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Calculating biotite formula from electron microprobe analysis data using a

machine learning method based on principal components regression

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DOI: 10.1016/j.lithos.2020.105371

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

Xiaoyan Li a,b, Chao Zhang a,b,⁎, Harald Behrens a, Francois Holtz a

a Institute of Mineralogy, Leibniz University Hannover, Callinstr. 3, 30167 Hannover, Germanyb State Key Laboratory of Continental Dynamics, Department of Geology, Northwest University, Xi'an 710069, China

⁎ Corresponding author at: State Key Laboratory of Conof Geology, Northwest University, Xi'an 710069, China.

E-mail address: zhangchao@nwu.edu.cn (C. Zhang).

https://doi.org/10.1016/j.lithos.2020.1053710024-4937/© 2020 Published by Elsevier B.V.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 15 October 2019Received in revised form 29 December 2019Accepted 7 January 2020Available online 10 January 2020

Keywords:BiotiteMineral formulaPrincipal component regressionOxidation stateTi-oxy substitution

We present a newmachine learning method for calculating biotite (sensu lato) structural formula from electronmicroprobe analysis (EMPA) data, which is based on principal components regression (PCR) of a datasetconsisting of 155 fully analyzed biotite references that have chemistry and crystal structure refinement. Thedataset is randomly grouped into a training set (75% in amount) and a test set (25% in amount). The trainingset is used to implement the structural formula and the test set is used to evaluate the performance of themodel. The resulting linear regression coefficient matrix is then applied to calculate mole proportions of cationsand anions of biotite samples using their compositional data fromEMPA. Through thismethod, the distribution ofthe different cations and anions in the different sites can be calculated, including the tetrahedral Fe3+, octahedralFe2+, octahedral Fe3+, OH and WO2− at the O(4) site. The O(4) site is assumed to be occupied by anions with arelation of 2 = F + Cl + OH + WO2−. Octahedral and interlayer vacancies could also be estimated in thismodel. The prediction quality for major elements is perfect with R2 N 0.95. The absolute errors in the estimatedoctahedral Fe2+, octahedral Al and OH at O(4) site are determined to be ±0.2 apfu (atom per formula unitbased 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 ma-jority falling in the range of ±20%. Compared to previous normalization schemes, our model shows significantimprovements in estimating Fe3+/ΣFe andWO2− atO(4) site. Ourmodel is capable for calculatingmineral formu-lae 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 usedfor performing calculation from EMPA data.

© 2020 Published by Elsevier B.V.

1. Introduction

Biotite (sensu lato) is a common rock-forming mineral that can bestable in a variety of geological chemical systems and under a widerange of temperature and pressure conditions. Its composition providesinformation on several intensive and extensive parameters of the mag-matic or hydrothermal systems inwhich it crystallizes, such as pressure,temperature, oxygen fugacity, andwater activity. For example, Fe3+/ΣFe(fraction of ferric iron) of biotite is employed to infer the oxidation stateduring its formation (Feeley and Sharp, 1996; Wones and Eugster,1965). Its Ti content can be used as a thermometer (Henry et al., 2005;Henry and Daigle, 2018), while its tetrahedral Al may indicate the for-mation pressure (Uchida et al., 2007). The presence of Fe3+ may havea significant effect on the partial melting of biotite. For example, an in-crease of oxygen fugacity, which is supposed to elevate Fe3+/ΣFe in

tinental Dynamics, Department

biotite, tends to increase the melting temperature of biotite (PatiñoDouce and Beard, 1995). Biotite melting could also lead to a reductionof iron from Fe3+ to Fe2+ and to CO2 production under graphite-bearing reducing conditions (Cesare et al., 2005). In addition, F and Clcontents in biotite are useful to decipher the F and Cl contents of acoexisting fluid or melt through empirical F-OH and Cl-OH exchangeequations (Munoz and Ludington, 1974; Munoz and Swenson, 1981;Rasmussen and Mortensen, 2013; Wang et al., 2018; Zhang et al.,2012; Zhu and Sverjensky, 1991; Zhu and Sverjensky, 1992).

