A review of nonparametric hypothesis tests of isotropy properties in spatial data. (English) Zbl 1442.62213

Summary: An important aspect of modeling spatially referenced data is appropriately specifying the covariance function of the random field. A practitioner working with spatial data is presented a number of choices regarding the structure of the dependence between observations. One of these choices is to determine whether or not an isotropic covariance function is appropriate. Isotropy implies that spatial dependence does not depend on the direction of the spatial separation between sampling locations. Misspecification of isotropy properties (directional dependence) can lead to misleading inferences, for example, inaccurate predictions and parameter estimates. A researcher may use graphical diagnostics, such as directional sample variograms, to decide whether the assumption of isotropy is reasonable. These graphical techniques can be difficult to assess, open to subjective interpretations, and misleading. Hypothesis tests of the assumption of isotropy may be more desirable. To this end, a number of tests of directional dependence have been developed using both the spatial and spectral representations of random fields. We provide an overview of nonparametric methods available to test the hypotheses of isotropy and symmetry in spatial data. We discuss important considerations in choosing a test, provide recommendations for implementing a test, compare several of the methods via a simulation study, and propose a number of open research questions. Several of the reviewed methods can be implemented in R using our package spTest, available on CRAN.


62M40 Random fields; image analysis
62G10 Nonparametric hypothesis testing
62M30 Inference from spatial processes


spTest; R; sm
Full Text: DOI arXiv Euclid


[1] Baczkowski, A. J. (1990). A test of spatial isotropy. In Compstat 277-282. Springer, Berlin.
[2] Baczkowski, A. J. and Mardia, K. V. (1987). Approximate lognormality of the sample semivariogram under a Gaussian process. Comm. Statist. Simulation Comput.16 571-585. · Zbl 0622.62110
[3] Baczkowski, A. J. and Mardia, K. V. (1990). A test of spatial symmetry with general application. Comm. Statist. Theory Methods19 555-572. · Zbl 0900.62502
[4] Bandyopadhyay, S., Lahiri, S. N. and Nordman, D. J. (2015). A frequency domain empirical likelihood method for irregularly spaced spatial data. Ann. Statist.43 519-545. · Zbl 1312.62120
[5] Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial Data. CRC Press, Boca Raton, FL. · Zbl 1358.62009
[6] Borgman, L. and Chao, L. (1994). Estimation of a multidimensional covariance function in case of anisotropy. Math. Geol.26 161-179. · Zbl 0970.86544
[7] Bowman, A. W. and Crujeiras, R. M. (2013). Inference for variograms. Comput. Statist. Data Anal.66 19-31. · Zbl 1471.62033
[8] Cabaña, E. M. (1987). Affine processes: A test of isotropy based on level sets. SIAM J. Appl. Math.47 886-891. · Zbl 0627.62088
[9] Chorti, A. and Hristopulos, D. T. (2008). Nonparametric identification of anisotropic (elliptic) correlations in spatially distributed data sets. IEEE Trans. Signal Process.56 4738-4751. · Zbl 1390.94137
[10] Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York. · Zbl 0799.62002
[11] Ecker, M. D. and Gelfand, A. E. (1999). Bayesian modeling and inference for geometrically anisotropic spatial data. Math. Geol.31 67-83.
[12] Ecker, M. D. and Gelfand, A. E. (2003). Spatial modeling and predication under stationary non-geometric range anisotropy. Environ. Ecol. Stat.10 165-178.
[13] Fuentes, M. (2005). A formal test for nonstationarity of spatial stochastic processes. J. Multivariate Anal.96 30-54. · Zbl 1073.62082
[14] Fuentes, M. (2006). Testing for separability of spatial-temporal covariance functions. J. Statist. Plann. Inference136 447-466. · Zbl 1077.62076
[15] Fuentes, M. (2007). Approximate likelihood for large irregularly spaced spatial data. J. Amer. Statist. Assoc.102 321-331. · Zbl 1284.62589
[16] Fuentes, M. and Reich, B. (2010). Spectral domain. In Handbook of Spatial Statistics 55-77. CRC Press, Boca Raton, FL.
