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REVIEW 3 major objections 5 minor 89 references

The paper claims a covariance-boosting Gaussian process can learn nonstationary variance functions from partially whitened residuals and meet 99.9% integrity on storm-time ionospheric data.

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2026-08-01 03:50 UTC pith:VF467V7G

load-bearing objection A genuinely new boosting construction with strong empirical work, but the known-correlation assumption is load-bearing and untested—worth refereeing, not desk-rejecting. the 3 major comments →

arxiv 2607.23018 v1 pith:VF467V7G submitted 2026-07-25 stat.ML cs.LGphysics.space-phstat.ME

Covariance-Boosted Gaussian Processes for Spatiotemporal Irregularities

Jeremy Ovadia This is my paper
classification stat.ML cs.LGphysics.space-phstat.ME
keywords Gaussian processescovariance boostingnonstationary kernelslatent functionsionospheric modelingSBASintegrity boundspartial whitening
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that a Gaussian process can safely adapt to localized irregularities without learning the correlation structure. Its claim: by iteratively boosting the signal and observation variance functions while keeping the correlation kernel fixed, and by inflating posterior uncertainties after the fit, a GP can become nonstationary and still meet strict 99.9% coverage (three-nines) integrity standards. Evidence includes simulated disturbances, standard benchmark datasets, and roughly 28 million ionospheric measurements over South America during four geomagnetic storms. If correct, the regional CBGP model would provide more accurate and better-calibrated ionospheric corrections for satellite-based augmentation systems than the currently operating local stationary GP, with lower broadcast uncertainty. The central mechanism is a partial-whitening step that exposes where the current covariance underestimates variation, feeding relative-error estimates back into the covariance.

Core claim

The central claim is that a Gaussian process can be made nonstationary—and still honest about its uncertainty—without learning the correlation kernel. Starting from a stationary 'weak learner' covariance that underestimates signal and observation variance, CBGP partially whitens the observations with the current covariance, fits two auxiliary GPs to the whitened residuals (one in the Gaussian domain and one in a chi-squared domain), and converts any excess variance into relative errors that multiplicatively boost the latent variance functions. After convergence, the variance functions are capped to prevent overfitting, and posterior uncertainties are inflated using an effective-sample-size s

What carries the argument

Partial whitening with the current covariance estimate: the matrix square root of the inverse covariance is applied to observations, and then auxiliary GP fits on the whitened residuals and their squares guide relative-error estimates for signal and observation variation. These relative errors multiplicatively update the diagonal variance functions in a gradient-descent-like loop, while the correlation matrix is held fixed. Post-fit, an effective sample number inflates posterior variances by a standard-error factor, and smooth thresholding caps the boosted variance functions to prevent overfitting.

Load-bearing premise

The whole procedure assumes the correlation shape—how quickly dependence dies with distance—is known in advance and never updated; if that shape is wrong, boosting the variance levels cannot fix the misfit.

What would settle it

Take a simulated spatial process with a known correlation length that varies by location (violating the fixed-correlation assumption), run CBGP with a nominal correlation matrix, and hold out observations. If more than 0.1% of held-out errors exceed 3.29 times the inflated uncertainty, or if the partially whitened residuals still show spatial autocorrelation after convergence, the central claim is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If CBGP is right, a nonstationary GP can meet strict integrity requirements without estimating correlation length scales, removing the most fragile part of adaptive covariance estimation.
  • Regional full-domain ionospheric modeling becomes viable: one per-epoch fit over all South American measurements is more accurate (roughly 10% lower RMSE) and produces lower broadcast uncertainties than local stationary fits for most grid points.
  • The integrity inflation mechanism—scaling posterior variance by an effective-standard-error factor tied to sampling density—automatically widens uncertainties in undersampled areas, which is exactly what safety-critical systems need.
  • On benchmark problems with heavy data deprivation, CBGP is competitive with or better than existing heteroskedastic and nonstationary GP models while meeting the three-nines criterion that some baselines miss.

