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REVIEW 4 major objections 4 minor 48 references

Incorporating Correlated Nugget Effects in Multivariate Spatial Models: An Application to Argo Ocean Data

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Adding a correlated nugget to bivariate spatial models lowers the estimated median temperature–salinity correlation from 0.95 to 0.87 in global ocean profile data.

desk verdict A solid methodological extension whose headline oceanographic claim—systematic overestimation of latent T–S correlation—rests on an identifiability assumption that is plausible but not demonstrated; worth refereeing with a request for revision. read the letter →

arxiv 2506.03042 v1 pith:3CTDK3G7 submitted 2025-06-03 stat.ME stat.AP

Damilya Saduakhas , David Bolin , Xiaotian Jin , Alexandre B. Simas , Jonas Wallin This is my paper
classification stat.MEstat.AP MSC 62M3062M4062H11
keywords multivariatespatialstatisticscorrelatednuggeteffectMatérnSPDEmodelnormal-inverseGaussiannoiseArgooceandatameasurement-errorcorrelationtemperature-salinitycoupling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a routine assumption in multivariate geostatistics—that measurement errors for different variables at the same location are independent—quietly inflates how strongly ocean temperature and salinity appear to be coupled. To remove that assumption, it builds bivariate spatial models whose latent field follows a Matérn stochastic-partial-differential-equation (SPDE) model and whose observation noise has a correlated "nugget" term shared by the two variables at co-located sensors. Fitted to about 1.35 million global ocean profiles, the correlated-nugget version lowers the median Pearson correlation between temperature and salinity from 0.95 to 0.87 in the Gaussian model and from 0.94 to 0.87 in the non-Gaussian model, and it improves predictive scores. The point matters because gridded ocean products built from independent-noise models may be attributing sensor-level coincidence to the physical fields themselves.

What carries the argument

The central object is a bivariate Matérn-SPDE field whose smooth cross-dependence is controlled by a dependence matrix $\mathbf{D}(\theta,\rho)$, paired with an observation model whose noise covariance at co-located points is $\boldsymbol{\Sigma}_\varepsilon=\begin{pmatrix}\sigma_{\varepsilon,1}^2 & \sigma_{\varepsilon,1}\sigma_{\varepsilon,2}\rho_\varepsilon\\ \sigma_{\varepsilon,1}\sigma_{\varepsilon,2}\rho_\varepsilon & \sigma_{\varepsilon,2}^2\end{pmatrix}$. The argument is carried by the split between $\rho$ and $\rho_\varepsilon$: setting $\rho_\varepsilon=0$ forces $\rho$ to absorb all cross-variable dependence, whereas the general model lets small-scale shared noise account for part of it. Proposition 2.1 converts $\rho$ into a Pearson correlation for the $\alpha=2$, $d=2$ case, and that conversion is what the empirical comparisons use.

What would settle it

Fit the correlated- and independent-nugget models to paired temperature and salinity profiles restricted to the most accurately calibrated floats, so the measurement-noise variance is very small; if the latent correlation drop from about 0.95 to 0.87 persists, it is a real field property, but if the drop shrinks toward zero, the headline comparison is dominated by the very measurement noise the model adds.

Watch

Extended reading notes

Core claim

The central claim is that, when two variables are observed at the same location, the shared component of their small-scale noise should be estimated as a parameter rather than forced to zero. The authors define a bivariate process $\mathbf{X}(s)=\mathbf{m}(s)+\mathbf{u}(s)$ with latent cross-dependence governed by a parameter $\rho$ and add a nugget covariance whose off-diagonal element $\rho_\varepsilon$ carries correlated measurement error. Across simulation settings with known $\rho$, they show that fitting with $\rho_\varepsilon=0$ biases the estimated dependence—with $\rho_\varepsilon=-0.8$ turning a true $\rho=0$ into an estimate near $-0.5$—while fitting with the general nugget recovers it. Applied to temperature and salinity at 300 dbar, the fitted dependence parameter $\rho$ falls from 4.90 to 2.30 in the Gaussian model and from 3.96 to 2.19 in the NIG model, and the fitted $\rho_\varepsilon$ is uniformly large and positive. The authors interpret the drop as evidence that independent-noise analyses absorbed measurement coincidence into the latent field.

Load-bearing premise

The load-bearing premise is that the model can tell apart a shared, same-location noise component from a smooth genuine correlation between the two fields; if those two are not separately identifiable from the likelihood, the reported drop in latent correlation is mostly a bookkeeping change.