[17] García-Soidán, P. (2007). Asymptotic normality of the Nadaraya-Watson semivariogram estimators. TEST16 479-503. · Zbl 1131.62075
[18] García-Soidán, P. H., Febrero-Bande, M. and González-Manteiga, W. (2004). Nonparametric kernel estimation of an isotropic variogram. J. Statist. Plann. Inference121 65-92. · Zbl 1038.62036
[19] Gneiting, T., Genton, M. and Guttorp, P. (2007). Geostatistical space-time models, stationarity, separability and full symmetry. In Statistical Methods for Spatio-Temporal Systems 151-175. Chapman & Hall/CRC Press, Boca Raton, FL. · Zbl 1282.86019
[20] Guan, Y., Sherman, M. and Calvin, J. A. (2004). A nonparametric test for spatial isotropy using subsampling. J. Amer. Statist. Assoc.99 810-821. · Zbl 1117.62348
[21] Guan, Y., Sherman, M. and Calvin, J. A. (2006). Assessing isotropy for spatial point processes. Biometrics62 119-125, 316. · Zbl 1091.62096
[22] Guan, Y., Sherman, M. and Calvin, J. A. (2007). On asymptotic properties of the mark variogram estimator of a marked point process. J. Statist. Plann. Inference137 148-161. · Zbl 1098.62121
[23] Guan, Y. T. (2003). Nonparametric methods of assessing spatial isotropy. Ph.D. thesis, Texas A&M Univ., College Station, TX.
[24] Hall, P. and Patil, P. (1994). Properties of nonparametric estimators of autocovariance for stationary random fields. Probab. Theory Related Fields99 399-424. · Zbl 0799.62102
[25] Irvine, K. M., Gitelman, A. I. and Hoeting, J. A. (2007). Spatial designs and properties of spatial correlation: Effects on covariance estimation. J. Agric. Biol. Environ. Stat.12 450-469. · Zbl 1306.62296
[26] Isaaks, E. H. and Srivastava, R. M. (1989). Applied Geostatistics, Vol. 2. Oxford Univ. Press, New York.
[27] Jona-Lasinio, G. (2001). Modeling and exploring multivariate spatial variation: A test procedure for isotropy of multivariate spatial data. J. Multivariate Anal.77 295-317. · Zbl 0997.62043
[28] Journel, A. G. and Huijbregts, C. J. (1978). Mining Geostatistics. Academic Press, New York.
[29] Jun, M. and Genton, M. G. (2012). A test for stationarity of spatio-temporal random fields on planar and spherical domains. Statist. Sinica22 1737-1764. · Zbl 1253.62070
[30] Kim, T. Y. and Park, J. (2012). On nonparametric variogram estimation. J. Korean Statist. Soc.41 399-413. · Zbl 1296.62118
[31] Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer, New York. · Zbl 1028.62002
[32] Lahiri, S. N. and Zhu, J. (2006). Resampling methods for spatial regression models under a class of stochastic designs. Ann. Statist.34 1774-1813. · Zbl 1246.62117
[33] Li, B., Genton, M. G. and Sherman, M. (2007). A nonparametric assessment of properties of space-time covariance functions. J. Amer. Statist. Assoc.102 736-744. · Zbl 1172.62311
[34] Li, B., Genton, M. G. and Sherman, M. (2008a). Testing the covariance structure of multivariate random fields. Biometrika95 813-829. · Zbl 1437.62528
[35] Li, B., Genton, M. G. and Sherman, M. (2008b). On the asymptotic joint distribution of sample space-time covariance estimators. Bernoulli14 228-248. · Zbl 1155.62010
[36] Lu, N. and Zimmerman, D. L. (2005). Testing for directional symmetry in spatial dependence using the periodogram. J. Statist. Plann. Inference129 369-385. · Zbl 1058.62078
[37] Maity, A. and Sherman, M. (2012). Testing for spatial isotropy under general designs. J. Statist. Plann. Inference142 1081-1091. · Zbl 1236.62052
[38] Matheron, G. (1961). Precision of exploring a stratified formation by boreholes with rigid spacing-application to a bauxite deposit. In International Symposium of Mining Research, University of Missouri, Vol. 1 (G. B. Clark, ed.) 407-22. Pergamon Press, Oxford.
[39] Matheron, G. (1962). Traité de géostatistique appliquée 1. Technip, Paris.
[40] Matsuda, Y. and Yajima, Y. (2009). Fourier analysis of irregularly spaced data on \(\mathbb{R}^d \). J. R. Stat. Soc. Ser. B. Stat. Methodol.71 191-217. · Zbl 1231.62169
[41] Modjeska, J. S. and Rawlings, J. O. (1983). Spatial correlation analysis of uniformity data. Biometrics 373-384. · Zbl 0527.62097
[42] Molina, A. and Feito, F. R. (2002). A method for testing anisotropy and quantifying its direction in digital images. Computers & Graphics26 771-784.