Where Pith is reading between the lines

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

  • Editor's inference: The partial-whitening diagnostic could double as a misspecification detector: if the whitened residuals' variance stays far from 1 or retains spatial autocorrelation after convergence, that directly signals that the fixed-correlation assumption is violated and the boosted variances are absorbing the wrong thing.
  • Editor's inference: Because the method assumes signal and observation variation rise together, it is likely to struggle in regimes where noise grows without signal growth—such as polar ionospheric patches, which the paper itself names. A simulation injecting anti-correlated variation would test this boundary.
  • Editor's inference: The effective sample number is computed from a fixed Gaussian kernel rather than estimated from data; in clustered or non-uniform sampling designs this may mis-scale the inflation, and a data-dependent effective sample size would be a natural robustness improvement.
  • Editor's inference: If regional CBGP generalizes, SBAS-style corrections elsewhere could abandon per-grid-point local fitting and use one full-region fit per epoch, but real-time scaling to larger networks would likely require sparse-GP approximations—an extension the paper mentions.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes Covariance-Boosted Gaussian Processes (CBGP), a nonstationary GP framework that fixes the correlation matrix ρ and iteratively boosts only the diagonal signal and observation variance functions. A sub-layer GP is fit to partially whitened observations; the resulting relative error estimates drive a gradient-descent-like update of σ_signal and σ_obs. After convergence, the latent functions are thresholded to avoid overfitting, and posterior variances are inflated using an effective-sample-size-based multiplier. The method is tested on a 1D synthetic problem with an injected irregularity, two benchmark datasets (motorcycle acceleration and Meuse River), and a large ionospheric dataset of roughly 28 million measurements over South America. The paper reports improved out-of-sample RMSE/MAE relative to a stationary GP and claims that all tested CBGP configurations empirically meet a three-nines (99.9%) coverage criterion under inflated uncertainty.

Significance. If the robustness issues are resolved, the paper makes a useful contribution: it offers a computationally feasible, integrity-oriented nonstationary GP construction and evaluates it on an unusually large and demanding real-world spatiotemporal dataset. The detailed algorithm and parameter tables make the method reproducible in principle, and the ionospheric application is relevant to SBAS safety-case research. The strengths are the breadth of out-of-sample testing and the honest enumeration of assumptions and limitations in Section 5. However, the central methodological claims rest on untested assumptions about the correlation structure and on heuristic signal/noise attribution, so the current evidence supports the empirical results for the specific configurations tested, not the general reliability claim.

major comments (3)
  1. [Sec. 3.3, Assumptions 4 and 7; Eqs. (27)–(33), (40)–(43)] The boosting procedure fixes ρ and only updates the diagonal variances (Eq. (17); Algorithm 1). The posterior mean and σ_infl inherit ρ, so Assumption 1 ('true correlation matrix known and well-specified') is load-bearing for every accuracy and coverage claim. The statement at the start of Section 4 that the test cases 'predominantly align' with the assumptions is asserted, not quantified. In the ionospheric experiment, ρ is a homogeneous exponential decay with Lρ = 8,000 km even though the region contains the EIA and storm-time gradients; no experiment varies Lρ or uses a spatially varying ρ. Section 5 concedes that correlation structure 'is usually only speculated or approximated.' A misspecified ρ would feed the residual variance of partially whitened observations into η_R and η_signal through Eqs. (27)–(33) and bias the latent functions and σ_infl. I request a sensitivity analysis in
  2. [Sec. 3.3, Assumptions 4 and 7; Eqs. (27)–(33), (40)–(43)] The split of var(ŷ^(j))−1 between signal and observation error is not identifiable from the partially whitened observations alone; any a^(j)∈[0,1] is consistent with Eq. (28). Assumption 4 postulates that the observation-variance error is a nonnegative multiple of the same deviation, and Assumption 7 selects a through a heuristic ratio involving the GP mean and gradient. The simulation in Section 4.1 only exercises the favorable co-located case (Eq. (48)), and in the real-data benchmarks the true attribution is unknown. The paper's own Section 5 identifies this heuristic estimation as the 'most significant area where improvement can be made.' Please add a controlled experiment with signal-only or observation-only irregularities, or with signal and noise increases that are not co-located, and show that the attribution error does not invalidate either the latent functions or the coverage c
  3. [Sec. 4, three-nines terminology; Eq. (44)] The claim that CBGP models 'meet a three-nines integrity standard' is based on empirical out-of-sample coverage on the tested storm periods and benchmarks (Tables 1, 2, 3, 6). This is a useful diagnostic but not an integrity guarantee: it does not bound the probability of future HMI under unseen conditions. In addition, the inflation constant z_infl used in Eq. (44) is the same 3.29 used to assess coverage, so the pass partly reflects the chosen inflation recipe. I recommend softening the wording (e.g., 'empirically satisfied three-nines coverage on the tested data') and reporting the empirical 99.9th percentile of |error|/σ_infl, which would allow readers to see the margin by which the models pass. This would strengthen the paper because the current tables show all models passing, hiding differences in conservatism.
minor comments (5)
  1. [General] There are several typos: 'perfomance' in the Introduction, 'Nonsationary' in the Algorithm 1 caption, and a duplicated '˜Z(j)χ2 and ˜Z(j)χ2' after Eq. (35). These should be corrected.
  2. [Sec. 2.3, Eq. (16)] Equation (16) is self-referential: χ_d,lowerbound appears on both sides. The minimizer should use an auxiliary variable (e.g., χ) on the right-hand side. As written, the definition is circular.
  3. [Algorithm 1] In the GPRFit call for the χ²-domain model, the manuscript writes H(Ŷ)², while Eq. (34) defines Zχ² = H((ŷ)²). Please clarify whether the square is inside or outside H, and make the algorithm consistent with the text.
  4. [Sec. 4.3 and Fig. 15] The conclusion states that the regional CBGP improves on the locally planar stationary GP 'for nearly every epoch,' but Figure 15 is difficult to parse without a numerical summary. Please report the proportion of epochs with improvement and the distribution of differences.
  5. [Availability] No code or data availability statement is provided. Given the complexity of Algorithm 1 and the scale of the ionospheric dataset, a code release or a detailed reproducibility appendix would substantially strengthen the paper.