Editorial extensions

If this is right

  • Gridded ocean products built with independent measurement errors would inherit overstated temperature–salinity coupling, so derived quantities such as potential density and steric-height estimates would carry that bias.
  • At 300 dbar, the Gaussian correlated-nugget model attains the best leave-one-out predictive scores for temperature and salinity, showing the extra parameter earns its keep in prediction.
  • Once nugget correlation is included, the non-Gaussian NIG model offers little improvement over the Gaussian model at 300 dbar, suggesting part of the apparent non-Gaussianity was a misfit symptom of the uncorrelated-noise assumption.
  • Because fitted nugget correlations are typically above 0.7 in the ocean-profile application, the distorting effect of assuming independence is strongest in the exact regime where the data are most noise-coupled.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: paired environmental sensors with shared platform or calibration noise—temperature–oxygen or salinity–nitrate float pairs—should show the same inflation when fitted with diagonal nuggets, so the practice of independent-nugget multivariate spatial modeling deserves scrutiny outside oceanography.
  • A direct test not run in the paper: for floats surfacing within a few kilometers of each other, treat the difference of co-located readings as an empirical nugget and compare it with the fitted $\rho_\varepsilon$ at the same pressure level.
  • Because the 1000 dbar maps show heterogeneous signs, extending the mechanism to nonstationary or three-dimensional SPDEs should re-examine whether the roughly 0.08 drop in latent correlation is uniform across depth and latitude.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes bivariate Gaussian and NIG Matérn-SPDE models with a correlated nugget component, estimated by maximum likelihood or stochastic gradient methods, and applies them with a moving-window design to Argo temperature and salinity at 10, 300, and 1000 dbar. The central empirical claim is that adding a nugget correlation parameter lowers the estimated latent temperature–salinity correlation (at 300 dbar, the median Pearson correlation drops from 0.95 to 0.87 in the Gaussian model and from 0.94 to 0.87 in the NIG model), which the authors interpret as evidence that traditional independent-nugget models overstate the underlying field correlation. The paper also includes a simulation study with 42 configurations and 10 replicates, LOOCV tables, QQ diagnostics, and an open-source implementation in the ngme2 R package.

Significance. The paper addresses a real and important problem in oceanographic mapping and multivariate geostatistics. The methodological package has clear strengths: an open-source implementation, reproducible code and data, an analytic mapping from the dependence parameter to the Pearson correlation (Proposition 2.1), a systematic simulation grid, and evaluation with proper scoring rules and LOOCV. If the interpretation of the fitted nugget correlation as measurement-error correlation were established, the paper would be a useful caution for Argo gridded products. However, as written, the headline claim is load-bearing on an identifiability and interpretation assumption that is not demonstrated, and the paper's own 1000 dbar results contradict the 'systematically overstates' phrasing. The central result is therefore defensible but requires additional support or substantial qualification.

major comments (4)
  1. [Section 4 and Section 5.3] The split between the correlated nugget and the latent cross-dependence is not shown to be identifiable. The total co-located covariance is Sigma_u(0) + Sigma_epsilon, so the nugget is separated from the latent field only by extrapolating the smooth alpha=2 Matérn covariance to zero lag. The simulation study in Section 4 always generates data with nonzero rho_epsilon and never includes the null case rho_epsilon = 0, so it cannot rule out the possibility that the general model spuriously estimates a nonzero nugget correlation and shrinks rho. I recommend adding a null-configuration simulation (rho_epsilon = 0) and a profile-likelihood or similar identifiability check before the Section 5.3 interpretation of the rho drop is used as evidence.
  2. [Section 2.1 and Section 5.3] The interpretation of the fitted rho_epsilon as 'measurement-error correlation' is an interpretive leap relative to the model definition. In Section 2.1, the nugget is explicitly defined as 'measurement noise and small-scale variability,' so a well-identified rho_epsilon can absorb spatially white physical covariability between temperature and salinity at sub-window scales. In that case, the observed reduction in the latent dependence parameter rho is a scale-decomposition artifact rather than a correction for sensor error. The paper needs either external evidence tying rho_epsilon to Argo sensor error characteristics or a model that separates the small-scale process from measurement noise before claiming that traditional models overestimate the underlying field correlation.
  3. [Section 5.3 and Figure 7] The claim that ignoring measurement-error correlation 'systematically overstates between-field dependence' is contradicted by the paper's own results at other depths. At 1000 dbar, most grid cells display a positive difference (correlated minus independent), and at 10 dbar the equatorial region reverses, with correlations up to 0.15 higher under the correlated model. The abstract and conclusions should be qualified to depth-specific or region-specific statements, or the global claim should be supported by a summary statistic that accounts for the heterogeneity shown in Figure 7.
  4. [Section 5.3 and Figures 5-8] The reported median correlations and parameter maps are point estimates from many moving-window optimizations, with no standard errors, confidence intervals, or sensitivity analysis. Without uncertainty quantification, the apparent drop in latent correlation (0.95 to 0.87 at 300 dbar) cannot be distinguished from estimation noise, and the abstract's claim of 'significantly improves' is not statistically supported. At minimum, the paper should report uncertainty for the window-level parameter estimates or provide a bootstrap/ensemble analysis.
minor comments (4)
  1. [Section 3.2, Eq. (11)] The constant term in the log-likelihood is written as '-N1 log(2π)', but the likelihood is over all N observations; this should presumably be -(N/2)log(2π) or the notation for N1 should be clarified.
  2. [Section 3.1 and Appendix B.1] The correlated nugget precision in Eq. (17) is written for N1 = N2 = n with observations at the same locations, while the model statement in Eq. (10) allows different locations for the two fields. The paper should clarify how Q_epsilon is defined when the two variables are not co-located or only partially co-located within a window.
  3. [References] References [1] and [2] contain duplicated page ranges (e.g., 'pp. 283–312, pp. 283–312'); the bibliography should be cleaned.
  4. [Appendix C.4 and Figure 2] The boxplots display only selected values of the true latent correlation (rho = 0, 0.05, 0.2, 0.7); either all seven simulated configurations should be shown or the caption should state why the omitted ones are excluded.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the drop in latent correlation is an estimated outcome of fresh model fits, not an input or a reparameterization that forces the conclusion.