The application of biotite composition as a tracer of geological condi-tions requires a correct description of the structural formula, whichhowever cannot be obtained directly fromdata obtained by electronmi-croprobe analysis (EMPA). Additional information, such as the Fe oxida-tion state and hydrogen content, is necessary. In literature, Fe3+/ΣFe ofbiotite is usuallymeasured byMössbauer spectroscopy ormicro-XANES(Cesare et al., 2003; Dyar, 2002; Dyar et al., 2001; Redhammer et al.,2000; Righter et al., 2002; Scordari et al., 2010). Recent developmentof the EMPA flank method provides an alternative in-situ method formeasuring Fe3+/ΣFe of biotite (Höfer and Brey, 2007; Li et al., 2019),

2 X. Li et al. / Lithos 356–357 (2020) 105371

but thismethod is far frombeing routine inmost EMPA laboratories dueto lack of well-calibrated standard samples. Measurement of OH con-tent in biotite is usually performed with FTIR spectroscopy, SIMS or C-H-N analysis (Dyar et al., 1991; Schingaro et al., 2007; Scordari et al.,2010). However, these techniques are usually time-consuming, de-manding high-quality reference materials, or even not easily accessiblefor themajority of geologists worldwide. Themost widely usedmethodin the literature for calculating the structural formula of biotite fromEMPA data is following a normalization scheme (e.g., assuming 22 pos-itive charges; ref. Rieder et al., 1998), with or without an estimation ofFe oxidation state and hydrogen content. However, the available nor-malization schemes in the literature (e.g., Dymek, 1983) are not robustenough and may result in large errors in the estimations of Fe3+/ΣFe orOH in biotite (see details below), which hinders or flaws greatly the useof biotite composition for petrological and geochemical applications.

With the development of computer science, various machine learn-ing algorithms have been widely used in geoscience researches. For ex-ample, partial least-squares regression (PLS) has been successfully usedto analyze the spectral data of XANES (X-ray absorptionnear-edge spec-troscopy) for evaluating the Fe3+/ΣFe of garnet (Dyar et al., 2012a). Inaddition, principal component regression (PCR) and PLSwere employedto acquire the response spectral intensity data of LIBS (remote laser-induced breakdown spectrometer) to improve the accuracy of elementidentification (Devangad et al., 2016; Dyar et al., 2012b). PCR is a robustmethod of machine learning (e.g., Merz and Pazzani, 1999), which hasbeen widely applied for signal processing and evaluation (e.g., Changet al., 2001). In this paper, we present a PCR-based machine learningmethod, trained and tested upon a large reference dataset of well-characterized biotites, for calculating biotite formula from routineEMPA data.

2. Biotite structure and cation assignment

Biotite (sensu lato), including phlogopite and biotite (sensu stricto),has a simplified formula of A1M3T4O10W2 (Brigatti and Guggenheim,2002; Rieder et al., 1998), where A represents the interlayer site com-monly occupied by K, Na, Ba, Ca, or vacancies; M refers to octahedralsites that are generally occupied by Mg, Fe2+, Fe3+, Al, Ti, Mn, Cr, Liand vacancies; T refers to the tetrahedral sites occupied by Si, Al, andFe3+; andWcorresponds to the anion site [hereafterO(4) site] occupiedby OH, WO2−, F and Cl. In pure biotite, the three octahedral sites are co-ordinated by 4 O2– and 2 OH−. One can distinguish twoM(2) sites withcis position of the OH groups and one M(1) site with trans position oftheOHgroups (Brigatti andGuggenheim, 2002). The values of cation as-signments are expressed in atom per formula unit (apfu).

3. Previous biotite normalization methods

Calculation of biotite formulae exclusively from EMPA data has longbeen recognized as a challenging task, and the most difficult problemsare the assignment of Fe2+ and Fe3+ for given total FeO content andthe estimation of OH and WO2−. The most popular scheme in literatureis the normalization of cations to 11 oxygen atoms (hereafter 11 oxygenmethod), which fixes the total cation charge to a value of 22 and as-sumes that all the iron is present as Fe2+. This procedure allows thepresence of octahedral vacancies but forbids the possible deprotonation(i.e. from OH– to WO2−) at the O(4) site. Therefore, the cation abun-dances could be overestimated (the overestimation is proportional tothe amount of Fe3+) or underestimated (the underestimation is propor-tional to the extent of deprotonation). Some problems related to thisscheme have been bypassed by assuming a fixed Fe3+/ΣFe valuebased on the coexistence of biotite with graphite or magnetite (e.g.Guidotti and Dyar, 1991).

