pith. sign in

arxiv: 2501.11448 · v5 · pith:KRVXN2DRnew · submitted 2025-01-20 · 📊 stat.CO · stat.ML

An accuracy-runtime trade-off comparison of scalable Gaussian process approximations for spatial data

Filippo Rambelli , Fabio Sigrist This is my paper

Pith reviewed 2026-05-23 05:16 UTC · model grok-4.3

classification 📊 stat.CO stat.ML
keywords Gaussian processesVecchia approximationspatial statisticsscalabilityaccuracy-runtime trade-offlikelihood evaluationparameter estimationprediction
0
0 comments X

The pith

Vecchia approximations deliver the best accuracy-runtime trade-off among scalable Gaussian process methods for spatial data.

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

The paper compares multiple approximations to Gaussian processes that aim to reduce the O(N^3) cost of exact inference on large spatial datasets. It measures performance on likelihood evaluation, parameter estimation, and prediction while tracking actual runtime, using both simulated examples and large real-world spatial datasets. The central finding is that Vecchia approximations achieve higher accuracy at given runtimes than competing methods in most tested conditions. This matters for any application where spatial data volume makes full Gaussian processes impractical but statistical reliability still matters.

Core claim

Through direct comparison on simulated and real spatial data, Vecchia approximations to Gaussian processes provide the best accuracy-runtime trade-off for likelihood evaluation, parameter estimation, and prediction tasks across most settings considered.

What carries the argument

Accuracy-runtime trade-off evaluation of Gaussian process approximations, with Vecchia methods as the leading performer on spatial data tasks.

If this is right

  • For large spatial datasets, Vecchia approximations can be used as the default choice when both accuracy and speed are required.
  • Alternative approximations become attractive only in narrow regimes where they match or exceed Vecchia performance on the three evaluation metrics.
  • Runtime measurements must be included alongside accuracy when ranking approximations, because pure accuracy rankings can favor slower methods.
  • The relative ranking of approximations can shift with dataset size and correlation structure, so empirical testing on the target domain remains necessary.

Where Pith is reading between the lines

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

  • The observed superiority may stem from how Vecchia approximations preserve local conditional independence structure that matches the Markov properties common in spatial fields.
  • Similar trade-off comparisons could be run on non-spatial data such as time series or graph-structured domains to test whether Vecchia remains competitive.
  • If Vecchia methods continue to dominate, software libraries for spatial statistics could prioritize their implementation and tuning over other approximation families.

Load-bearing premise

The selected simulated and real-world datasets represent typical spatial data, and the compared implementations allow fair measurement of both accuracy and runtime.

What would settle it

A new spatial dataset of comparable size where a non-Vecchia approximation achieves measurably higher accuracy at the same runtime as the best Vecchia variant.

Figures

Figures reproduced from arXiv: 2501.11448 by Fabio Sigrist, Filippo Rambelli.

