Calculating biotite formula from electron microprobe analysis data using a machine learning method based on principal components regression

We present a new machine learning method for calculating biotite (sensu lato) structural formula from electron microprobe analysis (EMPA) data, which is based on principal components regression (PCR) of a dataset consisting of 155 fully analyzed biotite references that have chemistry and crystal structure refinement. The dataset is randomly grouped into a training set (75% in amount) and a test set (25% in amount). The training set is used to implement the structural formula and the test set is used to evaluate the performance of the model. The resulting linear regression coefficient matrix is then applied to calculate mole proportions of cations and anions of biotite samples using their compositional data from EMPA. Through this method, the distribution of the different cations and anions in the different sites can be calculated, including the tetrahedral Fe3+, octahedral Fe2+, octahedral Fe3+, OH and WO2− at the O(4) site. The O(4) site is assumed to be occupied by anions with a relation of 2 = F + Cl + OH + WO2−. Octahedral and interlayer vacancies could also be estimated in this model. The prediction quality for major elements is perfect with R2 > 0.95. The absolute errors in the estimated octahedral Fe2+, octahedral Al and OH at O(4) site are determined to be ±0.2 apfu (atom per formula unit based on 11O + 2(F, Cl, OH, O)), while those in total Fe3+ and WO2− at O(4) site are approximately ±0.3 apfu. A funnel-shaped relationship between absolute error in Fe3+/ΣFe ratio and FeOT wt% is observed, with the majority falling in the range of ±20%. Compared to previous normalization schemes, our model shows significant improvements in estimating Fe3+/ΣFe and WO2− at O(4) site. Our model is capable for calculating mineral formulae of common igneous and hydrothermal biotites, but not suitable for those that have been modified in a post-formation oxidation or reduction process. A supplementary Excel spreadsheet is provided that can be easily used for performing calculation from EMPA data.