pith. sign in

arxiv: 2607.00051 · v1 · pith:6NRMSBQUnew · submitted 2026-06-30 · 📊 stat.AP · cs.LG

Spatio-Temporal Gaussian Process for Building Terrain-Incorporating Wind Power Curves

Ahmadreza Chokhachian , V. Roshan Joseph , Yu Ding This is my paper

Pith reviewed 2026-07-02 17:01 UTC · model grok-4.3

classification 📊 stat.AP cs.LG
keywords wind power curvesspatio-temporal Gaussian processterrain covariatesseparable kernelwind farmpredictive accuracynonparametric modeltemporal alignment
0
0 comments X

The pith

A spatio-temporal Gaussian process integrates terrain features to model wind turbine power curves more accurately.

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

The paper develops a nonparametric model that combines temporal environmental data with spatial terrain features for wind power curve modeling. Standard models overlook terrain effects on wind inflow. By creating a shared, smaller temporal covariate set from misaligned data, the model uses a separable kernel to capture dependencies. This results in higher predictive accuracy on real datasets and the ability to measure terrain's impact on turbine output. The approach addresses a key gap in wind farm operation modeling.

Core claim

The central discovery is a spatio-temporal Gaussian process model that uses a constructed shared representative temporal covariate set to align data and apply a separable kernel, thereby incorporating terrain covariates and improving predictions over baselines while quantifying their effects.

What carries the argument

The shared representative temporal covariate set, which aligns temporal inputs across turbines and reduces size by an order of magnitude to enable separable spatio-temporal kernels in the Gaussian process.

Load-bearing premise

The wind farm data lacking temporal alignment can be transformed into a shared representative temporal covariate set of much smaller size without losing essential information.

What would settle it

A test showing that the proposed model's predictive accuracy does not exceed that of baselines ignoring terrain, or that terrain impact quantification does not reveal distinct effects, on the real wind farm dataset.

Figures

Figures reproduced from arXiv: 2607.00051 by Ahmadreza Chokhachian, V. Roshan Joseph, Yu Ding.

Figure 1
Figure 1. Figure 1: (a) Nominal power curve of a wind turbine (Lee et al., 2015) and (b) IEC binning [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of structured missingness: (a) the ideal fully observed grid and (b) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: This procedure of data transformation offers an additional benefit. In the original [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: compares LK and OK under the LOTO setting with both training and testing on 2017 data. Both the methods employ separable covariance functions with the same kernels and parameters. While OK performs reasonably well for a few turbines, in most turbines, its RMSE is substantially larger than that of LK. An example is shown on the right panel, which clearly shows the mean reversion issue of OK. In the remainde… view at source ↗
Figure 5
Figure 5. Figure 5: ALE plots for temporal and spatial features. Gray lines represent conditional [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial distribution of turbines across the farm. Distant turbines (squares) [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Naive (red) and adjusted (blue) underperformance rate (%) for each turbine. [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
read the original abstract

Accurate modeling of wind turbine power curves is crucial for optimal wind farm operation. Nearly all existing power curve models focus on temporal variables such as wind speed and temperature while overlooking the influence of terrain covariates, which governs inflow wind conditions and thus also affects wind power production. This paper proposes a nonparametric spatio-temporal Gaussian process model that integrates temporal environmental covariates with spatial terrain features. The model falls in the category of spatial-temporal Gaussian process models with data on a grid. The challenge to be addressed is that the spatio-temporal modeling require certain temporal alignment among the data, a property that the wind farm data does not have. Our solution strategy is to construct a shared representative temporal covariate set which not only aligns the temporal inputs but also has a size an order of magnitude smaller than the original data size. With this transformation, our resulting model is able to employ a separable kernel structure that captures both spatial and temporal dependencies. Empirical analysis on a real wind farm dataset shows that our method improves predictive accuracy over existing baselines and can be used to quantify the various impact of the terrain characteristics on turbine performance.

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

1 major / 1 minor

Summary. The manuscript proposes a nonparametric spatio-temporal Gaussian process model for wind turbine power curves that incorporates both temporal environmental covariates and spatial terrain features. To address the lack of temporal alignment in wind farm data, the authors construct a shared representative temporal covariate set of size an order of magnitude smaller than the original data, enabling a separable kernel structure. The paper claims that this yields improved predictive accuracy over baselines on real wind farm data and permits quantification of terrain characteristic impacts on turbine performance.