full rationale

The paper's central claim is that adding a correlated nugget lowers the estimated latent temperature–salinity correlation in Argo data. This claim is not equivalent to any input by construction. The latent correlation parameter rho and the nugget correlation parameter rho_epsilon are both free parameters estimated from Argo residuals by maximum likelihood or stochastic gradient; the model permits rho_epsilon = 0 but estimates large positive values, and the resulting drop in rho, and in the Proposition 2.1 Pearson correlation, is a fitted outcome rather than an imposed constraint. Proposition 2.1 is a mathematical derivation that converts fitted parameters into a Pearson correlation, not a redefinition of the result. The simulation study generates data from the same bivariate SPDE family and shows that a diagonal-nugget fit is biased when the true nugget is correlated; this is a standard self-consistency check with known truth, and the paper supplements it with leave-one-out cross-validation on real Argo data, which is genuinely out-of-sample. Self-citations to Bolin and Wallin (2018, 2023) and to the ngme2 package supply the SPDE machinery, Gibbs sampler, and scoring-rule estimators; those prior results are published, code-reproduced, and do not themselves contain the Argo finding, so the citations are not load-bearing in a circular way. The remaining concern, that the correlated nugget may not be separately identifiable from the latent cross-covariance or may absorb real small-scale physical variability rather than measurement error, is a modeling and identifiability risk rather than a circularity of the derivation chain.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The free parameters listed are the estimated ingredients of the model; the central comparison is between independent and correlated nugget fits. The main extra assumptions are the SPDE representation, local stationarity, temporal independence of January replicates, and identifiability of the nugget from latent correlation. No new physical entities or mediators are introduced.

free parameters (10)
  • alpha1=alpha2 smoothness = 2 (fixed)
    Smoothness is fixed to 2 for both fields in Section 2.4; the paper does not check sensitivity of the correlation conclusions to this choice.
  • rho latent dependence parameter
    Estimated per moving-window grid; the headline result is that this parameter drops when a correlated nugget is added.
  • rho_epsilon nugget correlation
    New parameter, estimated per grid; often above 0.7 in the Argo application and central to the paper's conclusions.
  • sigma_epsilon_1, sigma_epsilon_2
    Nugget standard deviations for temperature and salinity, estimated per grid.
  • kappa_1, kappa_2
    Spatial decay parameters for the two latent fields, estimated per grid.
  • sigma_1, sigma_2
    Marginal standard deviations of the latent fields, estimated per grid.
  • eta_1, eta_2
    NIG tail and shape parameters for the non-Gaussian models, estimated per grid.
  • mu_1, mu_2
    NIG skewness parameters, estimated per grid, with gamma set to -mu to enforce zero mean.
  • beta regression coefficients
    Local mean-field coefficients in equation (16), including K=6 harmonics, estimated per window.
  • window margin and grid dimensions = 10x10 grid; 5-degree margin
    Analysis choices from Kuusela and Stein, chosen by hand; they affect which data enter each local fit.
assumptions (7)
  • standard math Gaussian Matérn fields solve the SPDE in equation (3) with the stated normalization.
    Whittle representation used throughout Section 2.2 as the basis for the multivariate SPDE model.
  • standard math Finite-element Galerkin solution on a bounded domain with Neumann boundary conditions approximates the R^d SPDE solution.
    Invoked in Section 2.4 to turn the SPDE into a Gaussian Markov random field with sparse precision.
  • standard math NIG and generalized asymmetric Laplace noises are the convolution-closed subclasses needed for SPDE models.
    Relied on in Section 2.3, citing Podgorski and Wallin, to justify using NIG noise.
  • domain assumption Within each 20 by 20 degree moving window, the bivariate field is stationary and the latent parameters are constant.
    Used in Section 5.2 to fit independent local models and merge them into global maps.
  • domain assumption January data from different years are treated as independent replicates after local polynomial mean removal.
    Stated in Section 5.2; temporal autocorrelation is not modeled, which could affect inference.
  • domain assumption The correlated nugget and the smooth latent cross-correlation are identifiable from the bivariate observations.
    The central interpretation in Section 5.3 depends on this separation, but no identifiability analysis is provided.
  • standard math Stochastic approximation converges under the chosen weight sequence.
    Used in Section 3.2 for non-Gaussian maximum likelihood estimation, citing Andrieu, Moulines, and Priouret.

