Polytopic vector analysis in igneous petrology: Application to lunar petrogenesis

Lunar samples represent a heterogeneous assemblage of rocks with complex inter-relationships that are difficult to decipher using standard petrogenetic approaches. These inter-relationships reflect several distinct petrogenetic trends as well as thermomechanical mixing of distinct components. Additional complications arise from the unequal quality of chemical analyses and from the fact that many samples (e.g., breccia clasts) are too small to be representative of the system from which they derived. Polytopic vector analysis (PVA) is a multi-variate procedure used as a tool for exploratory data analysis. PVA allows the analyst to classify samples and clarifies relationships among heterogenous samples with complex petrogenetic histories. It differs from orthogonal factor analysis in that it uses non-orthogonal multivariate sample vectors to extract sample endmember compositions. The output from a Q-mode (sample based) factor analysis is the initial step in PVA. The Q-mode analysis, using criteria established by Miesch and Klovan and Miesch, is used to determine the number of endmembers in the data system. The second step involves determination of endmembers and mixing proportions with all output expressed in the same geochemical variable as the input. The composition of endmembers is derived by analysis of the variability of the data set. Endmembers need not be present in the data set, nor is it necessary for their composition to be known a priori. A set of any endmembers defines a 'polytope' or classification figure (triangle for a three component system, tetrahedron for a four component system, a 'five-tope' in four dimensions for five component system, et cetera).