High charge cation substitutionmechanisms, especially involving Ti,have long been recognized to be associated with WO2− at the O(4) sitein biotite (e.g. Dymek, 1983). The prevailing Ti-oxy substitution

(Eqs. (1)–(2)) and Ti-vacancy substitution (Eq. (3)) in biotite can beexpressed by the following equations:

VI Mg; Fe;Mnð Þ 2þ þ 2WOH− ¼ VITi

4þ þ 2WO2− þH2 gð Þ ð1Þ

VI Al; Fe;Crð Þ 3þ þ WOH− ¼ VITi

4þ þ WO2− þ 1=2H2 gð Þ ð2Þ

2VI Mg; Feð Þ 2þ ¼ VITi4þ þ VI□ ð3Þ

As described in Section 2, there are two types of octahedral sites inthe biotite structure, including one M(1) site and two M(2) sites, andthe size of M(1) is larger than M(2) (Brigatti and Guggenheim, 2002).The divalent cations Mg and Fe can distribute orderly at both 1 M(1) and 2 M(2) octahedral sites. There are two controversial opinionsfor the occupation of Ti and other highly charged cations: (1) Ti onto1 M(1) site together with Al3+ and Fe3+ (White et al., 2014). (2) Tionto 2 M(2) sites while Fe3+ and Al3+ onto the M(1) site(Tajcmanova et al., 2009). Under high-temperature crystallization con-dition, Ti could incorporate into biotite structure by Eq. (1) that Ti4+

substitutes one divalent cation (Mg2+ or Fe2+) on the same octahedralsite and deprotonates two hydrogen ions (H+) from the hydroxyl sitesto keep the charge balance; (2) By Eq. (3) that Ti4+ incorporates intoone octahedral site accompanying the substitution of one divalent cat-ion (Mg2+ or Fe2+) in the adjacent octahedral site plus a vacancy to re-tain electroneutrality. Besides Fe3+, Al3+ and Cr3+ are usually alsosupposed to be introduced into octahedral sites by Tschermak-type ex-change (e.g. Dymek, 1983; Tajcmanova et al., 2009). These cations arealso assumed to be associated with Ti-oxy substitution through Eq. (2)(e.g. Dymek, 1983; Righter et al., 2002). By considering the relative im-portance of thesemechanisms or combination of them, several relation-ships between octahedral cations and WO2− have been established toestimate the WO2− content at the O(4) site.

Since Ti-oxy substitutions were considered to be a dominant mech-anism that contributes to the WO2− at the O(4) site, an improved nor-malization based on (11 + Ti) oxygen atoms was proposed based onthis observation (Waters and Charnley, 2002). Accordingly, Righteret al. (2002) proposed a correlation between WO2− at O(4) siteand the sum of octahedral highly-charged cations, expressed asWO2− = VIAl + VIFe3+ + Ti + Cr. Henry and Daigle (2018) suggesteda method for calculating the amount of WO2− at the O(4) site via 2*Tion the basis of 11 oxygen. The reliabilities of these two methods for es-timating WO2− are tested below (see details in Discussion).

The other commonly applied scheme is the normalization to 7 cat-ions (hereafter 7 cations method), which assumes that the tetrahedraland octahedral sites are completely occupied by cationswithout any va-cancy. One merit of this method is that it permits direct estimation ofFe3+ through the charge balance difference between the negative 22charges and the total positive charge. However, both the cation abun-dances and the acquired charge excess could be overestimated. All ofthese oversimplifications could result in incorrect estimation of Fe3+.Dymek (1983) created an iterative normalization procedure that allo-cates a vacancy for each Ti and 2 excess octahedral Al on the basis ofthe 7 cationsmethod, fromwhich the Fe3+ could be estimated by elim-inating charge excess. It is worth noting that, in the paper of Dymek(1983), no independent measurement was provided to evaluate the re-liability and uncertainty of the model. Applying the method of Dymek(1983), Yavuz and his colleagues worked out several software packages(Yavuz, 2003; Yavuz andÖztas, 1997) for calculating biotitemineral for-mula from EMPA data. However, as shown below by our tests (see de-tails in Discussion), estimation of Fe3+ using Dymek (1983)’s methodmay have large errors.