Circularity Check

0 steps flagged

No significant circularity: CBGP latent functions are estimated from training data by an explicit iterative update and tested out-of-sample; the known-ρ assumption is an acknowledged limitation, not a definitional reduction.

full rationale

The derivation chain is self-contained in the sense required here. Assumption 1 (known, well-specified correlation ρ), Assumption 4/7 (heuristic attribution a(j) between signal and observation variance) and the softplus clipping are explicit inputs; nothing in Equations (17)-(33) or Algorithm 1 defines a predicted quantity in terms of the benchmark used to test it. The latent functions σ_signal and σ_obs are fitted to the training observations via an iterative method-of-moments style update on partially-whitened residuals, which is standard empirical Bayes rather than a circular reduction. The three-nines claim is an empirical out-of-sample coverage rate: in the ionospheric experiment, Galileo IPPs are deprived while fitting on GPS (and vice versa), so σ_infl is computed before the held-out targets are observed. The same σ_infl is of course produced by the model whose coverage is being measured, but that is true of any GP uncertainty interval and does not make the pass rate an identity. The paper explicitly concedes the main limitation: correlation structure 'is usually only speculated or approximated in practical applications' (Section 5); that is a correctness/robustness risk, not circularity. The only potential self-citation touchpoint—Ovadia et al. (2025) for GNSS TEC retrieval—supports data preparation rather than the central methodological claim, and it is not invoked to rule out alternatives. The synthetic and benchmark experiments are also compared against independent implementations (stationary GP, hetGPy, Gibbs-Paciorek) on held-out data, further indicating the central result does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

10 free parameters · 8 axioms · 0 invented entities

The method has no new physical entities, but it rests on seven explicit assumptions, several of which are ad hoc. The free-parameter list captures the manually set knobs in Table 7 and Appendix B; the most load-bearing assumptions are known ρ (Assumption 1) and the non-negative partial-whitening covariance structure (Assumption 3).

free parameters (10)
  • Initial signal variation σ_signal,0 = √0.91 (simulated/ionosphere), 1 g (motorcycle), 0.1 (Meuse)
    Weak learner starting point; taken from WAAS parameters or set manually; validity depends on Assumption 6.
  • Initial observation variation σ_obs,0 = 0.3 (simulated/ionosphere), 1 g (motorcycle), 0.1 (Meuse)
    Manual starting value for observation noise; not learned.
  • Irregularity GP length scale L_η = 4,000 km (ionosphere), 16 ms (motorcycle), 1,000 m (Meuse)
    Length scale for ρ_η used in GP fits to partially-whitened observations; chosen by hand.
  • Effective-sample length scale L_eff = 800 km (ionosphere), 4 ms (motorcycle), 200 m (Meuse); ε_eff=0.25
    Controls N_eff via Equation (46); manual and no sensitivity analysis.
  • Initial confidence offset κ0 = -0.1257
    Sets initial κ in Algorithm 1; no formal justification.
  • Learning-rate scale ξ0 = 1 (simulated), 3 (other experiments)
    Scales dynamic learning rate in Equation (47); manual.
  • Threshold z_T and caps σ_s,max, σ_o,max = z_T=8; σ_s,max/σ_o,max = ∞/∞ for motorcycle, 2/2 for Meuse, 8m/1.2m for ionosphere
    Thresholding to prevent overfitting; set per problem by hand.
  • Inflation multiplier z_infl = 3.29 (three-nines tests), 1.96 (motorcycle)
    Multiplier for κ_infl in Equation (44); chosen to hit the target integrity level.
  • Softplus / Φ steepness γ_Ψ, γ_Φ = 4, 4
    Shape parameters of smooth thresholding functions; manual.
  • Convergence tolerance ε = 0.05
    Stops the boosting loop; no convergence guarantee beyond this threshold.
axioms (8)
  • domain assumption Assumption 1: The true correlation matrix ρ is known and well-specified.
    Invoked in Section 3.1 for all covariance approximations; the method never learns ρ.
  • standard math Assumption 2: The kernel function is non-negative, k≥0.
    Used to justify sign of covariance terms in Equations (24)-(26).
  • ad hoc to paper Assumption 3: All elements of cov(ŷ^(j)) are non-negative.
    Enables diagonal approximations in Equations (25)-(26) by discarding off-diagonal covariance terms.
  • ad hoc to paper Assumption 4: Observation variance error is a non-negative multiple of (var(ŷ)-1) times approximate observation variance.
    Equation (27)-(28); this is the heuristic that lets the method split excess variance into observation noise.
  • domain assumption Assumption 5: The GP posterior κ-quantile of Z_χ² approximates var(ŷ).
    Equations (34)-(37); relies on the GP fit being an adequate model of squared partially-whitened residuals.
  • domain assumption Assumption 6: Initial σ_signal and σ_obs are lower bounds on the true values.
    Section 3.3; justifies ever inflating but never deflating the variance functions.
  • ad hoc to paper Assumption 7: The heuristic estimate of a^(j) via Φ is reasonable when signal and observation errors are coincident.
    Equations (40)-(43); central to the signal/observation split and explicitly heuristic.
  • domain assumption Spatiotemporal irregularities are characterized by concurrent heightened signal and observation variation.
    Stated in Section 4 as the reason the test cases 'predominantly align' with the assumptions; limits the method's scope.

