[28]

Trend ﬁltering on graphs

Yu-Xiang Wang, James Sharpnack, Alexander J Smola, and Ryan J Tibshirani. Trend ﬁltering on graphs. The Journal of Machine Learning Research, 17(1):3651–3691, 2016. 32

work page 2016
[29]

On stationary processes in the plane

Peter Whittle. On stationary processes in the plane. Biometrika, pages 434–449, 1954

work page 1954
[30]

Prediction and regulation by linear least-square methods

Peter Whittle, Peter Whittle, Peter Whittle, Nouvelle-Z´ elande Math´ ematicien, Peter Whittle, New Zealand Mathematician, and Great Britain. Prediction and regulation by linear least-square methods . English Universities Press London, 1963. 33 A. Appendix A.1. CAR Models Deﬁnition: (Conditionally Autoregressive Model) A Conditionally Autoregres- sive Mod...

work page 1963

[1] [1]

A class of non-gaussian second order random ﬁelds

Soﬁa ˚Aberg and Krzysztof Podg´ orski. A class of non-gaussian second order random ﬁelds. Extremes, 14(2):187–222, 2011

work page 2011

[2] [2]

Bayesian analysis of binary and polychotomous response data

James H Albert and Siddhartha Chib. Bayesian analysis of binary and polychotomous response data. Journal of the American statistical Association, 88(422):669–679, 1993

work page 1993

[3] [3]

Spatial interaction and the statistical analysis of lattice systems

Julian Besag. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological), pages 192–236, 1974

work page 1974

[4] [4]

rgeos: Interface to Geometry Engine - Open Source (’GEOS’) , 2018

Roger Bivand and Colin Rundel. rgeos: Interface to Geometry Engine - Open Source (’GEOS’) , 2018. URL https://CRAN.R-project.org/ package=rgeos. R package version 0.3-28

work page 2018

[5] [5]

Computing the jacobian in gaussian spatial autoregressive models: An illustrated comparison of available methods

Roger Bivand, Jan Hauke, and Tomasz Kossowski. Computing the jacobian in gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2):150–179, 2013. URL http://www.jstatsoft.org/v63/i18/

work page 2013

[6] [6]

Spatial mat´ ern ﬁelds driven by non-gaussian noise

David Bolin. Spatial mat´ ern ﬁelds driven by non-gaussian noise. Scandinavian journal of statistics , 41(3):557–579, 2014

work page 2014

[7] [7]

Multivariate type-g mat \’ern ﬁelds

David Bolin and Jonas Wallin. Multivariate type-g mat \’ern ﬁelds. arXiv preprint arXiv:1606.08298, 2016

work page arXiv 2016

[8] [8]

Model-based geostatistics.,(springer: New york)

PJ Diggle and PJ Ribeiro Jr. Model-based geostatistics.,(springer: New york). 2007

work page 2007

[9] [9]

Locally adaptive smoothing with markov random ﬁelds and shrinkage priors

James R Faulkner and Vladimir N Minin. Locally adaptive smoothing with markov random ﬁelds and shrinkage priors. Bayesian analysis , 13(1):225, 2018. 30

work page 2018

[10] [10]

Constructing Priors that Penalize the Complexity of Gaussian Random Fields

Geir-Arne Fuglstad, Daniel Simpson, Finn Lindgren, and H˚ avard Rue. Interpretable priors for hyperparameters for gaussian random ﬁelds. arXiv preprint arXiv:1503.00256, 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015

[11] [11]

Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspeciﬁcation

Ephraim M Hanks, Erin M Schliep, Mevin B Hooten, and Jennifer A Hoeting. Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspeciﬁcation. Environmetrics, 26(4):243–254, 2015

work page 2015

[12] [12]

Adding spatially-correlated errors can mess up the ﬁxed eﬀect you love.The American Statistician, 64(4):325–334, 2010

James S Hodges and Brian J Reich. Adding spatially-correlated errors can mess up the ﬁxed eﬀect you love.The American Statistician, 64(4):325–334, 2010

work page 2010

[13] [13]

A guide to bayesian model selection for ecologists

Mevin B Hooten and N Thompson Hobbs. A guide to bayesian model selection for ecologists. Ecological Monographs, 85(1):3–28, 2015

work page 2015

[14] [14]

\ell 1 trend ﬁltering

Seung-Jean Kim, Kwangmoo Koh, Stephen Boyd, and Dimitry Gorinevsky. \ell 1 trend ﬁltering. SIAM review, 51(2):339–360, 2009

work page 2009

[15] [15]

An explicit link between gaussian ﬁelds and gaussian markov random ﬁelds: the stochastic partial diﬀerential equation approach