3X. Li et al. / Lithos 356–357 (2020) 105371

4. Method development

In order to establish a reliable protocol to calculate biotite structuralformula from EMPA data, we developed a new machine learningmethod based on principal components regression (PCR) of a large bio-tite dataset. Details of the dataset, the data processing protocol, the sta-tistical method, as well as the subsequent prediction quality anduncertainty, are described below.

4.1. Working dataset

Compositions and structures of biotite sensu lato that have been ac-curately characterized, including X-ray diffraction on single crystals,were collected as aworking dataset. The compositionswere determinedby EMPA, the Fe3+/ΣFe ratios were determined byMössbauer spectros-copy or micro-XANES, and the H2O contents were quantified by FTIR,SIMS or other independent methods (Supplementary Table S1). Biotitecompositions that clearly experienced post-formation oxidation or re-duction, such as those described in Matarrese et al. (2008), Lacalamitaet al. (2011) and Laurora et al. (2007), have been excluded from theworking dataset (see detailed explanation in Discussion).

The collected biotites in the working dataset have a broad chemicalrange for elements such as Si, Al, Mg, Fe, K, Ti and F: SiO2 (33–46 wt%),Al2O3 (8–22 wt%), MgO (5–28 wt%), FeOT (0–25 wt%), K2O (7–11 wt%),TiO2 (0–9 wt%) and F (0–6 wt%). The other elements are commonly atminor or trace levels: BaO (0–3 wt%), Na2O (0–1.2 wt%), Cr2O3

(0–1 wt%), MnO (0–1 wt%), Li2O (0–0.3 wt%), CaO (0–0.1 wt%), NiO(0–0.14 wt%) and Cl (0–0.4 wt%). The majority of biotites in this datasetare from volcanic rocks, covering both phlogopite and biotite with alarge variation in Mg/(Mg+ Fe) from ~0.25 to ~1 (Fig. 1a). Five samplesare low-Al ferri-phlogopite (Fig. 1a) from glimmerite andalkali‑carbonatite complex (Brigatti et al., 1996a, 1996b). The anionsOH, F and O2− are the dominant species at the O(4) site, with a traceamount of Cl (Fig. 1b). The dataset used in this study covers a majorpart of biotite compositional space, which is supposed to provide a reli-able statistical model for calculating biotite formula for biotite with awide compositional range. However, we suggest that application of thismethod would yield best reliability if the biotite compositions to beused fall into the range of reference biotite dataset collected in this study.

4.2. Data processing protocol

Step 1, for each reference biotite, themolar concentration of each ionis calculated (SiO2, TiO2, Al2O3, Cr2O3, FeOT, MnO, MgO, NiO, BaO, CaO,

Fig. 1. Illustrations of biotite compositions in the working dataset. (a) Plots inMg/(Mg+ Fe) vsdiagram.

Na2O, K2O, Li2O, F, Cl). The FeOT content is converted from the originalFe2O3 and FeOwhen both of them are provided. The element concentra-tions acquiredwith EMPA (Si, Ti, Al, Cr, Fe2+,Mn,Mg, Ba, Ca, Na, K, F, Cl),except Li (not measurable for routine EMPA), are summarized to form amatrix X (m by n), in which the row (m= 155) represents the numberof collected biotite and column (n = 13) indicates the number ofelements.

Step 2, the molar proportions of each site-assigned element are col-lected from structure-refined biotite formula given in literature, includ-ing T.Si, T.Al, T.Fe3+, M.Al, M.Mg, M.Fe2+, M.Fe3+, M.Ti, M.Cr, M.Mn, M.Ni, M.Li, A.K, A.Na, A.Ba, A.Ca,W·F,W·Cl,W.OH,W·O2−, where prefixes“T.”, “M.”, “A.” and “W.” mean tetrahedral, octahedral, interlayer and O(4) sites, respectively. These data are used as reference values to estab-lish our new calculation scheme.

Step 3, molar proportions of T.Si, T.Al, M.Mg, M.Fe2+, M.Ti, A.K, W·F,andW.OH are selected to make up a matrix Y with 155 rows and 8 col-umns. The selection criteria for these 8 parameters are based on:(1) selecting the dominant cations of each crystallographic site (Siand Al on tetrahedral sites; Mg, Fe2+ and Ti on octahedral sites; K oninterlayer sites; F and OH on O(4) sites); (2) leaving freedom forions which could reside in different sites or have various valence statesto keep the total amount of ions constant (therefore, only Fe2+ is se-lected at this step); (3) selecting OH (derived from direct measure-ment of H2O content) instead of WO2− (calculated through WO2− =2 – F − Cl).

