×
Anderes, Ethan B.; Stein, Michael L.

Estimating deformations of isotropic Gaussian random fields on the plane. (English) Zbl 1133.62077

Ann. Stat. 36, No. 2, 719-741 (2008).
Summary: This paper presents a new approach to the estimation of the deformation of an isotropic Gaussian random field on \(\mathbb R^{2}\) based on dense observations of a single realization of the deformed random field. Under this framework we investigate the identification and estimation of deformations. We then present a complete methodological package, from model assumptions to algorithmic recovery of the deformation, for the class of nonstationary processes obtained by deforming isotropic Gaussian random fields.

Cited in 32 Documents

MSC:

62M40 Random fields; image analysis
62M30 Inference from spatial processes
60G60 Random fields

Keywords:

quasiconformal maps; nonstationary random fields; diffeomorphisms with bounded distortion; complex dilations
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Ahlfors, L. V. (1966). Lectures on Quasiconformal Mappings . Van Nostrand, Toronto. · Zbl 0138.06002
[2] Anderes, E. B. and Stein, M. L. (2005). Estimating deformations of isotropic gaussian random fields on the plane. Technical report, Center for Integrating Statistical and Environmental Science. Available at http://galton.uchicago.edu/ cises/research/cises-tr27.pdf. · Zbl 1133.62077
[3] Chilès, J. and Delfiner, P. (1999). Geostatistics : Modeling Spatial Uncertainty . Wiley, New York. · Zbl 0922.62098
[4] Clerc, M. and Mallat, S. (2002). The texture gradient equation for recovering shape from texture. IEEE Trans. Pattern Analysis and Machine Intelligence 24 536-549.
[5] Clerc, M. and Mallat, S. (2003). Estimating deformations of stationary processes. Ann. Statist. 31 1772-1821. · Zbl 1052.62086 · doi:10.1214/aos/1074290327
[6] Cressie, N. (1993). Statistics for Spatial Data , rev. ed. Wiley, New York. · Zbl 0825.62477
[7] Damian, D., Sampson, P. and Guttorp, P. (2001). Bayesian estimation of semi-parametric non-stationary spatial covariance structures. Environmetrics 12 161-178.
[8] Gardiner, F. P. and Lakic, N. (2000). Quasiconformal Teichmüller Theory . Amer. Math. Soc., Providence, RI. · Zbl 0949.30002
[9] Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations . Johns Hopkins Univ. Press. · Zbl 0865.65009
[10] Guyon, X. and Perrin, O. (2000). Identification of space deformation using linear and superficial quadratic variations. Statist. Probab. Lett. 47 307-316. · Zbl 1054.60503 · doi:10.1016/S0167-7152(99)00171-6
[11] Iovleff, S. and Perrin, O. (2004). Estimating a nonstationary spatial structure using simulated annealing. J. Comput. Graph. Statist. 13 90-105. · doi:10.1198/1061860043100
[12] Krantz, S. G. (2004). Complex Analysis : The Geometric Viewpoint, 2nd ed. Mathematical Association of America, Washington, DC. · Zbl 1051.30001
[13] Krushkal’, S. L. (1979). Quasiconformal Mappings and Riemann Surfaces . Winston, Washington, DC. · Zbl 0479.30012
[14] Ławrynowicz, J. (1983). Quasiconformal Mappings in the Plane : Parametrical Methods . Springer, Berlin. · Zbl 0503.30013
[15] Lehto, O. and Virtanen, K. I. (1965). Quasiconformal Mappings in the Plane . Springer, New York. · Zbl 0267.30016
[16] Perrin, O. and Meiring, W. (1999). Identifiability for nonstationary spatial structure. J. Appl. Probab. 36 1244-1250. · Zbl 0993.60047 · doi:10.1239/jap/1032374771
[17] Rabiner, L. and Juang, B. (1993). Fundamentals of Speech Recognition . Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0762.62036
[18] Sampson, P. and Guttorp, P. (1992). Nonparametric estimation of nonstationary spatial covariance structure. J. Amer. Statist. Assoc. 87 108-119.
[19] Schmidt, A. and O’Hagan, A. (2003). Bayesian inference for nonstationary spatial covariance structure via spatial deformations. J. Roy. Statist. Soc. Ser. B 65 745-758. · Zbl 1063.62034 · doi:10.1111/1467-9868.00413
[20] Silverman, R. A. (1974). Complex Analysis with Applications . Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0348.30001
[21] Stein, M. L. (1999). Interpolation of Spatial Data : Some Theory for Kriging . Springer, New York. · Zbl 0924.62100
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.