Figure 1
Figure 1. Figure 1: Illustration of simulated training and test data with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logarithmic selling price over space (left) and illustration of training and test locations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Canopy height over space (left) and illustration of training and test locations (right) for the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Land surface temperature over space (left) and illustration of training and test locations [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Land surface temperature over space (left) and illustration of training and test locations [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Absolute mean difference between approximate and exact log-likelihood on simulated data [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Bias and MSE of the estimates for the range, marginal variance, and error term variance [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average RMSE, log-score, and KL divergence between exact and approximate predictions [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: RMSE, log-score and CRPS on the “interpolation” and “extrapolation” test sets of the house [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: KL divergence between exact and approximate predictions on the “interpolation” test (left) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: RMSE, log-score, and CRPS on the Laegern data set. [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Test RMSE, log-score, and CRPS on the MODIS 2016 data set. [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Negative log-likelihood on the MODIS 2016 data set. The parameters estimated by the [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Test RMSE, log-score, and CRPS on the MODIS 2023 data set. [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Negative log-likelihood on the MODIS 2023 data set. The parameters estimated by the [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Average negative log-likelihood on simulated data sets with a practical range of 0 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Bias and MSE of the estimates for the range, marginal variance, end error term variance [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Average RMSE and log-score on simulated data sets with a practical range of 0 [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Negative log-likelihood on the house price dataset. The parameters estimated via exact [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Negative log-likelihood on the Laegern data set. The parameters estimated by the Vecchia [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Average negative log-likelihood on simulated data sets with an anisotropic Mat´ern covari [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Bias and MSE of the estimates for the ranges, marginal variance, end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p023_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Average RMSE and log-score on simulated data sets with an anisotropic Mat´ern covariance [PITH_FULL_IMAGE:figures/full_fig_p023_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: RMSE between exact and approximate predictive means and variances on simulated data [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Absolute mean difference between approximate and exact log-likelihood on simulated data [PITH_FULL_IMAGE:figures/full_fig_p038_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Bias and MSE of the estimates for the GP range, GP variance end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p040_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p044_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: RMSE between exact and approximate predictive means and variances on simulated data [PITH_FULL_IMAGE:figures/full_fig_p048_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Absolute mean difference between approximate and exact log-likelihood on simulated data [PITH_FULL_IMAGE:figures/full_fig_p052_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Bias and MSE of the estimates for the GP range, GP variance end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p054_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p058_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: RMSE between exact and approximate predictive means and variances on simulated data [PITH_FULL_IMAGE:figures/full_fig_p062_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Average negative log-likelihood on simulated data sets with an effective range 0 [PITH_FULL_IMAGE:figures/full_fig_p073_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Bias and MSE of the estimates for the GP range, GP variance end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p075_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p079_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Average negative log-likelihood on simulated data sets with an effective range 0 [PITH_FULL_IMAGE:figures/full_fig_p083_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Bias and MSE of the estimates for the GP range, GP variance end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p085_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p089_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: Average negative log-likelihood on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p101_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: Bias and MSE of the estimates for the GP range, GP variance end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p103_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p107_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: Average negative log-likelihood on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p111_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: Bias and MSE of the estimates for the GP range, GP variance end error term variance [PITH_FULL_IMAGE:figures/full_fig_p113_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p117_44.png] view at source ↗
Figure 45
Figure 45. Figure 45: Average negative log-likelihood on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p121_45.png] view at source ↗
Figure 46
Figure 46. Figure 46: Bias and MSE of the estimates for the GP range, GP variance end error term variance [PITH_FULL_IMAGE:figures/full_fig_p123_46.png] view at source ↗
Figure 47
Figure 47. Figure 47: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p127_47.png] view at source ↗
Figure 48
Figure 48. Figure 48: Average negative log-likelihood on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p131_48.png] view at source ↗
Figure 49
Figure 49. Figure 49: Bias and MSE of the estimates for the GP range, GP variance end error term variance on [PITH_FULL_IMAGE:figures/full_fig_p133_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p137_50.png] view at source ↗
Figure 51
Figure 51. Figure 51: RMSE between exact and approximate predictive means and variances on the house price [PITH_FULL_IMAGE:figures/full_fig_p143_51.png] view at source ↗
Figure 52
Figure 52. Figure 52: RMSE, log-score and CRPS on the “interpolation” and “extrapolation” test sets of the [PITH_FULL_IMAGE:figures/full_fig_p146_52.png] view at source ↗
Figure 53
Figure 53. Figure 53: Negative log-likelihood on the house price data set including covariates. The parameters [PITH_FULL_IMAGE:figures/full_fig_p148_53.png] view at source ↗
Figure 54
Figure 54. Figure 54: KL divergence between exact and approximate predictions on the “interpolation” test [PITH_FULL_IMAGE:figures/full_fig_p150_54.png] view at source ↗
Figure 55
Figure 55. Figure 55: RMSE between exact and approximate predictive means and variances on the house price [PITH_FULL_IMAGE:figures/full_fig_p150_55.png] view at source ↗
Figure 56
Figure 56. Figure 56: Average negative log-likelihood on simulated data sets with an effective range 0 [PITH_FULL_IMAGE:figures/full_fig_p160_56.png] view at source ↗
Figure 57
Figure 57. Figure 57: Bias and MSE of the estimates for the GP range, GP variance end error term variance [PITH_FULL_IMAGE:figures/full_fig_p162_57.png] view at source ↗
Figure 58
Figure 58. Figure 58: Average RMSE and log-score on simulated data sets with an effective range of 0 [PITH_FULL_IMAGE:figures/full_fig_p166_58.png] view at source ↗
Figure 59
Figure 59. Figure 59: RMSE, log-score and CRPS on the “interpolation” and “extrapolation” test sets of the house [PITH_FULL_IMAGE:figures/full_fig_p170_59.png] view at source ↗
Figure 60
Figure 60. Figure 60: Negative log-likelihood on the house price data set in the single-core setting. The parameters [PITH_FULL_IMAGE:figures/full_fig_p172_60.png] view at source ↗
Figure 61
Figure 61. Figure 61: KL divergence between exact and approximate predictions on the “interpolation” test (left) [PITH_FULL_IMAGE:figures/full_fig_p174_61.png] view at source ↗
Figure 62
Figure 62. Figure 62: RMSE between exact and approximate predictive means and variances on the house price [PITH_FULL_IMAGE:figures/full_fig_p174_62.png] view at source ↗
read the original abstract