Step 4, a linear equation is established between thematrix X and thematrix Y:

Y ¼ Xβ þ ε ð4Þ

in which β denotes the regression coefficient matrix acquired throughprincipal component regression (PCR) and ε dominates the vector ofrandom errors with expected values equal to zero. The overall matrixlinear Eq. (4) can be expressed as the following equation:

T:Si ⋯ W:OH⋮ ⋱ ⋮

⋯

0@

1A

155�7

¼Si ⋯ Cl⋮ ⋱ ⋮

⋯

0@

1A

155�13

�β1;1 ⋯ β1;7⋮ ⋱ ⋮

β13;1 ⋯ β13;7

0@

1A

13�7

þ β0;1 … β0;7� �

1�7 ð5Þ

As a result, there are 8 corresponding linear equations for each col-umn Y (T.Si, T.Al, M.Mg, M.Fe2+, M.Ti, A.K, W·F, and W.OH). For

. IVAl/(IVAl + IVSi) binary diagram (after Deer et al., 1962). (b) Plots in F-OH-WO2− ternary

4 X. Li et al. / Lithos 356–357 (2020) 105371

example, the unfolded linear equation for predicted cells on thefirst col-umn of Y (T.Si) could be expressed as the following equation where thesubscripts i represent the row index of X and Y matrix:

T:Sii;1 ¼ β0;1 þ β1;1Sii;1 þ β2;1Tii;2 þ β3;1Ali;3 þ⋯þ β13;1Cli;13 ð6Þ

Step 5, once the regression coefficient matrix obtained by principalcomponent regression has been evaluated and accepted, the 8 site-assigned cations and anions (T.Si, T.Al, M.Mg, M.Fe2+, M.Ti, A.K, W·F,andW.OH) are calculated directly from EMPA data through the 8 linearequations. The comparisons of reference values and predicted values atthis step are illustrated in Fig. 2.

Fig. 2. Correlation between predicted and reference ions for theworking dataset. Predictions forformula unit (apfu).

Step 6, a correlation coefficient (denoted by γ) is obtained from thefollowing equation:

γ ¼ T:Si=Si ð7Þ

The reason why the selected coefficient is based on Si is that silica isalways a major component in biotite that can be measured by EMPAwith high accuracy. Hereafter, the mole proportion of M.Cr, M.Mn, M.Ni, M.Li, A.Na, A.Ba, A.Ca, and W·Cl, is calculated through multiplyingγ through their mole amount derived from EMPA data. An examplefor Cr is expressed as

M:Cr ¼ γ � Cr ð8Þ

these 8 ions are calculateddirectly from regression coefficients of PCR. Plots are in atomper

5X. Li et al. / Lithos 356–357 (2020) 105371

Step 7, the assignments for M.Al and M.Fe3+ are calculated from themole proportion difference between total cation number and the valuescalculated in steps 5 and 6. For T.Fe3+ and W·O2−, the calculation isbased on mass balance and full assignment at O(4) site respectively.

M:Al ¼ γ � Al−T:Al ð9Þ

T:Fe3þ ¼ 4−T:Si−T:Al ð10Þ

M:Fe3þ ¼ γ � Fe−T:Fe3þ−M:Fe2þ ð11Þ

W:O2− ¼ 2−W:OH−W:F−W:Cl ð12Þ

Step 8, if negative values for a given cation or anion are obtained foran occupation site, which would reflect a propagated error from EMPA

Fig. 3. Correlation between predicted and reference ions for the working dataset. Predictions ffrom regression coefficients of PCR (see Fig. 1) and stoichiometry of biotite. Plots are in atom p

andmodel uncertainty, this value is forced to be zero. Octahedral and in-terlayer vacancies are both allowed to be present and can be calculatedusing the ideal site number (3 for octahedral sites and 1 for interlayersites) minus all present cations on each site respectively. The compari-sons between reference values and predicted ones at steps 6 to 8 are il-lustrated in Fig. 3. After this procedure, the distribution of the differentcations and anions in the different sites can be calculated (Supplemen-tary Table S1).

