Compositional analysis of archaeological glasses

At CoDaWork'03 we presented work on the analysis of archaeological glass composi- tional data. Such data typically consist of geochemical compositions involving 10-12 variables and approximates completely compositional data if the main component, sil- ica, is included. We suggested that what has been termed `crude' principal component analysis (PCA) of standardized data often identi ed interpretable pattern in the data more readily than analyses based on log-ratio transformed data (LRA). The funda- mental problem is that, in LRA, minor oxides with high relative variation, that may not be structure carrying, can dominate an analysis and obscure pattern associated with variables present at higher absolute levels. We investigate this further using sub- compositional data relating to archaeological glasses found on Israeli sites. A simple model for glass-making is that it is based on a `recipe' consisting of two `ingredients', sand and a source of soda. Our analysis focuses on the sub-composition of components associated with the sand source. A `crude' PCA of standardized data shows two clear compositional groups that can be interpreted in terms of di erent recipes being used at di erent periods, re ected in absolute di erences in the composition. LRA analysis can be undertaken either by normalizing the data or de ning a `residual'. In either case, after some `tuning', these groups are recovered. The results from the normalized LRA are di erently interpreted as showing that the source of sand used to make the glass di ered. These results are complementary. One relates to the recipe used. The other relates to the composition (and presumed sources) of one of the ingredients. It seems to be axiomatic in some expositions of LRA that statistical analysis of compositional data should focus on relative variation via the use of ratios. Our analysis suggests that absolute di erences can also be informative

On the use of principal components in contemporany population genetics: a case study

In human Population Genetics, routine applications of principal component techniques are often required. Population biologists make widespread use of certain discrete classifications of human samples into haplotypes, the monophyletic units of phylogenetic trees constructed from several single nucleotide bimorphisms hierarchically ordered. Compositional frequencies of the haplotypes are recorded within the different samples. Principal component techniques are then required as a dimension-reducing strategy to bring the dimension of the problem to a manageable level, say two, to allow for graphical analysis. Population biologists at large are not aware of the special features of compositional data and normally make use of the crude covariance of compositional relative frequencies to construct principal components. In this short note we present our experience with using traditional linear principal components or compositional principal components based on logratios, with reference to a specific dataset