In Chapter 6 we introduce additional aspects of the geostatistical approach presented in the preceding chapters that were not necessary for its theoretical development, but that are essential for the practical application of the method to compositional data. We discuss how to treat zeros in compositional data sets; how to model the required cross-covariances; how to compute expected values and estimation variances for the original, constrained variables; and how to build and interpret confidence intervals for estimated values. As mentioned in Section 2.1, data sets with many zeros are as troublesome in compositional analysis as they are in standard multivariate analysis. In our approach, the additional restriction for compositional data is that zero values are not admissible for modeling. The justification for this restriction can be given using arithmetic arguments. A transformation that uses logarithms cannot be performed on zero values. This is the case for the logratio transformation that leads to the definition of an additive logistic normal distribution, as introduced by Aitchison (1986, p. 113). It is also the case for the additive logistic skew-normal distribution defined in Mateu-Figueras et al. (1998), following previous results by Azzalini and Dalla Valle (1996). The centered logratio transformation and the family of multivariate Box-Cox transformations discussed in Andrews et al. (1971), Rayens and Srinivasan (1991), and Barceló- Vidal (1996) also call for the restriction of zero values. This restriction is certainly a wellspring of discussion, albeit surprisingly so, as nobody would complain about eliminating zeros either by simple suppression of samples or by substitution with reasonable values when dealing with a sample from a lognormal distribution in the univariate case. Recall that the logarithm of zero is undefined and the sample space of the lognormal distribution is the positive real line, excluding the origin. In order to present our position on how to deal with zeros as clearly as possible, let us assume that only one of our components has zeros in some of the samples. Those cases where more than one variable is affected can be analyzed by methods described below.
The problem of estimation of a coregionalization of size q using cokriging will be discussed in this chapter. Cokriging—a multivariate extension of kriging—is the usual procedure applied to multivariate regionalized problems within the framework of geostatistics. Its fundament is a distribution-free, linear, unbiased estimator with minimum estimation variance, although the absence of constraints on the estimator is an implicit assumption that the multidimensional real space is the sample space of the variables under consideration. If a multivariate normal distribution can be assumed for the vector random function, then the simple kriging estimator is identical with the conditional expectation, given a sample of size N. See Journel (1977, pp. 576-577), Journel (1980, pp. 288-290), Cressie (1991, p. 110), and Diggle, Tawn, and Moyeed (1998, p. 300) for further details. This estimator is in general the best possible linear estimator, as it is unbiased and has minimum estimation variance, but it is not very robust in the face of strong departures from normality. Therefore, for the estimation of regionalized compositions other distributions must also be taken into consideration. Recall that compositions cannot follow a multivariate normal distribution by definition, their sample space being the simplex. Consequently, regionalized compositions in general cannot be modeled under explicit or implicit assumptions of multivariate Gaussian processes. Here only the multivariate lognormal and additive logistic normal distributions will be addressed. Besides the logarithmic and additive logratio transformations, others can be applied, such as the multivariate Box-Cox transformation, as stated by Andrews et al. (1971), Rayens and Srinivasan (1991), and Barcelo-Vidal (1996). Furthermore, distributions such as the multiplicative logistic normal distribution introduced by Aitchison (1986, p. 131) or the additive logistic skew-normal distribution defined by Azzalini and Dalla Valle (1996) can be investigated in a similar fashion. References to the literature for the fundamental principles of the theory discussed in this chapter were given in Chapter 2. Among those, special attention is drawn to the work of Myers (1982), where matrix formulation of cokriging was first presented and the properties included in the first section of this chapter were stated.
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