For calculating themineral formulae from biotite EMPA data, a user-friendly Excel spreadsheet is provided as Supplementary Table S1. OnlyEMPA data in wt% are needed as input, and all the calculation stepsmentioned above will be performed automatically by using the cali-brated regression coefficient obtained from the reference dataset. Theinput of trace element oxides, such as NiO and Li2O, is optional. The ab-sence of data for these elements would not affect the final formula

or these ions are based on constraints from the 8 major ions which are calculated directlyer formula unit (apfu).

6 X. Li et al. / Lithos 356–357 (2020) 105371

quality since these cations have not been used in the regression proce-dure. However, a value for the primary 13 oxide concentrations (seestep 1) should be provided otherwise the omitted input will be consid-ered to be zero, which might lead to a biased formula.

4.3. Statistical method

When applying statistical analysis on mineral chemistry datasets,closure and the associated multicollinearity are inevitable issues thatneed to be addressed carefully. Closure is typically present in chemicalsystems with a constant sum. The phenomenon of closure on mineralchemical data refers to the competition among atomic species for crys-tallographic positions, whichwould exaggerate the correlation betweenthe different elements of mineral (Young et al., 1997). Furthermore, be-cause a specific crystallographic position has a fixed number (e.g. 3 oc-tahedral sites) and because element exchanges at such sites arecontrolled by charge balance, correlation between relevant ions arealso strongly affected by closure (Young et al., 1997). Multicollinearityis the technical term for the situationwhere a pair of predictor variableshave a substantial correlation with each other and/or between multiplepredictors at once (Kuhn and Johnson, 2013). Since biotite has manycrystallographic sites and can accommodate a large number of cations,the issue of closure can also induce multicollinearity problems aboutits chemical composition. For instance, Mg and Fe exhibit a conspicuousnegative correlation due to the competition on the octahedral site, Tiand WO2− have a significant positive correlation due to the Ti-Oxy sub-stitution mechanism, Ti and Mg have polytype correlation due to com-bined effects. These covariate/multicollinearity features would lead tobiased estimation when performingmultivariate analysis through ordi-nary least-squares regression method (Aitchison, 1984; Yoo et al.,2014).

A statistical dimension reduction method, namely principal compo-nent analysis (PCA), was introduced to identify and evaluate the validcombined substitution mechanisms in the biotite and amphibole com-position space (Hewitt and Abrecht, 1986;Labotka, 1983 ; Waters andCharnley, 2002 ; Young et al., 1997). In practice, the mineral composi-tion is first transformed into one additive endmember plus a linear in-dependent exchange composition set applying the method ofcomposition space (Spear et al., 1982; Thompson Jr, 1982), which canavoid the effect of closure from stoichiometry. Later, this new composi-tional space is analyzed by PCA to find the ordered principal compo-nents according to the extent of variance (Aitchison, 1983; Filzmoser,1999). The obtained principal components, which are linear combina-tions of the multiple cation correlations, indicate the substitutions thatminimize the distortions of the crystal structure and reflects actuallythe underlying physical-chemical control on biotite structure(Labotka, 1983; Waters and Charnley, 2002; Young et al., 1997).

In this study, we performed principal component regression (PCR)to solve the potential problems of closure and multicollinearity on bio-tite formula calculation. For a general linear regression model definedby Eq. (4), the process of PCR is composed of two parts: (1) finding prin-cipal components through principal component analysis (PCA) on ma-trix X; (2) obtaining regression coefficient β through the regression ofmatrix Y on the acquired principal components (Jolliffe, 2002; Mevikand Wehrens, 2007). In Eq. (4), the right-side matrix X holds multi-predictor variables, and the left side matrix Y represents multi-response variables. Bothmatrixes are describing the biotite compositionin a different form, but actually provide information on the samecrystal-chemical constraints. This means that each principal componentcapturing the variance of X is associated with a component that cap-tures the relevant variance of Y. Therefore, this treatment perfectly coin-cides with the core hypothesis of PCR, which assumes that thedirections in which the predictors show the largest variation are theexact directions associated with the responsible variables (Kuhn andJohnson, 2013). In addition, in our PCR performance, there is no needto transform the two matrixes (X and Y in Eq. (2)) simultaneously

into new compositional spaces in order to eliminate closure (Chayesand Trochimczyk, 1978; Davis, 2002). Indeed, we also applied an alter-native method of data processing on the same dataset, namely partialleast squares regression (PLSR), whichmaximizes the covariancematrixbetween X and Y and requires less components than PCR (Wold et al.,2001). However, no significant improvement of predicted biotite for-mula has been observed, and therefore the simple PCR is preferred inthis study according to the theory of Occam's razor.