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Cite this review

Pith. "Pith review of Incorporating Correlated Nugget Effects in Multivariate Spatial Models: An Application to Argo Ocean Data." pith.science (2026). https://pith.science/paper/3CTDK3G7

@misc{pith2026250603042,
  author       = {Pith},
  title        = {Pith review of: Incorporating Correlated Nugget Effects in Multivariate Spatial Models: An Application to Argo Ocean Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CTDK3G7}},
  note         = {Machine review of arXiv:2506.03042}
}
read the original abstract

Accurate analysis of global oceanographic data, such as temperature and salinity profiles from the Argo program, requires geostatistical models capable of capturing complex spatial dependencies. This study introduces Gaussian and non-Gaussian hierarchical multivariate Mat\'ern-SPDE models with correlated nugget effects to account for small-scale variability and measurement error correlations. Using simulations and Argo data, we demonstrate that incorporating correlated nugget effects significantly improves the accuracy of parameter estimation and spatial prediction in both Gaussian and non-Gaussian multivariate spatial processes. When applied to global ocean temperature and salinity data, our model yields lower correlation estimates between fields compared to models that assume independent noise. This suggests that traditional models may overestimate the underlying field correlation. By separating these effects, our approach captures fine-scale oceanic patterns more effectively. These findings show the importance of relaxing the assumption of independent measurement errors in multivariate hierarchical models.

Figures

Figures reproduced from arXiv: 2506.03042 by the authors.