Gaussian processes (GPs) are flexible, probabilistic, nonparametric models widely used in fields such as spatial statistics and machine learning. A drawback of Gaussian processes is their computational cost, with $O(N^3)$ time and $O(N^2)$ memory complexity, which makes them prohibitive for large data sets. Numerous approximation techniques have been proposed to address this limitation. In this work, we systematically compare the accuracy of different Gaussian process approximations with respect to likelihood evaluation, parameter estimation, and prediction, explicitly accounting for the computational time required. We analyze the trade-off between accuracy and runtime on multiple simulated and large-scale real-world data sets and find that Vecchia approximations consistently provide the best accuracy-runtime trade-off across most settings considered.

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.

Referee Report

2 major / 0 minor

Summary. The manuscript conducts an empirical comparison of multiple scalable Gaussian process approximations (including Vecchia, inducing point, and others) for spatial data, evaluating accuracy in likelihood evaluation, parameter estimation, and prediction while accounting for runtime. It concludes that Vecchia approximations provide the best accuracy-runtime trade-off across most of the simulated and real-world datasets considered.

Significance. If the experimental comparisons are shown to be robust, this would offer practical guidance to spatial statisticians and machine learning practitioners facing large-N GP problems, where computational cost is a primary barrier. The work is a benchmarking study with no parameter-free derivations or machine-checked proofs, so its value rests entirely on the quality and representativeness of the empirical results.

major comments (2)
  1. [Abstract] Abstract: the headline claim that Vecchia approximations 'consistently provide the best accuracy-runtime trade-off across most settings considered' rests on the unverified assumptions that the chosen datasets span typical ranges of correlation lengths, observation densities, and non-stationarity, and that all implementations were comparably optimized (language, vectorization, convergence tolerances). No quantitative description of the data-generating processes or code-level controls is supplied, so the ranking cannot be assessed for robustness.
  2. [Methods] Methods/Experimental design (inferred from abstract): without explicit reporting of how runtime was measured (wall-clock on identical hardware, same hyper-parameter optimization protocol, etc.) or how accuracy metrics were normalized across methods, the accuracy-runtime trade-off curves cannot be interpreted as a fair comparison; this directly affects the central empirical ranking.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on the robustness of our empirical study. We address each major comment below and have revised the manuscript to provide greater transparency on the experimental design and data-generating processes.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claim that Vecchia approximations 'consistently provide the best accuracy-runtime trade-off across most settings considered' rests on the unverified assumptions that the chosen datasets span typical ranges of correlation lengths, observation densities, and non-stationarity, and that all implementations were comparably optimized (language, vectorization, convergence tolerances). No quantitative description of the data-generating processes or code-level controls is supplied, so the ranking cannot be assessed for robustness.

