The Annals of Statistics

Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations

Wei-Liem Loh

Abstract

This article introduces a method for estimating the smoothness of a stationary, isotropic Gaussian random field from irregularly spaced data. This involves novel constructions of higher-order quadratic variations and the establishment of the corresponding fixed-domain asymptotic theory. In particular, we consider:

(i) higher-order quadratic variations using nonequispaced line transect data,

(ii) second-order quadratic variations from a sample of Gaussian random field observations taken along a smooth curve in $\mathbb{R}^{2}$,

(iii) second-order quadratic variations based on deformed lattice data on $\mathbb{R}^{2}$.

Smoothness estimators are proposed that are strongly consistent under mild assumptions. Simulations indicate that these estimators perform well for moderate sample sizes.

Article information

Source
Ann. Statist., Volume 43, Number 6 (2015), 2766-2794.

Dates
Revised: July 2015
First available in Project Euclid: 7 October 2015

https://projecteuclid.org/euclid.aos/1444222092

Digital Object Identifier
doi:10.1214/15-AOS1365

Mathematical Reviews number (MathSciNet)
MR3405611

Zentralblatt MATH identifier
1327.62482

Subjects
Primary: 62M30: Spatial processes
Secondary: 62M40: Random fields; image analysis

Citation

Loh, Wei-Liem. Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations. Ann. Statist. 43 (2015), no. 6, 2766--2794. doi:10.1214/15-AOS1365. https://projecteuclid.org/euclid.aos/1444222092

References

• [1] Adler, R. J. and Pyke, R. (1993). Uniform quadratic variation for Gaussian processes. Stochastic Process. Appl. 48 191–209.
• [2] Anderes, E. and Chatterjee, S. (2009). Consistent estimates of deformed isotropic Gaussian random fields on the plane. Ann. Statist. 37 2324–2350.
• [3] Anderes, E. B. and Stein, M. L. (2008). Estimating deformations of isotropic Gaussian random fields on the plane. Ann. Statist. 36 719–741.
• [4] Begyn, A. (2005). Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10 691–717 (electronic).
• [5] Benassi, A., Cohen, S., Istas, J. and Jaffard, S. (1998). Identification of filtered white noises. Stochastic Process. Appl. 75 31–49.
• [6] Chan, G. and Wood, A. T. A. (2000). Increment-based estimators of fractal dimension for two-dimensional surface data. Statist. Sinica 10 343–376.
• [7] Chilès, J.-P. and Delfiner, P. (1999). Geostatistics. Modeling Spatial Uncertainty. Wiley, New York.
• [8] Cohen, S., Guyon, X., Perrin, O. and Pontier, M. (2006). Singularity functions for fractional processes: Application to the fractional Brownian sheet. Ann. Inst. Henri Poincaré Probab. Stat. 42 187–205.
• [9] Constantine, A. G. and Hall, P. (1994). Characterizing surface smoothness via estimation of effective fractal dimension. J. Roy. Statist. Soc. Ser. B 56 97–113.
• [10] Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.
• [11] Grenander, U. and Miller, M. I. (2007). Pattern Theory: From Representation to Inference. Oxford Univ. Press, Oxford.
• [12] Guyon, X. and León, J. (1989). Convergence en loi des $H$-variations d’un processus gaussien stationnaire sur ${\mathbf{R}}$. Ann. Inst. Henri Poincaré Probab. Stat. 25 265–282.
• [13] Hall, P. and Wood, A. (1993). On the performance of box-counting estimators of fractal dimension. Biometrika 80 246–252.
• [14] Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42 1079–1083.
• [15] Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge Univ. Press, Cambridge.
• [16] Im, H. K., Stein, M. L. and Zhu, Z. (2007). Semiparametric estimation of spectral density with irregular observations. J. Amer. Statist. Assoc. 102 726–735.
• [17] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Stat. 33 407–436.
• [18] Joshi, S. C. and Miller, M. I. (2000). Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Process. 9 1357–1370.
• [19] Kent, J. T. and Wood, A. T. A. (1997). Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 679–699.
• [20] Klein, R. and Giné, E. (1975). On quadratic variation of processes with Gaussian increments. Ann. Probab. 3 716–721.
• [21] Knuth, D. E. (1997). The Art of Computer Programming, 3rd ed. Fundamental Algorithms 1. Addison-Wesley, Reading, MA.
• [22] Lévy, P. (1940). Le mouvement brownien plan. Amer. J. Math. 62 487–550.
• [23] Loh, W.-L. (2015). Supplement to “Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations”. DOI:10.1214/15-AOS1365SUPP.
• [24] Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications. Ecole des Mines, Fontainebleau.
• [25] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
• [26] Stein, M. L. (2012). Simulation of Gaussian random fields with one derivative. J. Comput. Graph. Statist. 21 155–173.
• [27] Sylvester, J. J. (1857). On the partition of numbers. Quart. J. Math. 1 141–152.
• [28] Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in $[0,1]^{d}$. J. Comput. Graph. Statist. 3 409–432.