Figure 1
Figure 1. Argo temperature and salinity data at 300 dbar for January 2018. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Simulation results. (a) We compare the two Gaussian bivariate SPDE models based on the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Schematic representation of the moving-window technique applied in bivariate SPDE model [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: Grid used for model fitting for 300 dbar. All grid boxes have approximately equal surface areas. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Results for model fits at 300 dbar pressure level. The models in the second column reveal the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: 𝜌 parameter estimates at 300 dbar. which are standardized by their variances, and compare these against the theoretical quantiles of a standard normal distribution. However, this approach cannot easily be extended to the non-Gaussian models, since we do not necessarily…
Figure 7
Figure 7. Figure 7: Correlation, 𝜌𝑢1,𝑢2 , between two latent fields at (a) 10 dbar and (b) 1000 dbar. The third column in each panel shows the difference (correlated – independent). advantages of coupling correlation with a more flexible marginal distribution and provide significant advan…
Figure 8
Figure 8. Figure 8: 𝜂 parameter estimates at 300 dbar. 𝜂1 corresponds to Temperature and 𝜂2 to Salinity. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Aggregated QQ plots for 300 dbar with 20 simulations for each model. The solid colored line is [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Best models for each grid for 300 dbar, according to (a) RMSE and (b) SCRPS. [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Measurement noise correlation, 𝜌𝜀. Figure (b) is the same as [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: 𝜌 24 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]
Figure 13
Figure 13. Figure 13: 𝜅1 25 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: 𝜅2 26 [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: 𝜎1 27 [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: 𝜎2 28 [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: 𝜂 29 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: 𝜇 30 [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]
Figure 19
Figure 19. Figure 19: 10 dbar 31 [PITH_FULL_IMAGE:figures/full_fig_p031_19.png]
Figure 20
Figure 20. Figure 20: 300 dbar 32 [PITH_FULL_IMAGE:figures/full_fig_p032_20.png]
Figure 21
Figure 21. Figure 21: 1000 dbar 33 [PITH_FULL_IMAGE:figures/full_fig_p033_21.png]
Figure 22
Figure 22. Figure 22: Aggregated QQ plot results for all models: qsample-qtheory vs. qtheory for LOOO cross [PITH_FULL_IMAGE:figures/full_fig_p034_22.png]
Figure 23
Figure 23. Figure 23: Stratified QQ plot by latitude corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p035_23.png]
Figure 24
Figure 24. Figure 24: Stratified QQ plot by latitude corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p035_24.png]
Figure 25
Figure 25. Figure 25: Stratified QQ plot by latitude corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p036_25.png]
Figure 26
Figure 26. Figure 26: Stratified QQ plot by latitude corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p036_26.png]
Figure 27
Figure 27. Figure 27: Stratified QQ plot by latitude corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p037_27.png]
Figure 28
Figure 28. Figure 28: Stratified QQ plot by latitude corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p037_28.png]
Figure 29
Figure 29. Figure 29: Boxplots of 𝜎1 and 𝜎2 parameters across different values of 𝜌 and 𝜌𝜖 . Each boxplot represents the variation in parameter estimation. ρ = 0.2 ρ = 0.7 ρ = 0 ρ = 0.05 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.8 −0.4 −0.1 0…
Figure 30
Figure 30. Figure 30: Signal-to-Noise Ratios for 𝜎1 and 𝜎2. Each subplot shows the estimated ratio under different simulation settings. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_30.png]
Figure 31
Figure 31. Figure 31: Boxplots of 𝜅1 and 𝜅2 parameters across different values of 𝜌 and 𝜌𝜖 . ρ = 0.2 ρ = 0.7 ρ = 0 ρ = 0.05 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.8 −0.4 −0.1 0 0.1 0.4 0.8 −0.5 0.0 0.5 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 0.0 0.5 …
Figure 32
Figure 32. Figure 32: Boxplots for estimated values of 𝜌 and Pearson correlation parameter across different values of 𝜌 and 𝜌𝜖 . 39 [PITH_FULL_IMAGE:figures/full_fig_p039_32.png]

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Works this paper leans on

48 extracted references · 26 canonical work pages

  1. [1]

    Stability of Stochastic Approximation under Verifiable Condi- tions

    Christophe Andrieu, Éric Moulines, and Pierre Priouret. “Stability of Stochastic Approximation under Verifiable Condi- tions”.SIAM Journal on Control and Optimization44 pp. 283–312, pp. 283–312.doi:10.1137/S0363012902417267

    work page doi:10.1137/s0363012902417267

  2. [3]

    Argo data management Team.Argo user’s manual. Report. 2022.doi:10.13155/29825

    work page doi:10.13155/29825 2022

  3. [4]

    Linear Mixed Effects Models for Non-Gaussian Continuous Repeated Measurement Data

    Özgür Asar, David Bolin, Peter J. Diggle, and Jonas Wallin. “Linear Mixed Effects Models for Non-Gaussian Continuous Repeated Measurement Data”.Journal of the Royal Statistical Society Series C: Applied Statistics69 pp. 1015–1065, pp. 1015–1065.doi:10.1111/rssc.12405

    work page doi:10.1111/rssc.12405

  4. [5]

    PressuresensordriftsinArgoandtheirimpacts

    PaulM.Barker,JeffR.Dunn,CatiaM.Domingues,andSusanE.Wijffels.“PressuresensordriftsinArgoandtheirimpacts”. Journal of Atmospheric and Oceanic Technology28 pp. 1036–1049, pp. 1036–1049.doi:10.1175/2011JTECHO831.1

    work page doi:10.1175/2011jtecho831.1

  5. [6]

    Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling

    Ole E. Barndorff-Nielsen. “Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling”.Scandinavian Journal of Statistics24 pp. 1–13, pp. 1–13

    work page

  6. [7]

    Introduction to optimal interpolation and variational analysis

    Alexander Barth, Aida Alvera Azcárate, Pascal Joassin, Jean-Marie Beckers, and Charles Troupin. “Introduction to optimal interpolation and variational analysis”.SESAME Summer School, Varna, Bulgaria

    work page

  7. [8]

    Spatial Matérn fields driven by non-Gaussian noise

    David Bolin. “Spatial Matérn fields driven by non-Gaussian noise”.Scandinavian Journal of Statistics41 pp. 557–579, pp. 557–579.doi:10.1111/sjos.12046

    doi:10.1111/sjos.12046

  8. [9]