pith-pipeline@v1.3.0-alltime-deepseek · 35233 in / 15607 out tokens · 160833 ms · 2026-08-01T03:50:50.389863+00:00 · methodology

0 comments
read the original abstract

Nonstationary Gaussian process (GP) models are powerful tools for capturing input-dependent variability by adapting to observed data. However, with limited sampling and highly parameterized covariance structure, they are often prone to overfitting and overconfident uncertainty estimates, potentially leading to misleading predictions in safety-critical applications. Motivated by ionospheric modeling for satellite-based augmentation systems (SBAS), this paper proposes a Covariance-Boosted Gaussian Process (CBGP) framework centered upon boosting covariance priors to discover nonstationary latent functions for signal and observation variation that capture irregularities in the input domain. An additional layer of GP modeling of "partially-whitened" observations guides latent function relative error estimation that is used to iteratively update weak priors in a gradient descent-like procedure. Following boosting, restrictions are imposed upon prior covariances to prevent overfitting while posterior uncertainties are inflated to prevent model overconfidence. CBGP model efficacy and robustness are demonstrated through out-of-sample testing of both simulated and real-world applications that meet a three-nines integrity standard. The modeling of an extensive ionospheric storm dataset over South America suggests accurate and reliable means to compute SBAS ionospheric corrections in the most challenging space weather environment using regional models that are more informed and responsive than local fitting performed by currently-operating SBAS.

Figures

Figures reproduced from arXiv: 2607.23018 by Jeremy Ovadia.

Figure 1
Figure 1. Figure 1: Illustration of a nonstationary Gaussian process described by latent functions for signal [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Smooth thresholding functions Ψ (left) and Φ (right) guided by threshold parameter values and the identity function. 3.4 Post-Fit Uncertainty Inflation via Effective Standard Error After the prior covariance matrix is boosted using relative error estimates, we develop an approach for posterior uncertainty inflation as an inhomogeneous alternative to Rirreg inflation to meet in￾tegrity standards by inflatin… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic diagram of high-level procedures for Covariance-Boosted Gaussian Process [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: CBGP-estimated latent σsignal (top) and σobs (bottom) functions under nominal (left) and disturbed (right) conditions for the first 250 one-dimensional simulations along with respective true functions described by Equation (48) (green) and stationary estimates used as weak learners (turquoise). The orange curves correspond to the simulation used for Figures 5 and 6. Convergence of the iterative boosting sc… view at source ↗
Figure 5
Figure 5. Figure 5: Convergence and accuracy statistics of the iterative CBGP boosting scheme in Algorithm [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: GPR posteriors and whitened observations for stationary GP ( [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Complementary ECDF curves for the magnitudes of out-of-sample errors ( [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scatter plots comparing stationary GP and CBGP RMSEs across 5,000 one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Error statistics for Silverman’s motorcycle accident dataset experiment for deprivation [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: GP posterior means and uncertainties for computational tests with two different data [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: GP posterior means for five features from the Meuse River dataset. Concentrations for [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Set of Ionospheric Grid Points (IGPs) with GNSS receiver sites used for ionospheric [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Spatial visualizations of ionospheric truth at a single epoch ( [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Complementary ECDF curves for the magnitudes of out-of-sample errors ( [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Distributions of differences of magntiudes of out-of-sample errors from the locally planar [PITH_FULL_IMAGE:figures/full_fig_p035_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Proportions of inflated 99.9% uncertainties, represented by 3 [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗

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Reference graph

Works this paper leans on

89 extracted references · 11 canonical work pages

  1. [1]

