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Local approximation of variograms by covariance functions
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 1303–1304 www.elsevier.com/locate/stapro
Local approximation of variograms by covariance functions Martin Schlathera,, Tilmann Gneitingb,1 a
Institut fu¨r Statistik und Quantitative O¨konomik, Helmut-Schmidt-Universita¨t, Postfach 700822, 22008 Hamburg, Germany b Department of Statistics, University of Washington, Seattle, Washington 98195-4322, USA Received 6 January 2006 Available online 9 March 2006
abstract This letter considers stationary local approximations to intrinsically stationary random functions on Rd . For any continuous variogram g and any ball in Rd , there exists a covariance function with compact support whose respective variogram is arbitrarily close to g when restricted to the ball. r 2006 Elsevier B.V. All rights reserved. Keywords: Geostatistics; Locally equivalent stationary covariance; Variogram
Geostatistical techniques model spatial data as the realizations of random functions on Euclidean spaces, typically imposing some form of stationarity condition on the random process Z ¼ fZðxÞ : x 2 Rd g, where ZðxÞ is a scalar variable associated with the location x 2 Rd (Chile`s and Delfiner, 1999; Stein, 1999). The process Z is stationary if the first two moments exist and are invariant under translation. One then defines the covariance function, CðhÞ ¼ covðZðxÞ; Zðx þ hÞÞ for h 2 Rd . Classical geostatistical theory (Matheron, 1973) relies on a weaker assumption, the intrinsic hypothesis. The random function Z is called intrinsically stationary if the increment process I h ¼ fZðxÞ Zðx þ hÞ : x 2 Rd g is stationary for all lag vectors h 2 Rd . Then EðZðxÞ Zðx þ hÞÞ and EðZðxÞ Zðx þ hÞÞ2 do not depend on x, and we may define the variogram or centered variogram (Gneiting et al., 2001), gðhÞ ¼ 12varðZðxÞ Zðx þ hÞÞ for h 2 Rd . Any stationary process is intrinsically stationary, but the converse is not necessarily true, with a fractional Brownian surface being one such example. If the variogram g is bounded, then there exists a covariance function C such that gðhÞ ¼ Cð0Þ CðhÞ d
(1)
for all h 2 R . If the variogram is unbounded, no covariance function can be found such that (1) is true globally. However, a stationary process with covariance function C might exist such that the relationship (1) holds locally, on a ball centered at the origin of Rd . Then C is said to be a locally equivalent stationary Corresponding author. Tel.: +49 040 6541 3404; fax: +49 040 6541 2315. 1
E-mail addresses: [email protected] (M. Schlather), [email protected] (T. Gneiting). Supported by National Science Foundation award 0134264.
0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.02.002
ARTICLE IN PRESS M. Schlather, T. Gneiting / Statistics & Probability Letters 76 (2006) 1303–1304
1304
covariance (Chile`s and Delfiner, 1999, pp. 267–270). Locally equivalent stationary covariances exist for many but not for all variograms (Gneiting et al., 2001; Stein, 2001). In this light, Barry (2005) considered local approximations of variograms by covariance functions.2 Assuming regularity conditions, he proved that variograms g in R2 can be approximated by compactly supported covariance functions C, in the sense of the estimate (2) below. Barry asked whether analogous results hold in higher dimensions. The following result solves the problem in very general settings, including all continuous variograms, and all variograms that are sums of a continuous variogram and a nugget effect. Theorem. For all positive numbers and H and for any variogram g on Rd that satisfies supkhkoH gðhÞo1, there exists a covariance function C with compact support such that sup jgðhÞ ðCð0Þ CðhÞÞjo.
