×

zbMATH — the first resource for mathematics

Semi-parametric estimation of the variogram scale parameter of a Gaussian process with stationary increments. (English) Zbl 1461.60019
Summary: We consider the semi-parametric estimation of the scale parameter of the variogram of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based both on quadratic variations and the moment method. We provide asymptotic approximations of the mean and variance of this estimator, together with asymptotic normality results, for a large class of Gaussian processes. We allow for general mean functions, provide minimax upper bounds and study the aggregation of several estimators based on various variation sequences. In extensive simulation studies, we show that the asymptotic results accurately depict the finite-sample situations already for small to moderate sample sizes. We also compare various variation sequences and highlight the efficiency of the aggregation procedure.
MSC:
60G15 Gaussian processes
62F12 Asymptotic properties of parametric estimators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R.J. Adler and R. Pyke, Uniform quadratic variation for Gaussian processes. Stochastic Process. Appl. 48 (1993) 191-209 · Zbl 0783.60040
[2] J.-M. Azaïs and M. Wschebor, Level sets and Extrema of Random Processes and Fields. John Wiley & Sons, Inc., Hoboken, NJ (2009) · Zbl 1168.60002
[3] F. Bachoc, Cross validation and maximum likelihood estimations of hyper-parameters of Gaussian processes with model misspecification. Computat. Stat. Data Anal. 66 (2013) 55-69 · Zbl 06958972
[4] F. Bachoc, Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes. J. Multivariate Anal. 125 (2014) 1-35 · Zbl 1280.62100
[5] F. Bachoc, Asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case. Bernoulli 24 (2018) 1531-1575 · Zbl 1429.60035
[6] F. Bachoc and A. Lagnoux, Fixed-domain asymptotic properties of maximum composite likelihood estimators for Gaussian processes. J. Stat. Plann. Inference 209 (2020) 62-75 · Zbl 1441.62191
[7] J.M. Bates and C.W. Granger, The combination of forecasts. J. Oper. Res. Soc. 20 (1969) 451-468
[8] G. Baxter, A strong limit theorem for Gaussian processes. Proc. Am. Math. Soc. 7 (1956) 522-527 · Zbl 0070.36304
[9] Y. Cao and D.J. Fleet, Generalized product of experts for automatic and principled fusion of Gaussian process predictions, in Modern Nonparametrics 3: Automating the Learning Pipeline workshop at NIPS, Montreal. Preprint (2014)
[10] I. Clark, Practical Geostatistics, Vol. 3. Applied Science Publishers, London (1979) · Zbl 1038.86008
[11] J.-F. Coeurjolly, Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199-227 · Zbl 0984.62058
[12] S. Cohen and J. Istas, Fractional Fields and Applications, With a foreword by Stéphane Jaffard. Vol. 73 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2013)
[13] N. Cressie, Statistics for Spatial Data. John Wiley (1993) · Zbl 1347.62005
[14] N. Cressie and D.M. Hawkins, Robust estimation of the variogram: I. J. Int. Assoc. Math. Geol. 12 (1980) 115-125
[15] R. Dahlhaus, Efficient parameter estimation for self-similar processes. Ann. Stat. (1989) 1749-1766 · Zbl 0703.62091
[16] A. Datta, S. Banerjee, A.O. Finley, and A.E. Gelfand, Hierarchical nearest-neighbor Gaussian process models for large geostatistical datasets. J. Am. Stat. Assoc. 111 (2016) 800-812
[17] I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909-996 · Zbl 0644.42026
[18] M. David, Geostatistical Ore Reserve Estimation. Elsevier (2012)
[19] M.P. Deisenroth and J.W. Ng, Distributed Gaussian processes, In Proceedings of the 32nd International Conference on Machine Learning, Lille, France. JMLR: W&CP volume 37 (2015)
[20] R. Furrer, M.G. Genton and D. Nychka, Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Stat. 15 (2006) 502-523
[21] E.G. Gladyšev, A new limit theorem for stochastic processes with Gaussian increments. Teor. Verojatnost. Primenen. 6 (1961) 57-66
[22] U. Grenander, Abstract inference. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York (1981)
[23] X. Guyon and J. León, Convergence en loi des H-variations d’un processus gaussien stationnaire sur R. Ann. Inst. Henri Poincaré Probab. Statist. 25 (1989) 265-282 · Zbl 0691.60017
[24] P. Hall, N.I. Fisher and B. Hoffmann, On the nonparametric estimation of covariance functions. Ann. Stat. 22 (1994) 2115-2134 · Zbl 0828.62036
[25] P. Hall and P. Patil, Properties of nonparametric estimators of autocovariance for stationary random fields. Probab. Theory Related Fields 99 (1994) 399-424 · Zbl 0799.62102
[26] J. Han and X.-P. Zhang, Financial time series volatility analysis using Gaussian process state-space models, in 2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE (2015) 358-362
[27] J. Hensman and N. Fusi, Gaussian processes for big data. Uncertainty Artif. Intell. (2013) 282-290
[28] G.E. Hinton, Training products of experts by minimizing contrastive divergence. Neural Computat. 14 (2002) 1771-1800 · Zbl 1010.68111
[29] I. Ibragimov and Y. Rozanov, Gaussian Random Processes. Springer-Verlag, New York (1978) · Zbl 0392.60037
[30] J. Istas and G. Lang, Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Statist. 33 (1997) 407-436 · Zbl 0882.60032