    Abdu and I.S

    M.A. Abdu and I.S. Batista and A.J. Carrasco and C.G.M. Brum , abstract =. South Atlantic magnetic anomaly ionization: A review and a new focus on electrodynamic effects in the equatorial ionosphere , journal =. 2005 , note =. doi:https://doi.org/10.1016/j.jastp.2005.01.014 , url =

    work page doi:10.1016/j.jastp.2005.01.014 2005

  2. [2]

    and Torres-Arredondo, M

    Alvarez-Montoya, J. and Torres-Arredondo, M. and Sierra-P. Gaussian Process Modeling for Damage Detection in Composite Aerospace Structures by Using Discrete Strain Measurements , booktitle =. e-Journal of Nondestructive Testing , url =. 2019 , address =

    2019

  3. [3]

    Nelson and Jeremy Staum , journal =

    Bruce Ankenman and Barry L. Nelson and Jeremy Staum , journal =. Stochastic Kriging for Simulation Metamodeling , urldate =

  4. [4]

    2018 , issn =

    A brief review of equatorial ionization anomaly and ionospheric irregularities , journal =. 2018 , issn =. doi:10.26464/epp2018025 , url =

    work page doi:10.26464/epp2018025 2018

  5. [5]

    Advances in Neural Information Processing Systems , volume =

    BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization , author =. Advances in Neural Information Processing Systems , volume =. 2020 , url =

    2020

  6. [6]

    Technometrics , volume=

    Diagnostics for Gaussian Process Emulators , author=. Technometrics , volume=. 2009 , publisher=

    2009

  7. [7]

    Gramacy and Mike Ludkovski , title =

    Mickaël Binois and Robert B. Gramacy and Mike Ludkovski , title =. Journal of Computational and Graphical Statistics , volume =. 2018 , publisher =. doi:10.1080/10618600.2018.1458625 , URL =

    arXiv 2018

  8. [8]

    Journal of Statistical Software , volume =

    hetGP: Heteroskedastic Gaussian Process Modeling and Sequential Design in R , author =. Journal of Statistical Software , volume =. 2021 , doi =

    2021

  9. [9]

    NAVIGATION , volume =

    Blanch, Juan and Walter, Todd and Enge, Per , title =. NAVIGATION , volume =. doi:https://doi.org/10.1002/j.2161-4296.2002.tb00259.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.2161-4296.2002.tb00259.x , abstract =

    arXiv 2002

  10. [10]

    arXiv e-prints , keywords =

    Nonstationary Gaussian Process Surrogates. arXiv e-prints , keywords =. doi:10.48550/arXiv.2305.19242 , archivePrefix =. 2305.19242 , primaryClass =

    Pith/arXiv arXiv doi:10.48550/arxiv.2305.19242

  11. [11]

    Cawley and Nicola L

    Gavin C. Cawley and Nicola L. C. Talbot , title =. Journal of Machine Learning Research , year =

  12. [12]

    NeuroImage , volume=

    Individualized Gaussian process-based prediction and detection of local and global gray matter abnormalities in elderly subjects , author=. NeuroImage , volume=. 2014 , publisher=

    2014

  13. [13]

    , booktitle =

    Damianou, Andreas and Lawrence, Neil D. , booktitle =. Deep. 2013 , editor =

    2013

  14. [14]

    Thin and deep Gaussian processes , url =

    de Souza, Daniel Augusto and Nikitin, Alexander and John, ST and Ross, Magnus and \'. Thin and deep Gaussian processes , url =. Advances in Neural Information Processing Systems , editor =

  15. [15]

    and Kass, Robert E

    Dimatteo, Ilaria and Genovese, Christopher R. and Kass, Robert E. , title =. Biometrika , volume =. 2001 , month =. doi:10.1093/biomet/88.4.1055 , url =

    work page doi:10.1093/biomet/88.4.1055 2001

  16. [16]

    Scherer and Sebastian Trimpe , title =

    Christian Fiedler and Carsten W. Scherer and Sebastian Trimpe , title =. Proceedings of the AAAI Conference on Artificial Intelligence , volume =. 2021 , doi =

    2021

  17. [17]

    Gibbs , title =

    Mark N. Gibbs , title =

  18. [18]

    Journal of Machine Learning Research , volume =

    GPflow: A Gaussian Process Library using TensorFlow , author =. Journal of Machine Learning Research , volume =. 2017 , url =

    2017

  19. [19]

    Regression with Input-dependent Noise: A Gaussian Process Treatment , volume =

    Goldberg, Paul and Williams, Christopher and Bishop, Christopher , booktitle =. Regression with Input-dependent Noise: A Gaussian Process Treatment , volume =

  20. [20]

    and Maurin, M

    Goutelle, S. and Maurin, M. and Rougier, F. and Barbaut, X. and Bourguignon, L. and Ducher, M. and Maire, P. , journal=. The. 2008 , publisher=

    2008

  21. [21]

    Gramacy and Herbert K

    Robert B. Gramacy and Herbert K. H. Lee , journal =. Bayesian Treed Gaussian Process Models with an Application to Computer Modeling , urldate =

  22. [22]

    Proceedings of the 19th International Conference on Artificial Intelligence and Statistics , pages =