(2)
khkoH
Proof. Given any t40, Schoenberg’s theorem (Gneiting et al., 2001, Theorem 1) implies that C t ðhÞ ¼ t1 expðtgðhÞÞ is a covariance function on Rd . Let C 0 be a continuous correlation function on Rd with the property that C 0 ðhÞ ¼ 0 if khkX1, and note that C 0 ðshÞ ! 1 as s ! 0, uniformly on compact subsets of Rd . Given s40 and t40, consider the covariance function C s;t ðhÞ ¼ C t ðhÞC 0 ðshÞ on Rd , which has compact support. Using the Taylor expansion of the exponential function, we see that gðhÞ ðC s;t ð0Þ C s;t ðhÞÞ ! 0 for h 2 Rd as both s ! 0 and t ! 0. Since supkhkoH gðhÞ is finite, the convergence is uniform on the ball of radius H centered at the origin of Rd ; hence, the covariance function C s;t satisfies (2) if s and t are sufficiently small. & The above theorem and proof generalize verbatim to non-centered variograms (Gneiting et al., 2001). Any non-centered variogram decomposes as the sum of a centered variogram and a nonnegative quadratic form.3 References Barry, R.P., 2005. The local approximation of variograms in R2 by covariance functions. Statist. Probab. Lett. 74, 171–177. Chile`s, J.-P., Delfiner, P., 1999. Geostatistics. Modeling Spatial Uncertainty. Wiley, New York. Gneiting, T., Sasva´ri, Z., Schlather, M., 2001. Analogies and correspondences between variograms and covariance functions. Adv. Appl. Probab. 33, 617–630. Matheron, G., 1973. The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–468. Stein, M.L., 1999. Interpolation of Spatial Data. Some Theory for Kriging. Springer, New York. Stein, M.L., 2001. Local stationarity and simulation of self-affine intrinsic random functions. IEEE Trans. Inform. Theory 47, 1385–1390.
2 3
Note that Barry omits the factor 12 in the definition of the variogram;P clearly, this factor is immaterial. In Gneiting et al. (2001, p. 619), the quadratic form is given as QðhÞ ¼ di¼1 ai h2i ; this should read QðhÞ ¼ kAhk2 for a matrix A 2 Rdd .
Statistics & Probability Letters 76 (2006) 1303–1304 www.elsevier.com/locate/stapro
Local approximation of variograms by covariance functions Martin Schlathera,, Tilmann Gneitingb,1 a
Institut fu¨r Statistik und Quantitative O¨konomik, Helmut-Schmidt-Universita¨t, Postfach 700822, 22008 Hamburg, Germany b Department of Statistics, University of Washington, Seattle, Washington 98195-4322, USA Received 6 January 2006 Available online 9 March 2006
abstract This letter considers stationary local approximations to intrinsically stationary random functions on Rd . For any continuous variogram g and any ball in Rd , there exists a covariance function with compact support whose respective variogram is arbitrarily close to g when restricted to the ball. r 2006 Elsevier B.V. All rights reserved. Keywords: Geostatistics; Locally equivalent stationary covariance; Variogram
Geostatistical techniques model spatial data as the realizations of random functions on Euclidean spaces, typically imposing some form of stationarity condition on the random process Z ¼ fZðxÞ : x 2 Rd g, where ZðxÞ is a scalar variable associated with the location x 2 Rd (Chile`s and Delfiner, 1999; Stein, 1999). The process Z is stationary if the first two moments exist and are invariant under translation. One then defines the covariance function, CðhÞ ¼ covðZðxÞ; Zðx þ hÞÞ for h 2 Rd . Classical geostatistical theory (Matheron, 1973) relies on a weaker assumption, the intrinsic hypothesis. The random function Z is called intrinsically stationary if the increment process I h ¼ fZðxÞ Zðx þ hÞ : x 2 Rd g is stationary for all lag vectors h 2 Rd . Then EðZðxÞ Zðx þ hÞÞ and EðZðxÞ Zðx þ hÞÞ2 do not depend on x, and we may define the variogram or centered variogram (Gneiting et al., 2001), gðhÞ ¼ 12varðZðxÞ Zðx þ hÞÞ for h 2 Rd . Any stationary process is intrinsically stationary, but the converse is not necessarily true, with a fractional Brownian surface being one such example. If the variogram g is bounded, then there exists a covariance function C such that gðhÞ ¼ Cð0Þ CðhÞ d
(1)
for all h 2 R . If the variogram is unbounded, no covariance function can be found such that (1) is true globally. However, a stationary process with covariance function C might exist such that the relationship (1) holds locally, on a ball centered at the origin of Rd . Then C is said to be a locally equivalent stationary Corresponding author. Tel.: +49 040 6541 3404; fax: +49 040 6541 2315. 1
E-mail addresses: [email protected] (M. Schlather), [email protected] (T. Gneiting). Supported by National Science Foundation award 0134264.