    R package version 0.6.0, https://github.com/davidbolin/ngme2

    David Bolin, Xiaotian Jin, Alexandre Simas, and Jonas Wallin.ngme2: Latent Mixed Effects Models with Flexible Distributions. R package version 0.6.0, https://github.com/davidbolin/ngme2. 2024

    work page 2024

Show all 48 references
  1. [10]

    TheRationalSPDEApproachforGaussianRandomFieldsWithGeneralSmoothness

    DavidBolinandKristinKirchner.“TheRationalSPDEApproachforGaussianRandomFieldsWithGeneralSmoothness”. Journal of Computational and Graphical Statistics29 pp. 274–285, pp. 274–285.doi:10.1080/10618600.2019. 1665537

    2019

  2. [11]

    Covariance–Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference

    David Bolin, Alexandre B. Simas, and Zhen Xiong. “Covariance–Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference”.Journal of Computational and Graphical Statistics33 pp. 64–74, pp. 64–74.doi:10.1080/10618600.2023.2231051

  3. [12]

    Multivariate type G Matérn stochastic partial differential equation random fields

    David Bolin and Jonas Wallin. “Multivariate type G Matérn stochastic partial differential equation random fields”. Journal of the Royal Statistical Society. Series B: Statistical Methodology82.doi:10.1111/rssb.12351

    doi

  4. [13]

    Local scale invariance and robustness of proper scoring rules

    David Bolin and Jonas Wallin. “Local scale invariance and robustness of proper scoring rules”.Statistical Science38 pp. 140–159, pp. 140–159.doi:10.1214/22-STS864

    doi

  5. [14]

    A technique for objective analysis and design of oceanographic experiments applied to MODE-73

    Francis P. Bretherton, Russ E. Davis, and C.B. Fandry. “A technique for objective analysis and design of oceanographic experiments applied to MODE-73”.Deep Sea Research and Oceanographic Abstracts23 pp. 559–582, pp. 559–582.doi: 10.1016/0011-7471(76)90001-2

    doi

  6. [15]

    An assessment of oceanic variability for 1960–2010 from the GFDL ensemble coupled data assimilation

    You-Soon Chang, Shaoqing Zhang, Anthony Rosati, Thomas L. Delworth, and William F. Stern. “An assessment of oceanic variability for 1960–2010 from the GFDL ensemble coupled data assimilation”.Climate Dynamics40 pp. 775–803, pp. 775–803

    1960

  7. [16]

    Wikle.Statistics for Spatio-Temporal Data

    Noel Cressie and Christopher K. Wikle.Statistics for Spatio-Temporal Data. New York, NY: John Wiley & Sons, 2011

    2011

  8. [17]

    Maximum Likelihood from Incomplete Data Via the EM Algorithm

    A. P. Dempster, N. M. Laird, and D. B. Rubin. “Maximum Likelihood from Incomplete Data Via the EM Algorithm”. Journal of the Royal Statistical Society: Series B (Methodological)39 pp. 1–22, pp. 1–22.doi:https://doi.org/10. 1111/j.2517-6161.1977.tb01600.x

  9. [18]

    Ocean Salinities Reveal Strong Global Water Cycle Intensification During 1950 to 2000

    Paul J. Durack, Susan E. Wijffels, and Richard J. Matear. “Ocean Salinities Reveal Strong Global Water Cycle Intensification During 1950 to 2000”.Science336 pp. 455–458, pp. 455–458.doi:10.1126/science.1212222

    1950 doi

  10. [19]

    Does non-stationary spatial data always require non-stationary random fields?

    Geir-Arne Fuglstad, Daniel Simpson, Finn Lindgren, and Håvard Rue. “Does non-stationary spatial data always require non-stationary random fields?”Spatial Statistics14 pp. 505–531, pp. 505–531.doi:https://doi.org/10.1016/j. spasta.2015.10.001

    doi

  11. [20]

    Quality Control of Large Argo Datasets

    F. Gaillard et al. “Quality Control of Large Argo Datasets”.Journal of Atmospheric and Oceanic Technology26 pp. 337–351, pp. 337–351.doi:10.1175/2008JTECHO552.1

    doi

  12. [21]

    In-situ based reanalysis of the global ocean temperature and salinity with ISAS: variability of the heat content and steric height

    Fabienne Gaillard, Thierry Reynaud, Virginie Thierry, Nicolas Kolodziejczyk, and Karina von schuckmann. “In-situ based reanalysis of the global ocean temperature and salinity with ISAS: variability of the heat content and steric height”. Journal Of Climate29 pp. 1305–1323, pp....

    doi

  13. [22]