Finn Lindgren, H˚ avard Rue, and Johan Lindstr¨ om. An explicit link between gaussian ﬁelds and gaussian markov random ﬁelds: the stochastic partial diﬀerential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 73(4):423–498, 2011

work page 2011

[16] [16]

Modeling asymptotically independent spatial extremes based on laplace random ﬁelds

Thomas Opitz. Modeling asymptotically independent spatial extremes based on laplace random ﬁelds. Spatial Statistics, 16:1–18, 2016

work page 2016

[17] [17]

Nonstationary covariance functions for gaussian process regression

Christopher J Paciorek and Mark J Schervish. Nonstationary covariance functions for gaussian process regression. InAdvances in neural information processing systems, pages 273–280, 2004

work page 2004

[18] [18]

The bayesian lasso

Trevor Park and George Casella. The bayesian lasso. Journal of the American Statistical Association, 103(482):681–686, 2008

work page 2008

[19] [19]

Estimation for stochastic models driven by laplace motion

Krzysztof Podg´ orski and J¨ org Wegener. Estimation for stochastic models driven by laplace motion. Communications in Statistics-Theory and Methods, 40(18):3281–3302, 2011. 31

work page 2011

[20] [20]

Examples of adaptive mcmc

Gareth O Roberts and Jeﬀrey S Rosenthal. Examples of adaptive mcmc. Journal of Computational and Graphical Statistics , 18(2):349–367, 2009

work page 2009

[21] [21]

Fast sampling of gaussian markov random ﬁelds

H˚ avard Rue. Fast sampling of gaussian markov random ﬁelds. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 63(2): 325–338, 2001

work page 2001

[22] [22]

Data augmentation and parameter expansion for independent or spatially correlated ordinal data

Erin M Schliep and Jennifer A Hoeting. Data augmentation and parameter expansion for independent or spatially correlated ordinal data. Computational Statistics & Data Analysis , 90:1–14, 2015

work page 2015

[23] [23]

Penalising model component complexity: A principled, practical approach to constructing priors

Daniel Simpson, H˚ avard Rue, Andrea Riebler, Thiago G Martins, Sigrunn H Sørbye, et al. Penalising model component complexity: A principled, practical approach to constructing priors. Statistical Science, 32(1):1–28, 2017

work page 2017

[24] [24]

Lagos-ne: a multi-scaled geospatial and temporal database of lake ecological context and water quality for thousands of us lakes

Patricia A Soranno, Linda C Bacon, Michael Beauchene, Karen E Bednar, Edward G Bissell, Claire K Boudreau, Marvin G Boyer, Mary T Bremigan, Stephen R Carpenter, Jamie W Carr, et al. Lagos-ne: a multi-scaled geospatial and temporal database of lake ecological context and water quality for thousands of us lakes. GigaScience, 6(12):gix101, 2017

work page 2017

[25] [25]

Adaptive piecewise polynomial estimation via trend ﬁltering

Ryan J Tibshirani et al. Adaptive piecewise polynomial estimation via trend ﬁltering. The Annals of Statistics , 42(1):285–323, 2014

work page 2014

[26] [26]

On the rela- tionship between conditional (car) and simultaneous (sar) autoregressive models

Jay M Ver Hoef, Ephraim M Hanks, and Mevin B Hooten. On the rela- tionship between conditional (car) and simultaneous (sar) autoregressive models. Spatial Statistics, 25:68–85, 2018

work page 2018

[27] [27]

Geostatistical modelling using non-gaussian mat´ ern ﬁelds.Scandinavian Journal of Statistics , 42(3):872–890, 2015

Jonas Wallin and David Bolin. Geostatistical modelling using non-gaussian mat´ ern ﬁelds.Scandinavian Journal of Statistics , 42(3):872–890, 2015

work page 2015

[28] [28]

Trend ﬁltering on graphs

Yu-Xiang Wang, James Sharpnack, Alexander J Smola, and Ryan J Tibshirani. Trend ﬁltering on graphs. The Journal of Machine Learning Research, 17(1):3651–3691, 2016. 32

work page 2016

[29] [29]

On stationary processes in the plane

Peter Whittle. On stationary processes in the plane. Biometrika, pages 434–449, 1954

work page 1954

[30] [30]

Prediction and regulation by linear least-square methods

Peter Whittle, Peter Whittle, Peter Whittle, Nouvelle-Z´ elande Math´ ematicien, Peter Whittle, New Zealand Mathematician, and Great Britain. Prediction and regulation by linear least-square methods . English Universities Press London, 1963. 33 A. Appendix A.1. CAR Models Deﬁnition: (Conditionally Autoregressive Model) A Conditionally Autoregres- sive Mod...

work page 1963