Significance. If the data-reduction step is shown to preserve essential information, the work would supply a practical route to include terrain effects in wind-power modeling, an aspect typically omitted. The approach relies on standard GP techniques applied after a compression step; its value therefore rests on demonstrating that the compression does not erase turbine-specific temporal structure or terrain-time interactions.

major comments (1)
  1. [Abstract / modeling approach] Abstract and modeling section: the construction of the shared representative temporal covariate set is asserted to align inputs while preserving essential spatio-temporal information, yet no reconstruction error bound, sensitivity analysis, or verification that turbine-specific temporal structure and terrain-time interactions survive the reduction is supplied. This step is load-bearing for both the reported accuracy gains and the terrain-impact conclusions.
minor comments (1)
  1. [Abstract] Abstract: the claim of improved predictive accuracy is stated without any quantitative metrics, baseline details, error bars, or validation protocol, making it impossible to assess the magnitude or robustness of the improvement from the abstract alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments. The major comment correctly identifies a gap in validation of the data-reduction step, which we address by committing to additional empirical analyses in revision.

read point-by-point responses

  1. Referee: [Abstract / modeling approach] Abstract and modeling section: the construction of the shared representative temporal covariate set is asserted to align inputs while preserving essential spatio-temporal information, yet no reconstruction error bound, sensitivity analysis, or verification that turbine-specific temporal structure and terrain-time interactions survive the reduction is supplied. This step is load-bearing for both the reported accuracy gains and the terrain-impact conclusions.

    Authors: We agree that the manuscript asserts preservation of essential information without supplying a reconstruction error bound, sensitivity analysis, or explicit verification that turbine-specific temporal structure and terrain-time interactions are retained. This is a substantive point given the central role of the reduction. In the revised manuscript we will add an empirical sensitivity analysis that varies the size of the representative set and reports effects on predictive accuracy and terrain-coefficient estimates. We will also add side-by-side visualizations of temporal covariate trajectories for a subset of turbines before and after reduction to demonstrate retention of dominant patterns. A general theoretical reconstruction-error bound is difficult to obtain because the representative set is chosen data-dependently; we will therefore state this limitation explicitly and rely on the added empirical checks. revision: yes

Circularity Check

0 steps flagged

No circularity: modeling choices and empirical results remain independent of fitted outputs

full rationale

The paper presents a standard separable spatio-temporal GP after an explicit preprocessing step that constructs a shared temporal covariate grid. This step is described as a pragmatic alignment device rather than a quantity derived from or fitted to the final predictions; the reported accuracy gains are obtained by comparing the resulting model against baselines on held-out data. No equation equates a model output to a transformation of itself, no parameter is fitted on a subset and then relabeled as a prediction of a related quantity, and no load-bearing claim rests on a self-citation whose content is itself unverified. The derivation chain therefore stays self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; full text required for ledger construction.

pith-pipeline@v0.9.1-grok · 5724 in / 1048 out tokens · 29457 ms · 2026-07-02T17:01:05.466222+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

98 extracted references · 98 canonical work pages

  1. [1]

    Chokhachian, Ahmadreza and Katzfuss, Matthias and Ding, Yu , journal=. Fast. 2026 , pages=

    work page 2026

  2. [2]

    Journal of Statistical Software , author=

    het. Journal of Statistical Software , author=. 2021 , pages=

    work page 2021

  3. [3]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , year=

    Graddiv: Adversarial robustness of randomized neural networks via gradient diversity regularization , author=. IEEE Transactions on Pattern Analysis and Machine Intelligence , year=

    work page

  4. [4]

    Advances in Neural Information Processing Systems 18 , pages =

    Edward Snelson and Zoubin Ghahramani , title =. Advances in Neural Information Processing Systems 18 , pages =. 2005 , publisher =

    work page 2005

  5. [5]

    Proceedings of the 11th International Conference on Artificial Intelligence and Statistics , pages =

    Edward Snelson and Zoubin Ghahramani , title =. Proceedings of the 11th International Conference on Artificial Intelligence and Statistics , pages =. 2007 , publisher =

    work page 2007

  6. [6]

    Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics , editor =

    Titsias, Michalis , title =. Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics , editor =. 2009 , publisher =

    work page 2009

  7. [7]

    The Computer Journal , volume =

    Liu, Jie and Wang, Wei and Qiu, Weijian and Zuo, Gangyong , title =. The Computer Journal , volume =

    work page

  8. [8]

    Barber and Y

    S. Barber and Y. Ding , title =. Proceedings of the 2024 WindEurope Annual Event, Bilbao, Spain, March 20-22 , year =

    work page 2024

  9. [9]

    Ding and J

    Y. Ding and J. Tang and J. Z. Huang , title =. Proceedings of ASME Turbo Expo 2015: Turbine Technical Conference and Exposition (GT 2015), Montreal, Canada, June 15-19 , year =

    work page 2015

  10. [10]

