Software:Comparison of Gaussian process software

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Short description: Comparison of statistical analysis software that allows doing inference with Gaussian processes

This is a comparison of statistical analysis software that allows doing inference with Gaussian processes often using approximations.

This article is written from the point of view of Bayesian statistics, which may use a terminology different from the one commonly used in kriging. The next section should clarify the mathematical/computational meaning of the information provided in the table independently of contextual terminology.

Description of columns

This section details the meaning of the columns in the table below.

Solvers

These columns are about the algorithms used to solve the linear system defined by the prior covariance matrix, i.e., the matrix built by evaluating the kernel.

  • Exact: whether generic exact algorithms are implemented. These algorithms are usually appropriate only up to some thousands of datapoints.
  • Specialized: whether specialized exact algorithms for specific classes of problems are implemented. Supported specialized algorithms may be indicated as:
    • Kronecker: algorithms for separable kernels on grid data.[1]
    • Toeplitz: algorithms for stationary kernels on uniformly spaced data.[2]
    • Semisep.: algorithms for semiseparable covariance matrices.[3]
    • Sparse: algorithms optimized for sparse covariance matrices.
    • Block: algorithms optimized for block diagonal covariance matrices.
    • Markov: algorithms for kernels which represent (or can be formulated as) a Markov process.[4]
  • Approximate: whether generic or specialized approximate algorithms are implemented. Supported approximate algorithms may be indicated as:
    • Sparse: algorithms based on choosing a set of "inducing points" in input space,[5] or more in general imposing a sparse structure on the inverse of the covariance matrix.
    • Hierarchical: algorithms which approximate the covariance matrix with a hierarchical matrix.[6]

Input

These columns are about the points on which the Gaussian process is evaluated, i.e. [math]\displaystyle{ x }[/math] if the process is [math]\displaystyle{ f(x) }[/math].

  • ND: whether multidimensional input is supported. If it is, multidimensional output is always possible by adding a dimension to the input, even without direct support.
  • Non-real: whether arbitrary non-real input is supported (for example, text or complex numbers).

Output

These columns are about the values yielded by the process, and how they are connected to the data used in the fit.

  • Likelihood: whether arbitrary non-Gaussian likelihoods are supported.
  • Errors: whether arbitrary non-uniform correlated errors on datapoints are supported for the Gaussian likelihood. Errors may be handled manually by adding a kernel component, this column is about the possibility of manipulating them separately. Partial error support may be indicated as:
    • iid: the datapoints must be independent and identically distributed.
    • Uncorrelated: the datapoints must be independent, but can have different distributions.
    • Stationary: the datapoints can be correlated, but the covariance matrix must be a Toeplitz matrix, in particular this implies that the variances must be uniform.

Hyperparameters

These columns are about finding values of variables which enter somehow in the definition of the specific problem but that can not be inferred by the Gaussian process fit, for example parameters in the formula of the kernel.

  • Prior: whether specifying arbitrary hyperpriors on the hyperparameters is supported.
  • Posterior: whether estimating the posterior is supported beyond point estimation, possibly in conjunction with other software.

If both the "Prior" and "Posterior" cells contain "Manually", the software provides an interface for computing the marginal likelihood and its gradient w.r.t. hyperparameters, which can be feed into an optimization/sampling algorithm, e.g., gradient descent or Markov chain Monte Carlo.

Linear transformations

These columns are about the possibility of fitting datapoints simultaneously to a process and to linear transformations of it.

  • Deriv.: whether it is possible to take an arbitrary number of derivatives up to the maximum allowed by the smoothness of the kernel, for any differentiable kernel. Example partial specifications may be the maximum derivability or implementation only for some kernels. Integrals can be obtained indirectly from derivatives.
  • Finite: whether finite arbitrary [math]\displaystyle{ \mathbb R^n \to \mathbb R^m }[/math] linear transformations are allowed on the specified datapoints.
  • Sum: whether it is possible to sum various kernels and access separately the processes corresponding to each addend. It is a particular case of finite linear transformation but it is listed separately because it is a common feature.