[31] A. Journel and C. Huijbregts, Mining geostatistics, in Bureau De Recherches Geologiques Et Miniers, France Academic Pres Harcout Brace & Company, Publishers, London, San Diego, New York, Boston, Sidney, Toronto (1978)
[32] C.G. Kaufman, M.J. Schervish, and D.W. Nychka, Covariance tapering for likelihood-based estimation in large spatial data sets. J. Am. Stat. Assoc. 103 (2008) 1545-1555 · Zbl 1286.62072
[33] J.T. Kent and A.T.A. Wood, Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 (1997) 679-699 · Zbl 0889.62072
[34] S.N. Lahiri, Y. Lee and N. Cressie, On asymptotic distribution and asymptotic efficiency of least squares estimators of spatial variogram parameters. J. Stat. Plann. Inference 103 (2002) 65-85 · Zbl 0989.62049
[35] G. Lang and F. Roueff, Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inference Stoch. Process. 4 (2001) 283-306 · Zbl 1008.62081
[36] F. Lavancier and P. Rochet. A general procedure to combine estimators. Computat. Stat. Data Anal. 94 (2016) 175-192 · Zbl 06918660
[37] P. Lévy, Le mouvement brownien plan. Am. J. Math. 62 (1940) 487-550 · Zbl 0024.13906
[38] D. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley (1979) · Zbl 0458.93001
[39] J. Mateu, E. Porcu, G. Christakos and M. Bevilacqua, Fitting negative spatial covariances to geothermal field temperatures in Nea Kessani (Greece). Environmetrics 18 (2007) 759-773
[40] G. Matheron, Traité de géostatistique appliquée, Tome I, Vol. 14 of Editions Technip, Paris. Mémoires du Bureau de Recherches Géologiques et Minières (1962)
[41] E. Pardo-Igúzquiza and P.A. Dowd, AMLE3D: a computer program for the inference of spatial covariance parameters by approximate maximum likelihood estimation. Comput. Geosci. 23 (1997) 793-805
[42] O. Perrin, Quadratic variation for Gaussian processes and application to time deformation. Stochastic Process. Appl. 82 (1999) 293-305 · Zbl 0997.60038
[43] G. Pólya, Remarks on characteristic functions, in Proceedings of the First Berkeley Symposium on Mathematical Statistics and Probability, August 13-18, 1945 and January 27-29, 1946, edited by J. Neyman. Statistical Laboratory of the University of California, Berkeley. University of California Press, Berkeley, CA (1949) 115-123
[44] C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006) · Zbl 1177.68165
[45] F. Richard, Anisotropy of Hölder Gaussian random fields: characterization, estimation, and application to image textures. Stat. Comput. 28 (2018) 1155-1168 · Zbl 1430.62215
[46] F. Richard and H. Biermé, Statistical tests of anisotropy for fractional Brownian textures. application to full-field digital mammography. J. Math. Imaging Vis. 36 (2010) 227-240
[47] L. Risser, F. Vialard, R. Wolz, M. Murgasova, D. Holm and D. Rueckert, ADNI: Simultaneous multiscale registration using large deformation diffeomorphic metric mapping. IEEE Trans. Med. Imaging 30 (2011) 1746-1759
[48] O. Roustant, D. Ginsbourger and Y. Deville, DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51 (2012)
[49] H. Rue and L. Held, Gaussian Markov Random Fields Theory and Applications. Chapman & Hall (2005) · Zbl 1093.60003
[50] D. Rullière, N. Durrande, F. Bachoc and C. Chevalier, Nested Kriging predictions for datasets with a large number of observations. Stat. Comput. 28 (2018) 849-867 · Zbl 1384.62246
[51] G. Samorodnitsky and M. Taqqu, Non-Gaussian Stable Processes: Stochastic Models with Infinite Variance. Chapman ft Hall, London (1994) · Zbl 0925.60027
[52] T.J. Santner, B.J. Williams and W.I. Notz, The Design and Analysis of Computer Experiments. Springer Series in Statistics. Springer-Verlag, New York (2003) · Zbl 1041.62068
[53] D. Slepian, On the zeros of Gaussian noise, in Proc. Sympos. Time Series Analysis (Brown Univ., 1962). Wiley, New York (1963) 104-115
[54] M. Stein, Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York (1999) · Zbl 0924.62100
[55] M.L. Stein, Limitations on low rank approximations for covariance matrices of spatial data. Spatial Stat. 8 (2014) 1-19
[56] M.L. Stein, Z. Chi and L.J. Welty, Approximating likelihoods for large spatial data sets. J. Roy. Stat. Soc.: Ser. B (Stat. Methodol.) 66 (2004) 275-296 · Zbl 1062.62094
[57] V. Tresp, A Bayesian committee machine. Neural Computat. 12 (2000) 2719-2741
[58] B. van Stein, H. Wang, W. Kowalczyk, T. Bäck and M. Emmerich, Optimally weighted cluster Kriging for big data regression, in International Symposium on Intelligent Data Analysis . Springer (2015) 310-321
[59] C. Varin, N. Reid and D. Firth, An overview of composite likelihood methods. Stat. Sinica 21 (2011) 5-42 · Zbl 05849508
[60] A.V. Vecchia, Estimation and model identification for continuous spatial processes. J. Roy. Stat. Soc.: Ser. B (Methodological) 50 (1988) 297-312
[61] Y. Wu, J.M. Hernández-Lobato and Z. Ghahramani, Gaussian process volatility model, in Advances in Neural Information Processing Systems 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger. Curran Associates Inc. (2014) 1044-1052
[62] H. Zhang and Y. Wang, Kriging and cross-validation for massive spatial data. Environmetrics 21 (2010) 290-304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.