    Non-Stationary Gaussian Process Regression with Hamiltonian Monte Carlo , author =. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics , pages =. 2016 , editor =

    2016

  23. [23]

    and Riddell, A

    Johnston, G. and Riddell, A. and Hausler, G. , title =. Springer Handbook of Global Navigation Satellite Systems , year =. doi:10.1007/978-3-319-42928-1_33 , url =

    work page doi:10.1007/978-3-319-42928-1_33

  24. [24]

    2016 , howpublished =

    Kaggle , title =. 2016 , howpublished =

    2016

  25. [25]

    Oates , title =

    Toni Karvonen and Chris J. Oates , title =. Journal of Machine Learning Research , year =

  26. [26]

    Proceedings of the 24th International Conference on Machine Learning , pages =

    Kersting, Kristian and Plagemann, Christian and Pfaff, Patrick and Burgard, Wolfram , title =. Proceedings of the 24th International Conference on Machine Learning , pages =. 2007 , isbn =. doi:10.1145/1273496.1273546 , abstract =

    arXiv 2007

  27. [27]

    C Holmes , title =

    Hyoung-Moon Kim and Bani K Mallick and C. C Holmes , title =. Journal of the American Statistical Association , volume =. 2005 , publisher =. doi:10.1198/016214504000002014 , URL =

    work page doi:10.1198/016214504000002014 2005

  28. [28]

    1965 , publisher=

    Survey Sampling , author=. 1965 , publisher=

    1965

  29. [29]

    The Journal of Portfolio Management , volume=

    Big Data in Real Estate? From Manual Appraisal to Automated Valuation , author=. The Journal of Portfolio Management , volume=. 2017 , publisher=

    2017

  30. [30]

    Journal of Property Research , volume =

    Bastian Krämer and Moritz Stang and Vanja Doskoč and Wolfgang Schäfers and Tobias Friedrich , title =. Journal of Property Research , volume =. 2023 , publisher =. doi:10.1080/09599916.2023.2206823 , URL =

    arXiv 2023

  31. [31]

    Journal of Machine Learning Research , volume=

    Learning the kernel matrix with semidefinite programming , author=. Journal of Machine Learning Research , volume=. 2004 , url=

    2004

  32. [32]

    and Smola, Alex J

    Le, Quoc V. and Smola, Alex J. and Canu, St\'. Heteroscedastic Gaussian process regression , year =. Proceedings of the 22nd International Conference on Machine Learning , pages =. doi:10.1145/1102351.1102413 , abstract =

    arXiv

  33. [33]

    Advances in Neural Information Processing Systems 32 , editor =

    Armin Lederer and Jonas Umlauft and Sandra Hirche , title =. Advances in Neural Information Processing Systems 32 , editor =. 2019 , publisher =

    2019

  34. [34]

    Lejeune and M

    R. Lejeune and M. B. El-Arini and P. Doherty and J. Klobuchar and E. De Paula and F. Rodrigues and A. Canavitsas , title =. Proceedings of the 16th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS/GNSS 2003) , year =

    2003

  35. [35]

    When Gaussian Process Meets Big Data: A Review of Scalable GPs , year=

    Liu, Haitao and Ong, Yew-Soon and Shen, Xiaobo and Cai, Jianfei , journal=. When Gaussian Process Meets Big Data: A Review of Scalable GPs , year=

  36. [36]

    An Overview of Gaussian process Regression for Volatility Forecasting , year=

    Liu, Bingqing and Kiskin, Ivan and Roberts, Stephen , booktitle=. An Overview of Gaussian process Regression for Volatility Forecasting , year=

  37. [37]

    Advances in Neural Information Processing Systems (NeurIPS) , year =

    Automatic Construction and Natural-Language Description of Nonparametric Regression Models , author =. Advances in Neural Information Processing Systems (NeurIPS) , year =

  38. [38]

    Proceedings of the 39th International Conference on Machine Learning , series =

    Additive Gaussian Processes Revisited , author =. Proceedings of the 39th International Conference on Machine Learning , series =. 2022 , publisher =

    2022

  39. [39]

    Journal of Geodesy , keywords =

    The JPL-GIM algorithm and products: multi-GNSS high-rate global mapping of total electron content. Journal of Geodesy , keywords =. doi:10.1007/s00190-024-01860-3 , adsurl =

    doi:10.1007/s00190-024-01860-3

  40. [40]

    Krigeage d'un panneau rectangulaire par sa p

    Matheron, Georges , year=. Krigeage d'un panneau rectangulaire par sa p

  41. [41]

    and Stolle, C

    Matzka, J. and Stolle, C. and Yamazaki, Y. and Bronkalla, O. and Morschhauser, A. , title =. Space Weather , volume =. doi:https://doi.org/10.1029/2020SW002641 , url =. https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2020SW002641 , note =

    doi:10.1029/2020sw002641

  42. [42]