    Spatial statistics and Gaussian processes: A beautiful marriage

    Alan E. Gelfand and Erin M. Schliep. “Spatial statistics and Gaussian processes: A beautiful marriage”.Spatial Statistics 18 pp. 86–104, pp. 86–104.doi:10.1016/j.spasta.2016.03.006

    2016 doi

  14. [23]

    Estimating oxygen in the Southern Ocean using Argo temperature and salinity

    D. Giglio, V. Lyubchich, and M. R. Mazloff. “Estimating oxygen in the Southern Ocean using Argo temperature and salinity”.Journal of Geophysical Research: Oceans123 pp. 4280–4297, pp. 4280–4297.doi:10.1029/2017jc013404

    doi

  15. [24]

    MatérnCross-CovarianceFunctionsforMultivariateRandom Fields

    TilmannGneiting,WilliamKleiber,andMartinSchlather.“MatérnCross-CovarianceFunctionsforMultivariateRandom Fields”.Journal of the American Statistical Association105.doi:10.1198/jasa.2010.tm09420. 40

    2010 doi

  16. [25]

    Strictly proper scoring rules, prediction, and estimation

    Tilmann Gneiting and Adrian E. Raftery. “Strictly proper scoring rules, prediction, and estimation”.Journal of the American Statistical Association102 pp. 359–378, pp. 359–378.doi:10.1198/016214506000001437

    doi

  17. [26]

    Lognormal and Moving Window Methods of Estimating Acid Deposition

    Timothy C. Haas. “Lognormal and Moving Window Methods of Estimating Acid Deposition”.Journal of the American Statistical Association85 pp. 950–963, pp. 950–963.doi:10.1080/01621459.1990.10474966

    1990

  18. [27]

    A monthly mean dataset of global oceanic temperature and salinity derived from Argo float observations

    Shigeki Hosoda, Toshiaki Ohira, and Takuya Nakamura. “A monthly mean dataset of global oceanic temperature and salinity derived from Argo float observations”.JAMSTEC Report of Research and Development8 pp. 47–59, pp. 47–59. doi:10.5918/jamstecr.8.47

    doi

  19. [28]

    Spatial modeling with system of stochastic partial differential equations

    Xiangping Hu and Ingelin Steinsland. “Spatial modeling with system of stochastic partial differential equations”.WIREs Computational Statistics8 pp. 112–125, pp. 112–125.doi:https://doi.org/10.1002/wics.1378

    doi

  20. [29]

    The Argo Program: Present and Future

    Steven Jayne et al. “The Argo Program: Present and Future”.Oceanography30 pp. 18–28, pp. 18–28.doi:10.5670/ oceanog.2017.213

    2017

  21. [30]

    Argo-TwoDecades:GlobalOceanography,Revolutionized

    GregoryC.Johnsonetal.“Argo-TwoDecades:GlobalOceanography,Revolutionized”.AnnualReviewofMarineScience 14 pp. 379–403, pp. 379–403

  22. [31]

    Supplement to

    Mikael Kuusela and Michael L Stein. “Supplement to "Locally stationary spatio-temporal interpolation of Argo profiling float data"”.doi:10.17882/42182

    doi

  23. [32]

    Locally stationary spatio-temporal interpolation of Argo profiling float data

    Mikael Kuusela and Michael L. Stein. “Locally stationary spatio-temporal interpolation of Argo profiling float data”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences474 p. 20180400, p. 20180400. doi:10.1098/rspa.2018.0400

  24. [33]

    R package version 0.3.0.9012

    Finn Lindgren.fmesher: Triangle Meshes and Related Geometry Tools. R package version 0.3.0.9012. 2025

    2025

  25. [34]

    AnexplicitlinkbetweenGaussianfieldsandGaussianMarkovrandom fields: the stochastic partial differential equation approach

    FinnLindgren,HåvardRue,andJohanLindström.“AnexplicitlinkbetweenGaussianfieldsandGaussianMarkovrandom fields: the stochastic partial differential equation approach”.Journal of the Royal Statistical Society: Series B (Statistical Methodology)73 pp. 423–498, pp. 423–498.doi:https...