    Authors: We agree that the abstract claim benefits from explicit support. The revised manuscript now includes a dedicated paragraph in Section 4.1 quantifying the simulated data-generating processes (correlation lengths ranging from 0.05 to 5 times the spatial domain diameter, observation densities from 10 to 500 points per unit area, and controlled non-stationarity via spatially varying length-scale functions). All methods were coded in R with equivalent vectorization and the same L-BFGS optimizer and convergence tolerances; these implementation controls are now summarized in Section 3.2 with pointers to the public repository. The abstract has been lightly edited to reference these details. revision: yes

  2. Referee: [Methods] Methods/Experimental design (inferred from abstract): without explicit reporting of how runtime was measured (wall-clock on identical hardware, same hyper-parameter optimization protocol, etc.) or how accuracy metrics were normalized across methods, the accuracy-runtime trade-off curves cannot be interpreted as a fair comparison; this directly affects the central empirical ranking.

    Authors: We have expanded the Methods section (now Section 3.3) to state that all runtimes are wall-clock measurements on identical hardware (Intel Xeon E5-2680 v4, 64 GB RAM) under the same hyper-parameter optimization protocol (L-BFGS with maximum 200 iterations and identical gradient tolerances). Accuracy metrics (negative log-likelihood, parameter RMSE, and predictive RMSE) were computed using the same definitions for every method without post-hoc normalization that would favor one approach. These additions directly address interpretability of the trade-off curves. revision: yes

Circularity Check

0 steps flagged

Empirical benchmarking study contains no derivations or load-bearing self-referential steps

full rationale

The paper is a systematic empirical comparison of GP approximations on simulated and real-world spatial datasets, evaluating accuracy-runtime trade-offs for likelihood, estimation, and prediction. No equations, parameter fits, or uniqueness theorems are presented as derivations; the central claim rests directly on observed experimental rankings across the tested settings. No self-citation chains, ansatzes, or renamings of known results appear in the load-bearing positions. This is the standard case of a self-contained benchmarking study whose validity depends on data representativeness and implementation fairness rather than any internal definitional reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an empirical comparison study; no free parameters, axioms, or invented entities are introduced.

pith-pipeline@v0.9.0 · 5646 in / 864 out tokens · 35218 ms · 2026-05-23T05:16:39.222296+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A new framework for non-stationary spatio-temporal data fusion of multi-fidelity models

    stat.CO 2026-05 unverdicted novelty 7.0

    A decomposed multi-fidelity covariance formulation allows Vecchia approximation on latent processes and GLS mean removal to deliver scalable, fully likelihood-based fusion of noisy low-fidelity and accurate high-fidel...

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    18 Q. Pan, S. Abdulah, M. G. Genton, D. E. Keyes, H. Ltaief, and Y. Sun. GPU-Accelerated Vecchia Approximations of Gaussian Processes for Geospatial Data using Batched Matrix Computations. InISC High Performance 2024 Research Paper Proceedings (39th International Conference), pages 1–12. Prometeus GmbH,

    work page 2024

  2. [2]

    F. D. Schneider, F. Morsdorf, B. Schmid, O. L. Petchey, A. Hueni, D. S. Schimel, and M. E. Schaepman. Data by Schneider et al. (2017) Mapping Functional Diversity from Remotely Sensed Morphological and Physiological Forest Traits. 3

    work page 2017

  3. [3]

    19 Appendix A.1 Additional figures Figure 16: Average negative log-likelihood on simulated data sets with a practical range of 0.2 and a sample sizeN= 100,000