    Journal of Statistical Software , year =

    Zammit-Mangion, Andrew and Cressie, Noel , title =. Journal of Statistical Software , year =

    work page

  11. [11]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Fixed rank kriging for very large spatial data sets , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    work page

  12. [12]

    2000 , pages =

    Olea, Ricardo , title =. 2000 , pages =

    work page 2000

  13. [13]

    Journal of Statistical Planning and Inference , volume=

    Energy statistics: A class of statistics based on distances , author=. Journal of Statistical Planning and Inference , volume=

    work page

  14. [14]

    A. M. Sempreviva and S. E. Larsen and N. G. Mortensen and I. Troen , title =. Boundary-Layer Meteorology , volume =

    work page

  15. [15]

    Han and J

    X. Han and J. Guo and P. Wang , title =. IET Generation, Transmission and Distribution , volume =

    work page

  16. [16]

    Tian and A

    W. Tian and A. Ozbay and H. Hu , title =. Procedia Engineering , volume =

    work page

  17. [17]

    Mallick , title =

    Yu Ding and Abhinav Prakash and Se Yoon Lee and Xin Liu and Lei Liu and Bani K. Mallick , title =. Renewable Energy , volume =

    work page

  18. [18]

    , title=

    Thorndike, Robert L. , title=. Psychometrika , year=

    work page

  19. [19]

    Technometrics , volume=

    Limit kriging , author=. Technometrics , volume=

    work page

  20. [20]

    Chris K. I. Williams and Edwin V. Bonilla and Kian M. Chai , title =. Advances in Neural Information Processing Systems , year =

    work page

  21. [21]

    K. B. Petersen and M. S. Pedersen. The Matrix Cookbook. 2012

    work page 2012

  22. [22]

    Journal of the American Statistical Association , volume =

    Lee, Giwhyun and Ding, Yu and Genton, Marc G and Xie, Le , title =. Journal of the American Statistical Association , volume =

    work page

  23. [23]

    Roshan , title =

    Vakayil, Akhil and Joseph, V. Roshan , title =. Technometrics , volume =

    work page

  24. [24]

    Roshan , title =

    Mak, Simon and Joseph, V. Roshan , title =. The Annals of Statistics , volume =

    work page

  25. [25]

    IEEE Transactions on Sustainable Energy , volume =

    Bessa, Ricardo J and Miranda, Vladimiro and Botterud, Audun and Wang, Jianhui and Constantinescu, Emil M , title =. IEEE Transactions on Sustainable Energy , volume =

    work page

  26. [26]

    Technometrics , volume =

    Prakash, Abhinav and Tuo, Rui and Ding, Yu , title =. Technometrics , volume =

    work page

  27. [27]

    IEEE Transactions on Sustainable Energy , volume =

    Prakash, Abhinav and Lee, Se Yoon and Liu, Xin and Liu, Lei and Mallick, Bani K and Ding, Yu , title =. IEEE Transactions on Sustainable Energy , volume =

    work page

  28. [28]

    and Finley, Andrew O

    Banerjee, Sudipto and Gelfand, Alan E. and Finley, Andrew O. and Sang, Huiyan , title =. Journal of the Royal Statistical Society: Series B , volume =

    work page

  29. [29]

    and Datta, Abhirup and Finley, Andrew O

    Heaton, Matthew J. and Datta, Abhirup and Finley, Andrew O. and Furrer, Reinhard and Guinness, Joseph and Guhaniyogi, Rajarshi and Gerber, Florian and Gramacy, Robert B. and Hammerling, Dorit and Katzfuss, Matthias and others , title =. Journal of Agricultural, Biological and Environmental Statistics , volume =

    work page

  30. [30]

    , title =

    Stein, Michael L. , title =. Journal of the American Statistical Association , volume =

    work page

  31. [31]

    , title =

    Genton, Marc G. , title =. Environmetrics , volume =

    work page

  32. [32]

    Automatica , volume =

    Todescato, Marco and Carron, Andrea and Carli, Ruggero and Pillonetto, Gianluigi and Schenato, Luca , title =. Automatica , volume =

    work page

  33. [33]

    Applied Stochastic Models in Business and Industry , volume =

    Lambardi di San Miniato, Michele and Tugnoli, Marco and Zorzi, Mattia , title =. Applied Stochastic Models in Business and Industry , volume =

    work page

  34. [34]

    Bayesian Analysis , volume =

    Gu, Mengyang and Li, Hanmo , title =. Bayesian Analysis , volume =

    work page

  35. [35]