Comparison table

Name License Language Solvers Input Output Hyperparameters Linear transformations Name
Exact Specialized Approximate ND Non-real Likelihood Errors Prior Posterior Deriv. Finite Sum
PyMC Apache Python Yes Kronecker Sparse ND No Any Correlated Yes Yes No Yes Yes PyMC
Stan BSD, GPL custom Yes No No ND No Any Correlated Yes Yes No Yes Yes Stan
scikit-learn BSD Python Yes No No ND Yes Bernoulli Uncorrelated Manually Manually No No No scikit-learn
fbm[7] Free C Yes No No ND No Bernoulli, Poisson Uncorrelated, Stationary Many Yes No No Yes fbm
GPML[8][7] BSD MATLAB Yes No Sparse ND No Many i.i.d. Manually Manually No No No GPML
GPstuff[7] GNU GPL MATLAB, R Yes Sparse, Markov Sparse ND No Many Correlated Many Yes First RBF No Yes GPstuff
GPy[9] BSD Python Yes No Sparse ND No Many Uncorrelated Yes Yes No No No GPy
GPflow[9] Apache Python Yes No Sparse ND No Many Uncorrelated Yes Yes No No No GPflow
GPyTorch[10] MIT Python Yes Toeplitz, Kronecker Sparse ND No Many Uncorrelated Yes Yes First RBF Manually Manually GPyTorch
GPvecchia[11] GNU GPL R Yes No Sparse, Hierarchical ND No Exponential family Uncorrelated No No No No No GPvecchia
pyGPs[12] BSD Python Yes No Sparse ND Graphs, Manually Bernoulli i.i.d. Manually Manually No No No pyGPs
gptk[13] BSD R Yes Block? Sparse ND No Gaussian No Manually Manually No No No gptk
celerite[3] MIT Python, Julia, C++ No Semisep.[lower-alpha 1] No 1D No Gaussian Uncorrelated Manually Manually No No No celerite
george[6] MIT Python, C++ Yes No Hierarchical ND No Gaussian Uncorrelated Manually Manually No No Manually george
neural-tangents[14][lower-alpha 2] Apache Python Yes Block, Kronecker No ND No Gaussian No No No No No No neural-tangents
DiceKriging[15] GNU GPL R Yes No No ND No? Gaussian Uncorrelated SCAD RBF MAP No No No DiceKriging
OpenTURNS[16] GNU LGPL Python, C++ Yes No No ND No Gaussian Uncorrelated Manually (no grad.) MAP No No No OpenTURNS
UQLab[17] Proprietary MATLAB Yes No No ND No Gaussian Correlated No MAP No No No UQLab
ooDACE [18] Proprietary MATLAB Yes No No ND No Gaussian Correlated No MAP No No No ooDACE
DACE Proprietary MATLAB Yes No No ND No Gaussian No No MAP No No No DACE
GpGp MIT R No No Sparse ND No Gaussian i.i.d. Manually Manually No No No GpGp
SuperGauss GNU GPL R, C++ No Toeplitz[lower-alpha 3] No 1D No Gaussian No Manually Manually No No No SuperGauss
STK GNU GPL MATLAB Yes No No ND No Gaussian Uncorrelated Manually Manually No No Manually STK
GSTools GNU LGPL Python Yes No No ND No Gaussian No No No No No No GSTools
PyKrige BSD Python Yes No No 2D,3D No Gaussian i.i.d. No No No No No PyKrige
GPR Apache C++ Yes No Sparse ND No Gaussian i.i.d. Some, Manually Manually First No No GPR
celerite2 MIT Python No Semisep.[lower-alpha 1] No 1D No Gaussian Uncorrelated Manually[lower-alpha 4] Manually No No Yes celerite2
GPJax Apache Python Yes No Sparse ND Graphs Bernoulli No Yes Yes No No No GPJax
Stheno MIT Python Yes Low rank Sparse ND No Gaussian i.i.d. Manually Manually Approximate No Yes Stheno
Name License Language Exact Specialized Approximate ND Non-real Likelihood Errors Prior Posterior Deriv. Finite Sum Name
Solvers Input Output Hyperparameters Linear transformations

