Spatial covariance structure
For any component in time series analysis (Natke 1983), the concept of covariance between components of a spatially distributed random vector Z(u) leads to: direct covariances, Cov[Zi(u),Zj(u)]; shifted covariances or spatial covariances, Cov [Zi(u), Zj-(u+ h)], also known as cross-covariance functions; and autocovariance functions, Cov[Zi(u),Zi(u + h)]. The direct covariances may be thought of as a special case of the cross-covariance functions (for h = 0), and the same holds for the autocovariance functions (for i = j), so there is no need for a separate discussion. To simplify the exposition, hereafter the term function is dropped, and only the terms cross-covariance and autocovariance are used. Pawlowsky (1984) stated that if the vector random function constitutes an r-composition, then the problem of spurious spatial correlations appears. This is evident from the fact that at each point of the domain W, as in the nonregionalized case, the natural sample space of an r-composition is the D-simplex. This aspect will be discussed in Section 3.1.1. Aitchison (1986) discussed the problematic nature of the covariance analysis of nonregionalized compositions. He circumvents the problem of spurious correlations by using the fact that the ratio of two arbitrary components of a basis is identical to the ratios of the corresponding components of the associated composition. To avoid working with ratios, which is always difficult, Aitchison takes logarithms of the ratios. Then dependencies among variables of a composition can be examined in real space by analyzing the covariance structure of the log-quotients. The advantages of using this approach are not only numerical or related to the facility of subsequent mathematical operations. Essentially they relate to the fact that the approach consists of a projection of the original sample space, the simplex SD, onto a new sample space, namely real space IRD-1. Thus the door is open to many available methods and models based on the multivariate normal distribution. Recall that the multivariate normal distribution requires the sample space to be precisely the multidimensional, unconstrained real space. For this kind of model, strictly speaking, this is equivalent to saying that you need unconstrained components of the random vector to be analyzed.