    Journal of the Royal Statistical Society: Series B , volume=

    Visualizing the effects of predictor variables in black box supervised learning models , author=. Journal of the Royal Statistical Society: Series B , volume=

    work page

  36. [36]

    Fragoulis , title =

    A. Fragoulis , title =. Proceedings of the 35th Aerospace Sciences Meeting and Exhibit , year =

    work page

  37. [37]

    , booktitle =

    Wilson, Andrew Gordon and Gilboa, Elad and Nehorai, Arye and Cunningham, John P. , booktitle =. Fast kernel learning for multidimensional pattern extrapolation , volume =

    work page

  38. [38]

    International Conference on Machine Learning (ICML) , pages=

    Andrew Gordon Wilson and Hannes Nickisch , title =. International Conference on Machine Learning (ICML) , pages=

    work page

  39. [39]

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

    Oliver Hamelijnck and William Wilkinson and Niki Loppi and Arno Solin and Theodoros Damoulas , title =. Advances in Neural Information Processing Systems (NeurIPS) , year =

    work page

  40. [40]

    Lin, Jihao Andreas and Ament, Sebastian and Balandat, Maximilian and Bakshy, Eytan , booktitle=. Scaling

    work page

  41. [41]

    It is all in the noise:

    Rakitsch, Barbara and Lippert, Christoph and Borgwardt, Karsten and Stegle, Oliver , booktitle =. It is all in the noise:

    work page

  42. [42]

    Efficient

    Luttinen, Jaakko and Ilin, Alexander , booktitle=. Efficient

    work page

  43. [43]

    1991 , publisher=

    Statistics for Spatial Data , author=. 1991 , publisher=

    work page 1991

  44. [44]

    2011 , publisher=

    Statistics for Spatio-Temporal Data , author=. 2011 , publisher=

    work page 2011

  45. [45]

    Rasmussen, Carl Edward and Williams, Christopher KI , year=

    work page

  46. [46]

    Yarin Saatçi , title =

    work page

  47. [47]

    Electricity Generation from Wind , year =

    work page

  48. [48]

    2025 , note =

    Xgboost: Extreme Gradient Boosting , author =. 2025 , note =

    work page 2025

  49. [49]

    2021 , note =

    DSWE: Data Science for Wind Energy , author =. 2021 , note =

    work page 2021

  50. [50]

    2005 , note =

    IEC , title =. 2005 , note =

    work page 2005

  51. [51]

    2013 , note =

    IEC , title =. 2013 , note =

    work page 2013

  52. [52]

    Apley, D. W. and Zhu, J. (2020). Visualizing the effects of predictor variables in black box supervised learning models. Journal of the Royal Statistical Society: Series B , 82:1059--1086

    work page 2020

  53. [53]

    E., Finley, A

    Banerjee, S., Gelfand, A. E., Finley, A. O., and Sang, H. (2008). G aussian predictive process models for large spatial data sets. Journal of the Royal Statistical Society: Series B , 70:825--848

    work page 2008

  54. [54]

    and Ding, Y

    Barber, S. and Ding, Y. (2024). Improving data sharing in practice – power curve benchmarking case study. In Proceedings of the 2024 WindEurope Annual Event, Bilbao, Spain, March 20-22

    work page 2024

  55. [55]

    J., Miranda, V., Botterud, A., Wang, J., and Constantinescu, E

    Bessa, R. J., Miranda, V., Botterud, A., Wang, J., and Constantinescu, E. M. (2012). Time adaptive conditional kernel density estimation for wind power forecasting. IEEE Transactions on Sustainable Energy , 3:660--669

    work page 2012

  56. [56]

    and Gramacy, R

    Binois, M. and Gramacy, R. B. (2021). het G P : Heteroskedastic G aussian process modeling and sequential design in r. Journal of Statistical Software , 98(13):1–44

    work page 2021

  57. [57]

    Chokhachian, A., Katzfuss, M., and Ding, Y. (2026). Fast G aussian process approximations for autocorrelated data. INFORMS Journal on Data Science , page 0

    work page 2026

  58. [58]

    and Johannesson, G

    Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. Journal of the Royal Statistical Society Series B: Statistical Methodology , 70:209--226

    work page 2008

  59. [59]

    and Wikle, C

    Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data . John Wiley and Sons, Hoboken, NJ, USA

    work page 2011

  60. [60]

    Cressie, N. A. C. (1991). Statistics for Spatial Data . John Wiley and Sons, New York, NY, USA

    work page 1991

  61. [61]

    Ding, Y. (2019). Data Science for Wind Energy . CRC Press, Boca Raton, FL, USA

    work page 2019

  62. [62]

