Mathematical Geosciences

, 43:819 | Cite as

Tests of Significance for Structural Correlations in the Linear Model of Coregionalization



In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures (one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed tests are generally valid.


Coregionalization analysis Effective sample sizes Sill matrices Uncertainty of estimation Validity and power of statistical tests 


  1. Bonferroni CE (1937) Teoria statistica delle classi e calcolo delle probabilita. In: Volume in Onore di Riccardo dalla Volta, Universita di Firenza, pp 1–62 Google Scholar
  2. Cressie NAC (1993) Statistics for spatial data. Wiley, New York Google Scholar
  3. Dutilleul P (1993) Modifying the t test for assessing the correlation between two spatial processes. Biometrics 49:305–314 CrossRefGoogle Scholar
  4. Dutilleul P (2008) A note on sufficient conditions for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. Commun Stat, Theory Methods 37:137–145 CrossRefGoogle Scholar
  5. Dutilleul P, Herman M, Avella-Shaw T (1998) Growth rate effects on correlations among ring width, wood density and mean tracheid length in Norway spruce (Picea abies (L.) Karst). Can J For Res 28:56–68 CrossRefGoogle Scholar
  6. Dutilleul P, Pelletier B, Alpargu G (2008) Modified F tests for assessing the multiple correlation between one spatial process and several others. J Stat Plan Inference 138:1402–1415 CrossRefGoogle Scholar
  7. Dutilleul P, Pinel-Alloul B (1996) A doubly multivariate model for statistical analysis of spatio-temporal environmental data. EnvironMetrics 7:551–566 CrossRefGoogle Scholar
  8. Goovaerts P (1992) Factorial kriging analysis: A useful tool for exploring the structure of multivariate spatial soil information. J Soil Sci 43:597–619 CrossRefGoogle Scholar
  9. Goovaerts P, Webster R (1994) Scale-dependent correlation between topsoil copper and cobalt concentrations in Scotland European. J Soil Sci 45:79–95 CrossRefGoogle Scholar
  10. Goulard M (1989) Inference in a regionalization model. In: Armstrong M (ed) Geostatistics, vol 1. Kluwer Academic, Dordrecht, pp 397–408 Google Scholar
  11. Goulard M, Voltz M (1992) Linear coregionalization model: tools for estimation and choice of cross-variogram matrix. Math Geol 24:269–286 CrossRefGoogle Scholar
  12. Haining R (1990) Spatial data analysis in the social and environmental sciences. Cambridge University Press, Cambridge Google Scholar
  13. Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, New York Google Scholar
  14. Jenkins GM, Watts DG (1968) Spectral analysis and its applications. Holden-Day, San Francisco Google Scholar
  15. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, London Google Scholar
  16. Lark RM, Webster R (2001) Changes in variance and correlation of soil properties with scale and location: analysis using an adapted maximal overlap discrete wavelet transform. Eur J Soil Sci 52:547–562 CrossRefGoogle Scholar
  17. Larocque G, Dutilleul P, Pelletier B, Fyles JW (2007) Characterization and quantification of uncertainty in coregionalization analysis. Math Geol 39:263–288 CrossRefGoogle Scholar
  18. Li B, Genton MG, Sherman M (2008) Testing the covariance structure of multivariate random fields. Biometrika 95:813–829 CrossRefGoogle Scholar
  19. Maestre FT, Rodríguez F, Bautista S, Cortina J, Bellot J (2005) Spatial associations and patterns of perennial vegetation in a semi-arid steppe: A multivariate geostatistics approach. Plant Ecol 179:133–147 CrossRefGoogle Scholar
  20. Marchant BP, Lark RM (2007) Estimation of linear models of coregionalization by residual maximum likelihood. Eur J Soil Sci 58:1506–1513 CrossRefGoogle Scholar
  21. McBratney AB, Webster R (1986) Choosing functions for semi-variograms of soil properties and fitting them to sampling estimates. J Soil Sci 37:617–639 CrossRefGoogle Scholar
  22. Morris MD (1991) On counting the number of data pairs for semivariogram estimation. Math Geol 23:929–943 CrossRefGoogle Scholar
  23. Müller WG, Zimmerman DL (1999) Optimal designs for variogram estimation. EnvironMetrics 10:23–37 CrossRefGoogle Scholar
  24. Pelletier B, Dutilleul P, Larocque G, Fyles JW (2004) Fitting the linear model of coregionalization by generalized least squares. Math Geol 36:323–343 CrossRefGoogle Scholar
  25. Pelletier B, Dutilleul P, Larocque G, Fyles JW (2009a) Coregionalization analysis with a drift for multi-scale assessment of spatial relationships between ecological variables. 1. Estimation of drift and random components. Environ Ecol Stat 16:439–466 CrossRefGoogle Scholar
  26. Pelletier B, Dutilleul P, Larocque G, Fyles JW (2009b) Coregionalization analysis with a drift for multi-scale assessment of spatial relationships between ecological variables. 2. Estimation of correlations and coefficients of determination. Environ Ecol Stat 16:467–494 CrossRefGoogle Scholar
  27. Pinel-Alloul B, Guay C, Angeli N, Legendre P, Dutilleul P, Balvay G, Gerdeaux D, Guillard J (1999) Large-scale spatial heterogeneity of macrozooplankton in Lake Geneva. Can J Fish Aquat Sci 56:1437–1451 Google Scholar
  28. Platt T, Denman KL (1975) Spectral analysis in ecology. Ann Rev Ecolog Syst 6:189–210 CrossRefGoogle Scholar
  29. Priestley MB (1981) Spectral analysis and time series. Academic Press, London Google Scholar
  30. Sokal RR, Rohlf FJ (1995) Biometry: The principles and practice of statistics in biological research, 3rd edn. Freeman, New York Google Scholar
  31. Wackernagel H (1995) Multivariate geostatistics: An introduction with applications. Springer, Berlin Google Scholar
  32. Warrick AW, Myers D (1987) Optimization of sampling locations for variogram calculations:. Water Resour Res 23:496–500 CrossRefGoogle Scholar
  33. Whitcher B, Guttorp P, Percival DB (2000) Wavelet analysis of covariance with application to atmospheric time series. J Geophys Res 105:14941–14962 CrossRefGoogle Scholar
  34. Zhang XF, Van Eijkeren JCH, Heemink AW (1995) On the weighted least-squares method for fitting a semivariogram model. Comput Geosci 21:605–608 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2011

Authors and Affiliations

  1. 1.Department of Plant ScienceMcGill UniversityMacdonald CampusCanada
  2. 2.Department of Natural Resource SciencesMcGill UniversityMacdonald CampusCanada

Personalised recommendations