    Navigation: Journal of the Institute of Navigation , year=

    Principle of operation of NAVSTAR and system characteristics , author=. Navigation: Journal of the Institute of Navigation , year=

  43. [43]

    and Cawley, Gavin C

    Mohammed, Rekar O. and Cawley, Gavin C. Over-Fitting in Model Selection with Gaussian Process Regression. Machine Learning and Data Mining in Pattern Recognition. 2017. doi:10.1007/978-3-319-62416-7

    doi:10.1007/978-3-319-62416-7 2017

  44. [44]

    and Doerk, Gregory S

    Noack, Marcus M. and Doerk, Gregory S. and Li, Ruipeng and Streit, Jason K. and Vaia, Richard A. and Yager, Kevin G. and Fukuto, Masafumi , title =. Scientific Reports , issn =. 2020 , month =. doi:10.1038/s41598-020-74394-1 , url =

    doi:10.1038/s41598-020-74394-1 2020

  45. [45]

    and Luo, Hengrui and Risser, Mark D

    Noack, Marcus M. and Luo, Hengrui and Risser, Mark D. , title =. APL Machine Learning , volume =. 2024 , month =. doi:10.1063/5.0176963 , url =

    doi:10.1063/5.0176963 2024

  46. [46]

    Positioning and Applications

    A. Olabode and O. Abe and C. Cesaroni and A. Brack and T. Oluwadare and C. Nguyen and H. Schuh , title =. 2nd Symposium of IAG Commission 4 "Positioning and Applications" , year =

  47. [47]

    , title =

    O'Gara, David and Binois, Mickaël and Garnett, Roman and Hammond, Ross A. , title =. 2025 , publisher =. doi:10.21105/joss.07518 , url =

    work page doi:10.21105/joss.07518 2025

  48. [48]

    Adaptive hybrid-kernel function based Gaussian process regression for nonparametric modeling of ship maneuvering motion , journal =

    Zi-Lu Ouyang and Zao-Jian Zou and Lu Zou , keywords =. Adaptive hybrid-kernel function based Gaussian process regression for nonparametric modeling of ship maneuvering motion , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.oceaneng.2022.113373 , url =

    arXiv 2023

  49. [49]

    2025 , doi =

    Ovadia, Jeremy and Champlin, David and Pong,, Corey and Altshuler, Eric , title =. 2025 , doi =. https://navi.ion.org/content/72/4/navi.723.full.pdf , keywords =

    2025

  50. [50]

    Recurring Ionospheric Gradients, Irregularities, and Depletions Over New Zealand , doi =

    Ovadia, Jeremy and Altshuler, Eric and MacCallum, Sean and Reddan, Pat and Collett, Dave , year =. Recurring Ionospheric Gradients, Irregularities, and Depletions Over New Zealand , doi =

  51. [51]

    Nonstationary Covariance Functions for Gaussian Process Regression , url =

    Paciorek, Christopher and Schervish, Mark , booktitle =. Nonstationary Covariance Functions for Gaussian Process Regression , url =

  52. [52]

    2001 , publisher=

    High Availability: Design, Techniques, and Processes , author=. 2001 , publisher=

    2001

  53. [53]

    Nonstationary Gaussian Process Regression Using Point Estimates of Local Smoothness

    Plagemann, Christian and Kersting, Kristian and Burgard, Wolfram. Nonstationary Gaussian Process Regression Using Point Estimates of Local Smoothness. Machine Learning and Knowledge Discovery in Databases. 2008

    2008

  54. [54]

    PeerJ Computer Science , volume =

    PyMC: a modern, and comprehensive probabilistic programming framework , author =. PeerJ Computer Science , volume =. 2023 , doi =

    2023

  55. [55]

    A Unifying View of Sparse Approximate Gaussian Process Regression , journal =

    Joaquin Qui. A Unifying View of Sparse Approximate Gaussian Process Regression , journal =. 2005 , volume =

    2005

  56. [56]

    Infinite Mixtures of Gaussian Process Experts , url =

    Rasmussen, Carl and Ghahramani, Zoubin , booktitle =. Infinite Mixtures of Gaussian Process Experts , url =

  57. [57]

    Rasmussen, Carl Edward and Williams, Christopher K. I. , biburl =

  58. [58]

    Rikken, M. G. J. and Van Rijn, R. P. G. , title =. 1993 , type =

    1993

  59. [59]

    2020 , type =

    Minimum Operational Performance Standards for Global Positioning System/Satellite-Based Augmentation System Airborne Equipment , institution =. 2020 , type =

    2020

  60. [60]

    Proceedings of the 2017 International Technical Meeting of The Institute of Navigation , year =

    Takeyasu Sakai and Mitsunori Kitamura and Takahiro Aso and Kazuaki Hoshinoo , title =. Proceedings of the 2017 International Technical Meeting of The Institute of Navigation , year =