Notes

  1. ↑ 1.0 1.1 celerite implements only a specific subalgebra of kernels which can be solved in [math]\displaystyle{ O(n) }[/math].[3]
  2. ↑ neural-tangents is a specialized package for infinitely wide neural networks.
  3. ↑ SuperGauss implements a superfast Toeplitz solver with computational complexity [math]\displaystyle{ O(n\log^2n) }[/math].
  4. ↑ celerite2 has a PyMC3 interface.

References

  1. ↑ P. Cunningham, John; Gilboa, Elad; Saatçi, Yunus (Feb 2015). "Scaling Multidimensional Inference for Structured Gaussian Processes". IEEE Transactions on Pattern Analysis and Machine Intelligence 37 (2): 424–436. doi:10.1109/TPAMI.2013.192. PMID 26353252. 
  2. ↑ Leith, D. J.; Zhang, Yunong; Leithead, W. E. (2005). "Time-series Gaussian Process Regression Based on Toeplitz Computation of O(N²) Operations and O(N)-level Storage". Proceedings of the 44th IEEE Conference on Decision and Control. pp. 3711–3716. doi:10.1109/CDC.2005.1582739. ISBN 0-7803-9567-0. 
  3. ↑ 3.0 3.1 3.2 Foreman-Mackey, Daniel; Angus, Ruth; Agol, Eric; Ambikasaran, Sivaram (9 November 2017). "Fast and Scalable Gaussian Process Modeling with Applications to Astronomical Time Series". The Astronomical Journal 154 (6): 220. doi:10.3847/1538-3881/aa9332. Bibcode: 2017AJ....154..220F. 
  4. ↑ Sarkka, Simo; Solin, Arno; Hartikainen, Jouni (2013). "Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering". IEEE Signal Processing Magazine 30 (4): 51–61. doi:10.1109/MSP.2013.2246292. https://ieeexplore.ieee.org/document/6530736. Retrieved 2 September 2021. 
  5. ↑ Quiñonero-Candela, Joaquin; Rasmussen, Carl Edward (5 December 2005). "A Unifying View of Sparse Approximate Gaussian Process Regression". Journal of Machine Learning Research 6: 1939–1959. http://www.jmlr.org/papers/v6/quinonero-candela05a.html. Retrieved 23 May 2020. 
  6. ↑ 6.0 6.1 Ambikasaran, S.; Foreman-Mackey, D.; Greengard, L.; Hogg, D. W.; O’Neil, M. (1 Feb 2016). "Fast Direct Methods for Gaussian Processes". IEEE Transactions on Pattern Analysis and Machine Intelligence 38 (2): 252–265. doi:10.1109/TPAMI.2015.2448083. PMID 26761732. 
  7. ↑ 7.0 7.1 7.2 Vanhatalo, Jarno; Riihimäki, Jaakko; Hartikainen, Jouni; Jylänki, Pasi; Tolvanen, Ville; Vehtari, Aki (Apr 2013). "GPstuff: Bayesian Modeling with Gaussian Processes". Journal of Machine Learning Research 14: 1175−1179. http://jmlr.csail.mit.edu/papers/v14/vanhatalo13a.html. Retrieved 23 May 2020. 
  8. ↑ Rasmussen, Carl Edward; Nickisch, Hannes (Nov 2010). "Gaussian processes for machine learning (GPML) toolbox". Journal of Machine Learning Research 11 (2): 3011–3015. doi:10.1016/0002-9610(74)90157-3. PMID 4204594. 
  9. ↑ 9.0 9.1 Matthews, Alexander G. de G.; van der Wilk, Mark; Nickson, Tom; Fujii, Keisuke; Boukouvalas, Alexis; León-Villagrá, Pablo; Ghahramani, Zoubin; Hensman, James (April 2017). "GPflow: A Gaussian process library using TensorFlow". Journal of Machine Learning Research 18 (40): 1–6. http://jmlr.org/papers/v18/16-537.html. Retrieved 6 July 2020. 