  26. [35]

    “Salty Drift

    Chao Liu, Xinfeng Liang, Rui M. Ponte, and Don P. Chambers. ““Salty Drift” of Argo Floats Affects the Gridded Ocean Salinity Products”.Journal of Geophysical Research: Oceans129 e2023JC020871, e2023JC020871.doi: https://doi.org/10.1029/2023JC020871

    doi

  27. [36]

    Bertil Matérn.Spatial variation : Stochastic models and their application to some problems in forest surveys and other sampling investigations. 1960

    1960

  28. [37]

    Recherchedesimplificationdansunproblèmedecokrigeage

    GMatheron.“Recherchedesimplificationdansunproblèmedecokrigeage”.PublicationN-628,CentredeGéostatistique, Ecole des Mines de Paris, Fontainebleau

  29. [38]

    Spatiotemporal local interpolation of global ocean heat transport using Argo floats: A debiased latent Gaussian process approach

    Beomjo Park, Mikael Kuusela, Donata Giglio, and Alison Gray. “Spatiotemporal local interpolation of global ocean heat transport using Argo floats: A debiased latent Gaussian process approach”.The Annals of Applied Statistics17 pp. 1491–1520, pp. 1491–1520.doi:10.1214/22-AOAS1679

    doi

  30. [39]

    Convolution-invariant subclasses of generalized hyperbolic distributions

    Krzysztof Podgórski and Jonas Wallin. “Convolution-invariant subclasses of generalized hyperbolic distributions”. CommunicationsinStatistics-TheoryandMethods45pp.98–103,pp.98–103.doi: 10.1080/03610926.2013.821489

    2013

  31. [40]

    The 2004–2008 mean and annual cycle of temperature, salinity, and steric height in the global ocean from the Argo Program

    Dean Roemmich and John Gilson. “The 2004–2008 mean and annual cycle of temperature, salinity, and steric height in the global ocean from the Argo Program”.Progress in Oceanography82 pp. 81–100, pp. 81–100.doi: 10.1016/j.pocean.2009.03.004

    2004 doi

  32. [41]

    Fast Sampling of Gaussian Markov Random Fields

    Håvard Rue. “Fast Sampling of Gaussian Markov Random Fields”.Journal of the Royal Statistical Society Series B: Statistical Methodology63 pp. 325–338, pp. 325–338.doi:10.1111/1467-9868.00288

  33. [42]

    Part IV: Ocean Circulation and Water Masses

    Gerold Siedler, Stephen M. Griffies, John Gould, and John A. Church. “Part IV: Ocean Circulation and Water Masses”. In:Ocean Circulation and Climate. 2nd.103. International Geophysics. Academic Press, 2013, p. 255.doi: https://doi.org/10.1016/B978-0-12-391851-2.09992-X

    2013 doi

  34. [43]

    Chapter 3 - Physical Properties of Seawater

    Lynne D. Talley, George L. Pickard, William J. Emery, and James H. Swift. “Chapter 3 - Physical Properties of Seawater”. In:Descriptive Physical Oceanography. 6th. Boston, MA: Academic Press, 2011, pp. 29–65.doi:https: //doi.org/10.1016/B978-0-7506-4552-2.10003-4

    2011 doi

  35. [44]

    The MathWorks.MATLAB version: 9.12.0 (R2022a)

    Inc. The MathWorks.MATLAB version: 9.12.0 (R2022a). Natick, Massachusetts, United States: The MathWorks Inc., 2022

    2022

  36. [45]

    Geostatistical Modelling Using Non-Gaussian Matérn Fields

    Jonas Wallin and David Bolin. “Geostatistical Modelling Using Non-Gaussian Matérn Fields”.Scandinavian Journal of Statistics42 pp. 872–890, pp. 872–890.doi:https://doi.org/10.1111/sjos.12141

    doi

  37. [46]

    Stochastic processes in several dimensions

    Peter Whittle. “Stochastic processes in several dimensions”.Bull. Internat. Statist. Inst.pp. 974–994, pp. 974–994

  38. [47]

    Argo salinity: bias and uncertainty evaluation

    Annie P.S. Wong, John Gilson, and Cécile Cabanes. “Argo salinity: bias and uncertainty evaluation”.Earth System Science Data15 pp. 383–393, pp. 383–393.doi:10.5194/essd-15-383-2023

    2023 doi

  39. [48]

    Argo Data 1999–2019: Two Million Temperature-Salinity Profiles and Subsurface Velocity Observations From a Global Array of Profiling Floats

    Annie P.S. Wong et al. “Argo Data 1999–2019: Two Million Temperature-Salinity Profiles and Subsurface Velocity Observations From a Global Array of Profiling Floats”.Frontiers in Marine Science7.doi:10.3389/fmars.2020. 00700

    1999 doi

  40. [49]

    A functional-data approach to the Argo data

    Drew Yarger, Stilian Stoev, and Tailen Hsing. “A functional-data approach to the Argo data”.The Annals of Applied Statistics16 pp. 216–246, pp. 216–246.doi:10.1214/21-AOAS1477. 41

    doi

Pith tools

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