    2017

  61. [61]

    Deisenroth , booktitle =

    Hugh Salimbeni and Marc P. Deisenroth , booktitle =. Deeply Non-Stationary Gaussian Processes , url =

  62. [62]

    A. D. Sarma and D. Venkata Ratnam and D. Krishna Reddy , title =. IET Radar, Sonar & Navigation , volume =. 2009 , doi =

    2009

  63. [63]

    Proceedings of the 19th International Conference on Artificial Intelligence and Statistics , pages =

    Chained Gaussian Processes , author =. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics , pages =. 2016 , editor =

    2016

  64. [64]

    Duncan Shillan and Jonathan A. C. Sterne and Alan Champneys and Ben Gibbison , title =. Critical Care , year =. doi:10.1186/s13054-019-2564-9 , publisher =

    work page doi:10.1186/s13054-019-2564-9

  65. [65]

    European Journal of Operational Research , volume =

    Applications of stochastic modeling in air traffic management: Methods, challenges and opportunities for solving air traffic problems under uncertainty , author =. European Journal of Operational Research , volume =. 2021 , doi =

    2021

  66. [66]

    Journal of Machine Learning Research , year =

    Fabio Sigrist , title =. Journal of Machine Learning Research , year =

  67. [67]

    B. W. Silverman , journal =. Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting , urldate =

  68. [68]

    International Conference on Machine Learning (ICML) , year =

    Input Warping for Bayesian Optimization of Non-Stationary Functions , author =. International Conference on Machine Learning (ICML) , year =

  69. [69]

    Radio Science , volume =

    Sparks, Lawrence and Blanch, Juan and Pandya, Nitin , title =. Radio Science , volume =. doi:https://doi.org/10.1029/2011RS004667 , url =. https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2011RS004667 , abstract =

    work page doi:10.1029/2011rs004667

  70. [70]

    Radio Science , volume =

    Sparks, Lawrence and Blanch, Juan and Pandya, Nitin , title =. Radio Science , volume =. doi:https://doi.org/10.1029/2011RS004781 , url =. https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2011RS004781 , abstract =

    doi:10.1029/2011rs004781

  71. [71]

    2022 , doi =

    Sparks, Lawrence and Altshuler, Eric and Pandya, Nitin and Blanch,, Juan and Walter, Todd , title =. 2022 , doi =. https://navi.ion.org/content/69/1/navi.503.full.pdf , keywords =

    2022

  72. [72]

    and Seeger, Matthias , title =

    Srinivas, Niranjan and Krause, Andreas and Kakade, Sham M. and Seeger, Matthias , title =. Proceedings of the 27th International Conference on Machine Learning , year =

  73. [73]

    Journal of Geomagnetism and Geoelectricity , year = 1990, month = jan, volume =

    Ring current response to impulsive southward IMF: A cause of second development of the Dst index. Journal of Geomagnetism and Geoelectricity , year = 1990, month = jan, volume =. doi:10.5636/jgg.42.1325 , adsurl =

    work page doi:10.5636/jgg.42.1325 1990

  74. [74]

    Journal of Machine Learning Research , year =

    Teemu Härkönen and Sara Wade and Kody Law and Lassi Roininen , title =. Journal of Machine Learning Research , year =

  75. [75]

    H. J. Thiébaux and F. W. Zwiers , journal =. The Interpretation and Estimation of Effective Sample Size , urldate =

  76. [76]

    Spatial mapping with Gaussian processes and nonstationary Fourier features , journal =

    Jean-Francois Ton and Seth Flaxman and Dino Sejdinovic and Samir Bhatt , keywords =. Spatial mapping with Gaussian processes and nonstationary Fourier features , journal =. 2018 , note =. doi:https://doi.org/10.1016/j.spasta.2018.02.002 , url =

    work page doi:10.1016/j.spasta.2018.02.002 2018

  77. [77]

    Tsunoda , title =

    Roland T. Tsunoda , title =. Reviews of Geophysics , year =. doi:10.1029/RG026i004p00719 , publisher =

    work page doi:10.1029/rg026i004p00719

  78. [78]

    Gaussian Process modeling of large scale terrain , year=

    Vasudevan, Shrihari and Ramos, Fabio and Nettleton, Eric and Durrant-Whyte, Hugh and Blair, Allan , booktitle=. Gaussian Process modeling of large scale terrain , year=

  79. [79]

    Smart GNSS Integrity Monitoring for Road Vehicles: An Overview of AI Methods , year=

    Viveiros, Inês and Silva, Hélder and Andrade, Yuri and Pendão, Cristiano , journal=. Smart GNSS Integrity Monitoring for Road Vehicles: An Overview of AI Methods , year=

  80. [80]

    95 , institution =

    WAAS Performance Analysis Report No. 95 , institution =. 2025 , month = dec, url =

    2025

Showing first 80 references.