  10. ↑ Gardner, Jacob R; Pleiss, Geoff; Bindel, David; Weinberger, Kilian Q; Wilson, Andrew Gordon (2018). "GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration". Advances in Neural Information Processing Systems 31: 7576–7586. http://papers.nips.cc/paper/7985-gpytorch-blackbox-matrix-matrix-gaussian-process-inference-with-gpu-acceleration.pdf. Retrieved 23 May 2020. 
  11. ↑ Zilber, Daniel; Katzfuss, Matthias (January 2021). "Vecchia–Laplace approximations of generalized Gaussian processes for big non-Gaussian spatial data". Computational Statistics & Data Analysis 153: 107081. doi:10.1016/j.csda.2020.107081. ISSN 0167-9473. https://www.sciencedirect.com/science/article/pii/S0167947320301729. Retrieved 1 September 2021. 
  12. ↑ Neumann, Marion; Huang, Shan; E. Marthaler, Daniel; Kersting, Kristian (2015). "pyGPs — A Python Library for Gaussian Process Regression and Classification". Journal of Machine Learning Research 16: 2611–2616. http://jmlr.org/papers/v16/neumann15a.html. 
  13. ↑ Kalaitzis, Alfredo; Lawrence, Neil D. (May 20, 2011). "A Simple Approach to Ranking Differentially Expressed Gene Expression Time Courses through Gaussian Process Regression". BMC Bioinformatics 12 (1): 180. doi:10.1186/1471-2105-12-180. ISSN 1471-2105. PMID 21599902. 
  14. ↑ Novak, Roman; Xiao, Lechao; Hron, Jiri; Lee, Jaehoon; Alemi, Alexander A.; Sohl-Dickstein, Jascha; Schoenholz, Samuel S. (2020). "Neural Tangents: Fast and Easy Infinite Neural Networks in Python". International Conference on Learning Representations. 
  15. ↑ Roustant, Olivier; Ginsbourger, David; Deville, Yves (2012). "DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization". Journal of Statistical Software 51 (1): 1–55. doi:10.18637/jss.v051.i01. https://www.jstatsoft.org/v51/i01/. 
  16. ↑ Baudin, Michaël; Dutfoy, Anne; Iooss, Bertrand; Popelin, Anne-Laure (2015). "OpenTURNS: An Industrial Software for Uncertainty Quantification in Simulation". Handbook of Uncertainty Quantification. pp. 1–38. doi:10.1007/978-3-319-11259-6_64-1. ISBN 978-3-319-11259-6. 
  17. ↑ Marelli, Stefano; Sudret, Bruno (2014). "UQLab: a framework for uncertainty quantification in MATLAB". Vulnerability, Uncertainty, and Risk. Quantification, Mitigation, and Management: 2554–2563. doi:10.3929/ethz-a-010238238. https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/379365/eth-14488-01.pdf?sequence=1&isAllowed=y. Retrieved 28 May 2020. 
  18. ↑ Couckuyt, Ivo; Dhaene, Tom; Demeester, Piet (2014). "ooDACE toolbox: a flexible object-oriented Kriging implementation". Journal of Machine Learning Research 15: 3183–3186. http://www.jmlr.org/papers/volume15/couckuyt14a/couckuyt14a.pdf. Retrieved 8 July 2020. 

External links

  • [1] The website hosting C. E. Rasmussen's book Gaussian processes for machine learning; contains a (partially outdated) list of software.



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