Spatial Analysis of Compositional Data

Spurious correlation is known to be a problem in statistics since Pearson's early warnings in 1897. The same problems arise in spatial statistics: bias towards negative values and non-zero cross-covariances and cross-covariograms; singular matrices of intrinsic co-dispersion; co-kriged regionalised vectors of proportions that do not satisfy the constant sum constraint. A way out is to use log-ratio transformations: the spatial structure can be described in terms of direct variograms of each possible pairwise logratio; variation-variograms can be estimated even in case of missing components; they can be modelled with standard tools; both the data and the spatial structure model can be expressed in isometric logratio coordinates, and standard co-kriging techniques can be applied to obtain interpolated logratios. These can be back-transformed to compositions, delivering interpolated maps of each component that satisfy the required constraints. Moreover, the result does not depend on which logratio transformation was used for the computations. This approach and its potentialities is illustrated with a data set of soil geochemistry.