    ISSN 00359246. 19 Appendix A.1 Additional figures Figure 16: Average negative log-likelihood on simulated data sets with a practical range of 0.2 and a sample sizeN= 100,000. The true data-generating parameters are used on the left and “wrong” parameters obtained by doubling the true ones are used on the right. Figure 17: Bias and MSE of the estimates for...

    work page 2000

  4. [4]

    extrapolation

    0.43 36.3 1.27 11.3 13.6 6.99 Time (s) 3.37 15 23.6 49.3 19.3 16.1 Table 18: Average RMSE between means, RMSE between variances, KL divergence between exact and approximate predictions, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2 and a sample size of N=10,000....

    work page 1931

  5. [5]

    1.83e-04 4.17e-04 1.28e-03 6.7e-04 5.52e-03 5.45e-03 0.0178 KL 1214 4960 2035 4074 44168 2105 269 (SE

    work page 2035

  6. [6]

    extrapolation

    15.4 181 0.332 6.18 39.7 31.1 Time (s) 3.34 15.5 23.6 49.5 26.3 14.6 Table 30: Average RMSE between means, RMSE between variances, KL divergence between exact and approximate predictions, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.5 and a sample size of N=10,00...

    work page 2065

  7. [7]

    1.17e-06 2.78e-05 6.45e-05 4.64e-05 1.06e-03 6.74e-05 1.57e-04 KL 38.4 2094 3878 3245 4566 969 26771 (SE

    work page 2094

  8. [8]

    2.47e-05 1.83e-04 1.45e-04 1.94e-04 7.28e-03 3.3e-04 2.04e-04 KL 163 4421 8301 6057 8509 2062 70297 (SE

    work page 2062

  9. [9]

    1.42e-06 1.33e-04 2.98e-04 1.77e-04 1.94e-04 2.72e-04 KL 5.89 1910 4988 2482 886 4526 (SE

    work page 1910

  10. [10]

    extrapolation

    0.0485 0.0364 0.0484 0.0457 0.0478 0.0598 Time (s) 46.5 475 451 1981 475 695 Table 50: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2 and a sample size of N=100,000. 72 B.5 Simulated data withN= 100,000a...

    work page 1981

  11. [11]

    extrapolation

    0.0698 0.0527 0.0683 0.0621 0.0633 0.0847 Time (s) 46.4 510 453 1972 846 1207 Table 59: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.5 and a sample size of N=100,000. 82 B.6 Simulated data withN= 100,000...

    work page 1972

  12. [12]

    1.24e-07 0.0826 4.84e-04 5.09e-06 1.13e-06 1.24e-06 Time (s) 20.7 250 61.3 133 1719 1954 4 Bias -5.92e-05 1.22 0.0254 5.99e-03 1.85e-03 3.31e-03 (SE

    work page 1954

  13. [13]

    7.34e-04 1.84e-03 0.819 1.13e-03 9.83e-03 0.471 Time (s) 20.7 250 61.3 133 1719 1954 4 Bias -0.0131 -0.272 1.78 -0.0334 0.171 4.08 (SE

    work page 1954

  14. [14]

    2.45e-06 6.08e-04 5.78e-03 6.62e-04 5.02e-06 4.57e-06 Time (s) 20.7 250 61.3 133 1719 1954 4 Bias 2.55e-04 -0.178 0.112 -0.141 1.68e-03 1.94e-03 (SE

    work page 1954

  15. [15]

    extrapolation

    0.0153 0.0156 0.0155 0.0158 0.0174 0.0178 Time (s) 46.4 463 452 1908 131 737 Table 68: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.05 and a sample size of N=100,000. 92 B.7 Simulated data withN= 100,000...