    Ding, Y., Tang, J., and Huang, J. Z. (2015). Data analytics methods for wind energy applications. In Proceedings of ASME Turbo Expo 2015: Turbine Technical Conference and Exposition (GT 2015), Montreal, Canada, June 15-19

    work page 2015

  63. [63]

    Fragoulis, A. (1997). The complex terrain wind environment and its effects on the power output and loading of wind turbines. In Proceedings of the 35th Aerospace Sciences Meeting and Exhibit , page 934

    work page 1997

  64. [64]

    Genton, M. G. (2007). Separable approximations of space-time covariance matrices. Environmetrics , 18:681--695

    work page 2007

  65. [65]

    and Li, H

    Gu, M. and Li, H. (2022). G aussian orthogonal latent factor processes for large incomplete matrices of correlated data. Bayesian Analysis , 17:1219--1244

    work page 2022

  66. [66]

    Hamelijnck, O., Wilkinson, W., Loppi, N., Solin, A., and Damoulas, T. (2021). Spatio-temporal variational G aussian processes. In Advances in Neural Information Processing Systems (NeurIPS) , pages 23621--23633

    work page 2021

  67. [67]

    Han, X., Guo, J., and Wang, P. (2012). Adequacy study of a wind farm considering terrain and wake effect. IET Generation, Transmission and Distribution , 6:1001--1008

    work page 2012

  68. [68]

    J., Datta, A., Finley, A

    Heaton, M. J., Datta, A., Finley, A. O., Furrer, R., Guinness, J., Guhaniyogi, R., Gerber, F., Gramacy, R. B., Hammerling, D., Katzfuss, M., et al. (2019). A case study competition among methods for analyzing large spatial data. Journal of Agricultural, Biological and Environmental Statistics , 24:398--425

    work page 2019

  69. [69]

    IEC TS 61400-12-1 Ed

    IEC (2005). IEC TS 61400-12-1 Ed. 1: Power Performance Measurements of Electricity-Producing Wind Turbines . International Electrotechnical Commission, Geneva, Switzerland. Technical Specification

    work page 2005

  70. [70]

    IEC 61400-12-2 Ed

    IEC (2013). IEC 61400-12-2 Ed. 1.0: Wind Turbines—Part 12-2: Power Performance of Electricity-Producing Wind Turbines Based on Nacelle Anemometry . International Electrotechnical Commission, Geneva, Switzerland. International Standard

    work page 2013

  71. [71]

    Joseph, V. R. (2006). Limit kriging. Technometrics , 48:458--466

    work page 2006

  72. [72]

    Kumar, N., Prakash, A., and Ding, Y. (2021). DSWE: Data Science for Wind Energy . R package version 1.8.2. available at https://CRAN.R-project.org/package=DSWE

    work page 2021

  73. [73]

    Lambardi di San Miniato, M., Tugnoli, M., and Zorzi, M. (2022). Separable spatio-temporal kriging for fast virtual sensing. Applied Stochastic Models in Business and Industry , 38:806--829

    work page 2022

  74. [74]

    G., and Xie, L

    Lee, G., Ding, Y., Genton, M. G., and Xie, L. (2015). Power curve estimation with multivariate environmental factors for inland and offshore wind farms. Journal of the American Statistical Association , 110:56--67

    work page 2015

  75. [75]

    Lee, S., Kim, H., and Lee, J. (2022). Graddiv: Adversarial robustness of randomized neural networks via gradient diversity regularization. IEEE Transactions on Pattern Analysis and Machine Intelligence , 45:2645--2651

    work page 2022

  76. [76]

    A., Ament, S., Balandat, M., and Bakshy, E

    Lin, J. A., Ament, S., Balandat, M., and Bakshy, E. (2024). Scaling G aussian processes for learning curve prediction via latent K ronecker structure. In NeurIPS 2024 Workshop on Bayesian Decision-making and Uncertainty

    work page 2024

  77. [77]

    Liu, J., Wang, W., Qiu, W., and Zuo, G. (2025). G aussian process regression based on random projections and dynamic pseudo-input selection. The Computer Journal , 68:1682--1698

    work page 2025

  78. [78]

    and Ilin, A

    Luttinen, J. and Ilin, A. (2012). Efficient Gaussian process inference for short-scale spatio-temporal modeling. In Artificial Intelligence and Statistics, PMLR , pages 741--750

    work page 2012

  79. [79]

    and Joseph, V

    Mak, S. and Joseph, V. R. (2018). Support points. The Annals of Statistics , 46:2562--2592

    work page 2018

  80. [80]

    Petersen, K. B. and Pedersen, M. S. (2012). The matrix cookbook. Version 20051003

    work page 2012

Showing first 80 references.