Separate batches of differentiated magma are often delineated by looking at clusters of points on binary oxide diagrams. However, these are neither directly amenable to statistical analysis, nor do they use all the data available at the time to form the clusters. It is possible to numerically form the clusters using cluster analysis.
Hildreth and Fierstein  have recently looked at and summarized some of the geochemical data for the LVVR to understand the different magmatic systems. Their overview suggests the existence of numerous batches of evolved magma developing separately through time. Their conclusions are based primarily on careful inspection of binary plots. How valid is this generally accepted ad-hoc characterization and correlation? What is the relation between ad-hoc correlation and improvements in analytical techniques and mathematical or statistical interpretation? What if we could quantitatively investigate multidimensional plots of all chemical species simultaneously? Would the results be the same as ad-hoc correlation? Or, could we gain additional insight into the magmatic systems were we to consider the geochemistry in a more quantitative, holistic fashion?
New geochemical data in the present contribution consist of whole rock analyses of Mono-Inyo Craters rocks from flows, domes and pyroclastic deposits. The data were gathered during investigation of stratigraphic sections containing deposits of multiple eruptions, potentially from one of any number of vents in the 50-km long young volcanic belt. Most of the new analyses were performed on pyroclastic units exposed in hand-dug trenches. Some of the units were exposed in other man-made excavations or natural streamcuts.
New analyses reported herein were performed by SGS Minerals Services of Toronto, ON, Canada, or at the XRF Laboratory, Michigan State University (MSU). At SGS and MSU, major elements were analyzed by fusion XRF (scheme code XRF77), and a Bruker S4 Pioneer 4 kW wavelength dispersive X-ray fluorescence spectrometer, respectively, following each laboratory’s standard procedure for sample washing and crushing. To check for consistency, duplicates of randomly selected samples were analyzed to test for reproducibility. In most cases, the duplicates were within analytical error. Repeatability was almost always less than nominal analytical errors of 0.01% for major and minor elements, 20 ppm for Ba and 2 ppm for other trace elements. In a few instances the difference for major elements was > 0.01%, to a maximum of 0.04%.
Published sources were consulted for additional data [Sampson and Cameron, 1987; Bailey, 1989; Kelleher and Cameron, 1990; Hildreth et al., 2014; Hildreth and Fierstein, 2016]. The published sources included data on dome and lava flow samples, as well as a few pyroclastic samples. All of the final 483 new and published sample analyses of differentiated lavas and pyroclasts with silica > 63% used in the present study were thus analyzed by whole-rock XRF. 177 of the samples could be traced back to a source vent, and 91 had associated numerical ages. 281 of the samples had silica content > 69%, making them rhyolites.
Clustering required the data to be normalized in a way unfamiliar in the geochemical context. Many workers normalize major element XRF data to a 100% anhydrous basis. In the present case, only the data from Sampson and Cameron  and Kelleher and Cameron  had already been normalized in this familiar way. Pouget et al  pointed out that such normalization is not necessarily a good solution when certain analyses lack some minor oxides, such as MnO. Following our earlier work, therefore, all results reported herein are based on using data for which the major element oxides sum to at least 95 wt%. There are further reasons why it is not preferable to use this normalization for clustering.
Because clustering depends on a distance metric, those numbers associated with concentrations that are largest will be weighted most heavily by the algorithm and play the largest role in determining clusters. This means, for example, that Ba, because the values range up to several thousand, irrespective of the units, will be weighted much more heavily than any other element. We have therefore normalized in one of two ways to transform each datum for each element to a comparable interval. These normalization methods are based on the range or variance in the data. The first method normalizes the data according to the range: (xi - xmin) / (xmax - xmin),
where xi is a single measurement of the concentration of one species, and xmin and xmax the minimum and maximum values in the vector of that species. In this case, all data for a single element are in the range [0, 1]. The second method uses the variance: (xi - m) / V,
where m is the mean and V the variance in the data for a single chemical species. In this case, the data for a single element have m = 0, and s = 1, where s is the standard deviation. In this context, normalization of the major element compositional feature vectors to 100% results in dispersal of the variance, which may be originally concentrated in one species, to all other chemical species, thus potentially changing the value of the denominator in both equations, which is dependent on some measure of sample spread, anomalously for all major elements. This could of course result in the “washing out” of the weight of an element with particularly low, hence diagnostic, variance.
Starting from 16 or 22 clusters, the cluster analysis found that the best consensus results suggested the existence of seven to eleven clusters (see table lvvr_cluster_analysis.xlsx). There are a number critical ways in which we would like to look at the results:
- Understand whether the geochemical clustering correlates with either spatial (vent position) or temporal information (age).
- See whether use of different chemical groupings leads to samples being put in different clusters
- See how tightly bound each sample is within the different clusters
- See what the results say about the geologic implications of the geochemical data to determine whether any new insight can be gained by comparing the clustering results to the impressions of human operators