    work page 1908

  16. [16]

    extrapolation

    0.0177 0.0216 0.0199 0.0179 0.0341 0.0243 Time (s) 37.5 491 451 1942 386 448 Table 77: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an anisotropic Mat´ ern covariance function and a sample size N=100,000. 100 B.8 Simulated data...

    work page 1942

  17. [17]

    extrapolation

    0.027 0.0357 0.0229 0.0282 0.0295 0.0342 Time (s) 46.5 491 452 1919 496 1186 Table 86: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2, error variance of 0.1 and a sample size N=100,000. 110 B.9 Simulated...

    work page 1919

  18. [18]

    2.67e-05 1921 834 0.0742 8.66e-04 8.65e-06 Time (s) 2.17 65.8 1.48 28.5 457 343 12.1 2 Bias 2.82e-03 173 0.483 0.102 0.0493 -0.0138 (SE

    work page 1921

  19. [19]

    extrapolation

    0.029 0.0322 0.0273 0.0281 0.029 0.0519 Time (s) 46.3 479 453 1898 97.2 597 Table 95: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2, Mat´ ern smoothness of 0.5 and a sample size N=100,000. 120 B.10 Simu...

    work page 2000

  20. [20]

    2.83e-05 0.0215 4.3e-04 2.37e-04 3.57e-05 3.83e-05 Time (s) 4.17 88.5 44 96.5 1973 113 3 Bias 2.66e-04 0.164 0.027 0.0179 0.0117 0.0117 (SE

    work page 1973

  21. [21]

    0.022 5.39e-03 3.1 1.19 0.0943 0.787 Time (s) 4.17 88.5 44 96.5 1973 113 3 Bias 0.0332 -0.13 1.58 1.14 0.605 2.86 (SE

    work page 1973

  22. [22]

    1.52e-06 2.78e-03 8.39e-06 1.71e-04 1.46e-06 1.34e-06 Time (s) 4.17 88.5 44 96.5 1973 113 3 Bias -4.74e-04 -0.24 -2.74e-03 -0.0331 -2.78e-04 -4.47e-04 (SE

    work page 1973

  23. [23]

    extrapolation

    0.0343 0.037 0.0362 0.0381 0.0328 0.037 Time (s) 46.6 480 452 1974 936 2315 Table 104: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2, Mat´ ern smoothness of 2.5 and a sample size N=100,000. 130 B.11 Sim...

    work page 1974

  24. [24]

    extrapolation

    0.0337 0.037 0.0382 0.0371 0.0309 0.183 Time (s) 46.3 478 451 1944 98 599 Table 113: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2, misspecified Mat´ ern smoothness and a sample size N=100,000. 140 B.12...

    work page 1944

  25. [25]

    1.15e-05 0.315 9.98e-06 1.21e-04 4.18e-05 2.7e-05 Time (s) 135 4068 1986 2267 3690 6684 5 Bias -7.72e-04 1.74 -1.59e-03 8.01e-04 0.0137 6.08e-03 (SE

    work page 1986

  26. [26]

    9.95e-03 0.0111 0.0218 0.373 0.0684 1.36 Time (s) 135 4068 1986 2267 3690 6684 5 Bias -0.0205 -0.51 -9.38e-04 0.172 0.534 3.51 (SE

    work page 1986

  27. [27]

    1.76e-06 1.09e-03 5.51e-06 4.11e-05 1.8e-06 2.26e-06 Time (s) 135 4068 1986 2267 3690 6684 5 Bias 8.42e-04 -0.101 -6.51e-04 -7.94e-03 9.05e-04 1.65e-03 (SE

    work page 1986

  28. [28]

    extrapolation

    0.0485 0.0364 0.0484 0.0457 0.0478 0.0567 Time (s) 312 1710 2015 2294 477 1892 Table 145: Average RMSE and log-score on the test “extrapolation” set, standard errors and time needed for making predictions on the test “extrapolation” set on simulated data sets with an effective range of 0.2 and a sample size of N=100,000 in the single-core setting. 169 B.1...

    work page 2015