Biplots for compositional data derived from generalized joint diagonalization methods

Abstract

Biplots constructed from principal components of a compositional data set are an established means to explore its features. Principal Component Analysis (PCA) is also used to transform a set of spatial variables into spatially decorrelated factors. However, because no spatial structures are accounted for in the transformation the application of PCA is limited. In geostatistics and blind source separation a variety of different matrix diagonalization methods have been developed with the aim to provide spatially or temporally decorrelated factors. Just as PCA, many of these transformations are linear and so lend themselves to the construction of biplots. In this contribution we consider such biplots for a number of methods (MAF, UWEDGE and RJD transformations) and discuss how and if they can contribute to our understanding of relationships between the components of regionalized compositions. A comparison of the biplots with the PCA biplot commonly used in compositional data analysis for the case of data from the Northern Irish geochemical survey shows that the biplots from MAF and UWEDGE are comparable as are those from PCA and RJD. The biplots emphasize different aspects of the regionalized composition: for MAF and UWEDGE the focus is the spatial continuity, while for PCA and RJD it is variance explained. The results indicate that PCA and MAF combined provide adequate and complementary means for exploratory statistical analysis.

Keywords

• Semivariogram matrices
• Spatial decorrelation
• Structural analysis
• Geochemical data