[43] Nadaraya, E. A. (1964). On estimating regression. Theory of Probability & Its Applications9 141-142. · Zbl 0136.40902
[44] Nicolis, O., Mateu, J. and D’Ercole, R. (2010). Testing for anisotropy in spatial point processes. In Proceedings of the Fifth International Workshop on Spatio-Temporal Modelling 1990-2010. Publisher Unidixital, Santiago de Compostela.
[45] Pagano, M. (1971). Some asymptotic properties of a two-dimensional periodogram. J. Appl. Probab.8 841-847. · Zbl 0228.62053
[46] Park, M. S. and Fuentes, M. (2008). Testing lack of symmetry in spatial-temporal processes. J. Statist. Plann. Inference138 2847-2866. · Zbl 1140.62072
[47] Politis, D. N. and Sherman, M. (2001). Moment estimation for statistics from marked point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol.63 261-275. · Zbl 0979.62074
[48] Possolo, A. (1991). Subsampling a random field. In Spatial Statistics and Imaging (Brunswick, ME, 1988). Institute of Mathematical Statistics Lecture Notes—Monograph Series20 286-294. IMS, Hayward, CA. · Zbl 0768.62091
[49] Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic Press, New York. · Zbl 0537.62075
[50] Priestley, M. B. and Subba Rao, T. (1969). A test for non-stationarity of time-series. J. R. Stat. Soc. Ser. B. Stat. Methodol.31 140-149. · Zbl 0182.51403
[51] R Core Team (2014). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
[52] Scaccia, L. and Martin, R. J. (2002). Testing for simplification in spatial models. In COMPSTAT 2002 (Berlin) 581-586. Physica, Heidelberg. · Zbl 1446.62022
[53] Scaccia, L. and Martin, R. J. (2005). Testing axial symmetry and separability of lattice processes. J. Statist. Plann. Inference131 19-39. · Zbl 1075.62086
[54] Scaccia, L. and Martin, R. J. (2011). Model-based tests for simplification of lattice processes. J. Stat. Comput. Simul.81 89-107. · Zbl 1206.62159
[55] Schabenberger, O. and Gotway, C. A. (2004). Statistical Methods for Spatial Data Analysis. CRC Press, Boca Raton, FL. · Zbl 1068.62096
[56] Shao, X. and Li, B. (2009). A tuning parameter free test for properties of space-time covariance functions. J. Statist. Plann. Inference139 4031-4038. · Zbl 1183.62073
[57] Sherman, M. (1996). Variance estimation for statistics computed from spatial lattice data. J. R. Stat. Soc. Ser. B. Stat. Methodol.58 509-523. · Zbl 0855.62082
[58] Sherman, M. (2011). Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties. Wiley, Chichester. · Zbl 1277.62025
[59] Spiliopoulos, I., Hristopulos, D. T., Petrakis, M. P. and Chorti, A. (2011). A multigrid method for the estimation of geometric anisotropy in environmental data from sensor networks. Computers & Geosciences37 320-330.
[60] Stein, M. L. (1988). Asymptotically efficient prediction of a random field with a misspecified covariance function. Ann. Statist.16 55-63. · Zbl 0637.62088
[61] Stein, M. L., Chi, Z. and Welty, L. J. (2004). Approximating likelihoods for large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol.66 275-296. · Zbl 1062.62094
[62] Thon, K., Geilhufe, M. and Percival, D. B. (2015). A multiscale wavelet-based test for isotropy of random fields on a regular lattice. IEEE Trans. Image Process.24 694-708. · Zbl 1408.94644
[63] Vecchia, A. V. (1988). Estimation and model identification for continuous spatial processes. J. R. Stat. Soc. Ser. B. Stat. Methodol.50 297-312.
[64] Watson, G. S. (1964). Smooth regression analysis. Sankhyā Ser. A26 359-372. · Zbl 0137.13002
[65] Weller, Z. D. (2015b). SpTest: An R package implementing nonparametric tests of isotropy. J. Stat. Softw. To appear.
[66] Zhang, X., Li, B. and Shao, X. (2014). Self-normalization for spatial data. Scand. J. Stat.41 311-324. · Zbl 06298504
[67] Zimmerman, D. L. (1993). Another look at anisotropy in geostatistics. Math. Geol.25 453-470.
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.