Tests of Significance for Structural Correlations in the Linear Model of Coregionalization
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Tests of Significance for Structural Correlations in the Linear Model of Coregionalization Pierre Dutilleul · Bernard Pelletier
Received: 1 April 2009 / Accepted: 7 March 2011 / Published online: 28 July 2011 © International Association for Mathematical Geosciences 2011
Abstract 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 secondorder 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. Keywords Coregionalization analysis · Effective sample sizes · Sill matrices · Uncertainty of estimation · Validity and power of statistical tests P. Dutilleul () · B. Pelletier Department of Plant Science, McGill University, Macdonald Campus, Ste-Anne-de-Bellevue QC H9X 3V9, Canada e-mail: [email protected] B. Pelletier Department of Natural Resource Sciences, McGill University, Macdonald Campus, Ste-Anne-de-Bellevue QC H9X 3V9, Canada
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Math Geosci (2011) 43:819–846
1 Introduction Scale-dependent correlations can be defined, analyzed, and estimated in different ways for the same multivariate spatial or temporal dataset. In spectral analysis, the coherency statistic measures the correlation between the frequency components (i.e., fitted cosine and sine waves) of two observed patterns (Platt and Denman 1975; Priestley 1981). Alternatively, the cross spectrum computed with discrete or continuous wavelets provides a set of coefficients of correlation in another basis in mathematical sense (Whitcher et al. 2000; Lark and Webster 2001). In the distance-based approach, without orthogonal decomposition comparable to those performed with cosine and sine waves and the discrete wavelets, cross correlograms and cross variograms depict how the relationship between two random functions changes
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