## Annals of Statistics

### Adaptive estimation of stationary Gaussian fields

Nicolas Verzelen

#### Abstract

We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures like AIC are proved to achieve optimal model selection among autoregressive models. However, there exists no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection procedure for spatial fields. For all neighborhoods m in a given collection $\mathcal{M}$, this procedure first amounts to computing a covariance estimator of X within the GMRFs of neighborhood m. Then it selects a neighborhood ̂m by applying a penalization strategy. The so-defined method satisfies a nonasymptotic oracle-type inequality. If X is a GMRF, the procedure is also minimax adaptive to the sparsity of its neighborhood. More generally, the procedure is adaptive to the rate of approximation of the true distribution by GMRFs with growing neighborhoods.

#### Article information

Source
Ann. Statist., Volume 38, Number 3 (2010), 1363-1402.

Dates
First available in Project Euclid: 8 March 2010

https://projecteuclid.org/euclid.aos/1268056620

Digital Object Identifier
doi:10.1214/09-AOS751

Mathematical Reviews number (MathSciNet)
MR2662346

Zentralblatt MATH identifier
1189.62157

Subjects
Primary: 62H11: Directional data; spatial statistics
Secondary: 62M40: Random fields; image analysis

#### Citation

Verzelen, Nicolas. Adaptive estimation of stationary Gaussian fields. Ann. Statist. 38 (2010), no. 3, 1363--1402. doi:10.1214/09-AOS751. https://projecteuclid.org/euclid.aos/1268056620

#### References

• [1] Aykroyd, R. (1998). Bayesian estimation for homogeneous and inhomogeneous Gaussian random fields. IEEE Trans. Pattern Anal. Machine Intell. 20 533–539.
• [2] Besag, J. E. (1975). Statistical analysis of non-lattice data. Statistica 24 179–195.
• [3] Besag, J. E. (1977). Efficiency of pseudolikelihood estimation for simple Gaussian fields. Biometrika 64 616–618.
• [4] Besag, J. E. and Kooperberg, C. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733–746.
• [5] Besag, J. E. and Moran, P. A. P. (1975). On the estimation and testing of spatial interaction in Gaussian lattice processes. Biometrika 62 555–562.
• [6] Birgé, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc. (JEMS) 3 203–268.
• [7] Birgé, L. and Massart, P. (2007). Minimal penalties for Gaussian model selection. Probab. Theory Related Fields 138 33–73.
• [8] Boucheron, S., Bousquet, O., Lugosi, G. and Massart, P. (2005). Moment inequalities for functions of independent random variables. Ann. Probab. 33 514–560.
• [9] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
• [10] Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
• [11] Cressie, N. A. C. and Verzelen, N. (2008). Conditional-mean least-squares of Gaussian Markov random fields to Gaussian fields. Comput. Statist. Data Anal. 52 2794–2807.
• [12] Crouse, M., Nowak, R. and Baraniuk, R. (1998). Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Process. 46 886–902.
• [13] Dass, S. C. and Nair, V. N. (2003). Edge detection, spatial smoothing, and image reconstruction with partially observed multivariate data. J. Amer. Statist. Assoc. 98 77–89.
• [14] Edwards, D. (2000). Introduction to Graphical Modelling, 2nd ed. Springer, New York.
• [15] Gray, R. (2006). Toeplitz and Circulant Matrices: A Review, rev. ed. Now Publishers, Norwell, MA.
• [16] Guyon, X. (1987). Estimation d’un champ par pseudo-vraisemblance conditionnelle: Étude asymptotique et application au cas Markovien. In Spatial processes and spatial time series analysis (Brussels, 1985). Travaux Rech. 11 15–62. Publ. Fac. Univ. Saint-Louis, Brussels.
• [17] Guyon, X. (1995). Random Fields on a Network. Springer, New York.
• [18] Guyon, X. and Yao, J. (1999). On the underfitting and overfitting sets of models chosen by order selection criteria. J. Multivariate Anal. 70 221–249.
• [19] Hall, P., Fisher, N. and Hoffmann, B. (1994). On the nonparametric estimation of covariance functions. Ann. Statist. 22 2115–2134.
• [20] Hurvich, C. and Tsai, C.-L. (1989). Regression and time series model selection in small samples. Biometrika 76 297–307.
• [21] Im, H., Stein, M. and Zhu, Z. (2007). Semiparametric estimation of spectral density with irregular observations. J. Amer. Statist. Assoc. 102 726–735.
• [22] Kashyap, R. and Chellapa, R. (1984). Estimation and choice of neighbors in spatial-interaction models of images. IEEE Trans. Inform. Theory 29 60–72.
• [23] Lakshmanan, S. and Derin, H. (1993). Valid parameter space for 2-D Gaussian Markov random fields. IEEE Trans. Inform. Theory 39 703–709.
• [24] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Oxford Univ. Press, New York.
• [25] Massart, P. (2007). Concentration Inequalities and Model Selection. Lecture Notes in Math. 1896. Springer, Berlin.
• [26] McQuarrie, A. D. R. and Tsai, C.-L. (1998). Regression and Time Series Model Selection. World Scientific, River Edge, NJ.
• [27] Portilla, J., Strela, V., Wainwright, M. J. and Simoncelli, E. P. (2003). Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12 1338–1351.
• [28] Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation. Electron. J. Stat. 2 494–515.
• [29] Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability 104. Chapman & Hall/CRC, London.
• [30] Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 319–392.
• [31] Rue, H. and Tjelmeland, H. (2002). Fitting Gaussian Markov random fields to Gaussian fields. Scand. J. Statist. 29 31–49.
• [32] Shibata, R. (1980). Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Ann. Statist. 8 147–164.
• [33] Song, H.-R., Fuentes, M. and Ghosh, S. (2008). A comparative study of Gaussian geostatistical models and Gaussian Markov random field models. J. Multivariate Anal. 99 1681–1697.
• [34] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
• [35] Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563.
• [36] Verzelen, N. (2009). Technical Appendix to “Adaptive estimation of stationary Gaussian fields.” Available at arXiv:0908.4586.
• [37] Verzelen, N. (2010). Data-driven neighborhood selection of a Gaussian field. Comput. Statist. Data Anal. To appear.
• [38] Yu, B. (1997). Assouad, Fano and Le Cam. In Festschrift for Lucien Le Cam 423–435. Springer, New York.