0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.02.002
ARTICLE IN PRESS M. Schlather, T. Gneiting / Statistics & Probability Letters 76 (2006) 1303–1304
1304
covariance (Chile`s and Delfiner, 1999, pp. 267–270). Locally equivalent stationary covariances exist for many but not for all variograms (Gneiting et al., 2001; Stein, 2001). In this light, Barry (2005) considered local approximations of variograms by covariance functions.2 Assuming regularity conditions, he proved that variograms g in R2 can be approximated by compactly supported covariance functions C, in the sense of the estimate (2) below. Barry asked whether analogous results hold in higher dimensions. The following result solves the problem in very general settings, including all continuous variograms, and all variograms that are sums of a continuous variogram and a nugget effect. Theorem. For all positive numbers and H and for any variogram g on Rd that satisfies supkhkoH gðhÞo1, there exists a covariance function C with compact support such that sup jgðhÞ ðCð0Þ CðhÞÞjo.
(2)
khkoH
Proof. Given any t40, Schoenberg’s theorem (Gneiting et al., 2001, Theorem 1) implies that C t ðhÞ ¼ t1 expðtgðhÞÞ is a covariance function on Rd . Let C 0 be a continuous correlation function on Rd with the property that C 0 ðhÞ ¼ 0 if khkX1, and note that C 0 ðshÞ ! 1 as s ! 0, uniformly on compact subsets of Rd . Given s40 and t40, consider the covariance function C s;t ðhÞ ¼ C t ðhÞC 0 ðshÞ on Rd , which has compact support. Using the Taylor expansion of the exponential function, we see that gðhÞ ðC s;t ð0Þ C s;t ðhÞÞ ! 0 for h 2 Rd as both s ! 0 and t ! 0. Since supkhkoH gðhÞ is finite, the convergence is uniform on the ball of radius H centered at the origin of Rd ; hence, the covariance function C s;t satisfies (2) if s and t are sufficiently small. & The above theorem and proof generalize verbatim to non-centered variograms (Gneiting et al., 2001). Any non-centered variogram decomposes as the sum of a centered variogram and a nonnegative quadratic form.3 References Barry, R.P., 2005. The local approximation of variograms in R2 by covariance functions. Statist. Probab. Lett. 74, 171–177. Chile`s, J.-P., Delfiner, P., 1999. Geostatistics. Modeling Spatial Uncertainty. Wiley, New York. Gneiting, T., Sasva´ri, Z., Schlather, M., 2001. Analogies and correspondences between variograms and covariance functions. Adv. Appl. Probab. 33, 617–630. Matheron, G., 1973. The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–468. Stein, M.L., 1999. Interpolation of Spatial Data. Some Theory for Kriging. Springer, New York. Stein, M.L., 2001. Local stationarity and simulation of self-affine intrinsic random functions. IEEE Trans. Inform. Theory 47, 1385–1390.
2 3
Note that Barry omits the factor 12 in the definition of the variogram;P clearly, this factor is immaterial. In Gneiting et al. (2001, p. 619), the quadratic form is given as QðhÞ ¼ di¼1 ai h2i ; this should read QðhÞ ¼ kAhk2 for a matrix A 2 Rdd .