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

The curvature of the conical intersection seam: an approximate second-order analysis

We present a method for analyzing the curvature (second derivatives) of the conical intersection hyperline at an optimized critical point. Our method uses the projected Hessians of the degenerate states after elimination of the two branching space coordinates, and is equivalent to a frequency calculation on a single Born-Oppenheimer potential-energy surface. Based on the projected Hessians, we develop an equation for the energy as a function of a set of curvilinear coordinates where the degeneracy is preserved to second order (i.e., the conical intersection hyperline). The curvature of the potential-energy surface in these coordinates is the curvature of the conical intersection hyperline itself, and thus determines whether one has a minimum or saddle point on the hyperline. The equation used to classify optimized conical intersection points depends in a simple way on the first- and second-order degeneracy splittings calculated at these points. As an example, for fulvene, we show that the two optimized conical intersection points of C2v symmetry are saddle points on the intersection hyperline. Accordingly, there are further intersection points of lower energy, and one of C2 symmetry - presented here for the first time - is found to be the global minimum in the intersection space