Electronic Journal of Statistics

Local inversion-free estimation of spatial Gaussian processes

Hossein Keshavarz, XuanLong Nguyen, and Clayton Scott

Full-text: Open access


Maximizing the likelihood has been widely used for estimating the unknown covariance parameters of spatial Gaussian processes. However, evaluating and optimizing the likelihood function can be computationally intractable, particularly for large number of (possibly) irregularly spaced observations, due to the need to handle the inverse of ill-conditioned and large covariance matrices. Extending the “inversion-free” method of Anitescu, Chen and Stein [1], we investigate a broad class of covariance parameter estimation based on inversion-free surrogate losses and block diagonal approximation schemes of the covariance structure. This class of estimators yields a spectrum for negotiating the trade-off between statistical accuracy and computational cost. We present fixed-domain asymptotic properties of our proposed method, establishing $\sqrt{n}$-consistency and asymptotic normality results for isotropic Matern Gaussian processes observed on a multi-dimensional and irregular lattice. Simulation studies are also presented for assessing the scalability and statistical efficiency of the proposed algorithm for large data sets.

Article information

Electron. J. Statist., Volume 13, Number 2 (2019), 4224-4279.

Received: November 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document

Digital Object Identifier

Primary: 62M30: Spatial processes 62M40: Random fields; image analysis
Secondary: 60G15: Gaussian processes

Local inversion-free covariance estimation Gaussian process computationally scalable fixed-domain asymptotic analysis irregularly spaced observations

Creative Commons Attribution 4.0 International License.


Keshavarz, Hossein; Nguyen, XuanLong; Scott, Clayton. Local inversion-free estimation of spatial Gaussian processes. Electron. J. Statist. 13 (2019), no. 2, 4224--4279. doi:10.1214/19-EJS1592. https://projecteuclid.org/euclid.ejs/1571709694

Export citation


  • [1] E. Anderes. On the consistent separation of scale and variance for gaussian random fields., The Annals of Statistics, 38(2):870–893, 2010.
  • [2] M. Anitescu, J. Chen, and M. L. Stein. An inversion-free estimating equations approach for gaussian process models., Journal of Computational and Graphical Statistics, 26(1):98–107, 2017.
  • [3] M. Anitescu, J. Chen, and L. Wang. A matrix-free approach for solving the parametric gaussian process maximum likelihood problem., SIAM Journal on Scientific Computing, 34(1):A240–A262, 2012.
  • [4] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. A limited memory algorithm for bound constrained optimization., SIAM Journal on Scientific Computing, 16(5) :1190–1208, 1995.
  • [5] J. Chen. On the use of discrete laplace operator for preconditioning kernel matrices., SIAM Journal on Scientific Computing, 35(2):A577–A602, 2013.
  • [6] N. Cressie. Statistics for spatial data., Terra Nova, 4(5):613–617, 1992.
  • [7] R. Furrer, M. G. Genton, and D. Nychka. Covariance tapering for interpolation of large spatial datasets., Journal of Computational and Graphical Statistics, 15(3):502–523, 2006.
  • [8] A. E. Gelfand, P. Diggle, P. Guttorp, and M. Fuentes., Handbook of spatial statistics. CRC Press, 2010.
  • [9] C. Kaufman and B. Shaby. The role of the range parameter for estimation and prediction in geostatistics., Biometrika, 100(2):473–484, 2013.
  • [10] C. G. Kaufman, M. J. Schervish, and D. W. Nychka. Covariance tapering for likelihood-based estimation in large spatial data sets., Journal of the American Statistical Association, 103(484) :1545–1555, 2008.
  • [11] H. Keshavarz, C. Scott, and X. Nguyen. On the consistency of inversion-free parameter estimation for gaussian random fields., Journal of Multivariate Analysis, 150:245–266, 2016.
  • [12] H. Keshavarz, C. Scott, and X. Nguyen. Optimal change point detection in gaussian processes., Journal of Statistical Planning and Inference, 193:151–178, 2018.
  • [13] H. Keshavarz Shenastaghi. Detection and estimation in gaussian random fields: Minimax theory and efficient algorithms., 2017.
  • [14] M. Lee., Local properties of irregularly observed Gaussian fields, volume 74. 2012.
  • [15] M. Rudelson and R. Vershynin. Hanson-wright inequality and sub-gaussian concentration., Electronic Communications in Probability, 18, 2013.
  • [16] M. L. Stein., Interpolation of spatial data: some theory for kriging. Springer Science & Business Media, 2012.
  • [17] M. L. Stein, J. Chen, and M. Anitescu. Difference filter preconditioning for large covariance matrices., SIAM Journal on Matrix Analysis and Applications, 33(1):52–72, 2012.
  • [18] M. L. Stein, J. Chen, M. Anitescu, et al. Stochastic approximation of score functions for gaussian processes., The Annals of Applied Statistics, 7(2) :1162–1191, 2013.
  • [19] M. L. Stein, Z. Chi, and L. J. Welty. Approximating likelihoods for large spatial data sets., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(2):275–296, 2004.
  • [20] A. V. Vecchia. Estimation and model identification for continuous spatial processes., Journal of the Royal Statistical Society. Series B (Methodological), pages 297–312, 1988.
  • [21] D. Wang and W.-L. Loh. On fixed-domain asymptotics and covariance tapering in gaussian random field models., Electronic Journal of Statistics, 5:238–269, 2011.
  • [22] Z. Ying. Asymptotic properties of a maximum likelihood estimator with data from a gaussian process., Journal of Multivariate Analysis, 36(2):280–296, 1991.
  • [23] H. Zhang. Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics., Journal of the American Statistical Association, 99(465):250–261, 2004.