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Numerical examples and visualizations are presented herein as teaching aids for multivariate data analysis, spatial estimation using kriging and inverse distance methods, and the variogram as a standalone data analytical tool. Attention is focused on the practical application of these methods.

For more than a half-century, scientists have been developing a tool for linear unmixing utilizing collections of algorithms and computer programs that is appropriate for many types of data commonly encountered in the geologic and other science disciplines. Applications include the analysis of particle size data, Fourier shape coefficients and related spectrum, biologic morphology and fossil assemblage information, environmental data, petrographic image analysis, unmixing igneous and metamorphic petrographic variable and the unmixing and determination of oil sources, to name a few. Each of these studies used algorithms that were designed to use data whose row sums are constant. Non-constant sum data comprise what is a larger set of data that permeates many of our sciences. Many times, these data can be modeled as mixtures even though the row sums do not sum to the same value for all samples in the data. This occurs when different quantities of one or more end-member are present in the data. Use of the constant sum approach for these data can produce confusing and inaccurate results especially when the end-members need to be defined away from the data cloud. The approach to deal with these non-constant sum data is defined and called Hyperplanar Vector Analysis (HVA). Without abandoning over 50 years of experience, HVA merges the concepts developed over this time and extends the linear unmixing approach to more types of data. The basis for this development involves a translation and rotation of the raw data that conserves information (variability). It will also be shown that HVA is a more appropriate name for both the previous constant sum algorithms and future programs algorithms as well.

While this chapter is about Pearce element ratios, I’ve included some personal reflections as this book is a 50th Anniversary project of the IAMG. Pearce element ratios, Felix Chayes and the Chayes medal, came together on September 11, 2001. As the recipient of the Chayes Medal, I was in Cancún, Mexico on that fateful date to deliver a talk on Pearce element ratios. Pearce element ratios are designed to model processes of fractionation and accumulation in igneous systems. They are frequently used to extract information from analyses of rocks formed from melts produced by fractionation—volcanic suites. Rock bodies formed from the fractionated crystals—the cumulate rocks—have received practically no attention. From the standard paradigm describing the formation of cumulate rocks, based on studies of the Skaergaard Intrusion, one expects a predicted pattern of data points on a Pearce element ratio diagram. Points derived from the mean compositions of the units in the cumulate body should fall up-slope from the point representing the initial melt composition on a diagram that accounts for the cumulate assemblage. Points derived from the compositions of the inferred residual melts present at the beginning of crystallization of a unit in the rock body should fall down-slope from the point representing the initial magma. The distance between a point on the line of a Pearce element ratio diagram and the point representing the initial magma composition depends on (1) the size of the aliquot that crystallized to form the rock unit and (2) the ratio of crystals to melt in the mush that solidified to form the rock unit. Patterns extracted from computer simulations compared to analogous data points from units of the Skaergaard Intrusion indicate that the crystal mushes that formed the units of the Marginal Border Series had a smaller ratio of trapped melt to crystals than did coeval mushes forming the Upper Border Series. Simulation patterns further indicate that the LZa and UZa units of the Layered Series formed from assemblages with larger ratios of melt to crystals than did the respective coeval units, LZa* and UZa*, of the Marginal Border Series.

This note is to highlight the transformation of the names of and its flagship journal respectively into and .

This chapter is primarily concerned with the first 15 years of our existence (I was a member of the IAMG Founding Committee, and on the 1968–1972 and 1996–1980 IAMG Councils). Daniel Merriam and Richard Reyment are the principal fathers of the IAMG, and many other scientists have contributed significantly to its origin and early development. Personal contacts with them are briefly described. These comments are supplementary to those already provided in earlier chapters by Founding Members and others who have made significant contributions to the IAMG originally. Special attention is paid to inputs by prominent mathematical statisticians with an interest in geology. I am grateful to all pioneers who have helped to establish the IAMG and provided a climate encouraging younger scientists, including myself, to pursue careers in their field of interest.