Data processing is performed using open source GNU R software (RCore Team, 2018) and the PLS package (Mevik et al., 2019). As describedabove, we have collected a dataset consisting of 155 biotites, includingchemical composition and structural refinement. The dataset is firstsplit into a training set (75% in amount) and a test set (25% in amount)by a simple random sampling method (with no replacement) embed-ded in R software. The training set (n=116) is used to tune the predic-tive model while performing repeated 10-fold cross-validation toestimate the model performance. The hand-out test set (n = 39) isthen employed to evaluate the generalization of themodel. It is empha-sized that the initial regression coefficients obtained by PCR on the basisof principal components of X have been transformed back automaticallyby the PLS package into a coefficient matrix β in the original form of X.

4.4. Prediction quality and uncertainty

Compared to the original data, the predicted ion occupations in bio-tite from the training set and from the independent test set follow thesame 1:1 trend for all cations and anions (Figs. 2–5),whichdemonstratethe ability of ourmodel to be applied to a variety of biotite compositions.The first 6 cations and 2 anions plotted in Fig. 2 are calculated directlyfrom the regression coefficient equations obtained by PCR. The pre-dicted Mg, Ti, and F perfectly coincide to the reference data with R2 al-most equal to 1 (Fig. 2c, e and g). Tetrahedral Si and interlayer K bothhave an excellent R2 of 0.98 (Fig. 2a and f). The prediction quality of tet-rahedral Al and octahedral Fe2+ is slightly lower but is still satisfyingwith R2 of 0.93 and 0.95, respectively (Fig. 2b and d). The predictionquality of OH is lower with R2 of 0.90 (Fig. 2h).

Fig. 3 illustrates the prediction performance of other ions that arecalculated by multiplying the coefficient γ with their mole amount.Minor elements in biotite, such as Cr, Mn, Na, Ba, Ca, Li and Cl, alsohave good consistency with the reference dataset (R2 = 0.93–0.98,Fig. 3b–h). It is worth noting that, in some cases, our model predicts aslightly higher proportion for some elements present as trace (e.g. Mnand Cl). This overestimation is probably due to rounding of numericalvalues or lower precision of the data in the literature. The octahedralAl, which is calculated by the difference between total Al and the pre-dicted tetrahedral Al (Eq. (9)), has a relatively lower prediction qualitycompared with the reference data (R2 = 0.81, Fig. 2a). Similarly, thepredicted WO2− and total Fe3+, which are also calculated indirectly,have slightly lower R2 of 0.89 (Fig. 4a) and 0.90 (Fig. 5a) respectively.The predicated tetrahedral Al and octahedral Fe2+ generally havean error of ±0.2 apfu (Fig. 2b and d), and their complementary octahe-dral Al and total Fe3+ have relatively larger errors of ±0.3 apfu (Figs. 3aand 5a).

The prediction quality is closely associated with the quality of origi-nal data of cation assignment in the literature. In our model, the pre-dicted cation and anion occupancies that were derived from EMPAdata are in good agreementwith reference data, while for other compo-nents measured by other methods usually show lower quality. For in-stance, OH in the O(4) site in the literature could be derived throughseveral independent analytical methods, such as H2O content measuredby FTIR (Scordari et al., 2013), SIMS (Matarrese et al., 2008), C-H-N anal-ysis (Schingaro et al., 2007), or by empirical linear regression equationof cell parameter along c-axis versus OH (e.g. Lacalamita et al., 2012;Ventruti et al., 2008); therefore, the inherited analytical inconsistencieswill eventually propagate into the final prediction model, resulting inrelatively low prediction qualities for OH and WO2− (Figs. 2h and 4a).

Fig. 4. Correlation between predicted and reference WO2− for the working dataset.

7X. Li et al. / Lithos 356–357 (2020) 105371

Analogously, the Fe oxidation state of biotite in the reference datasethas been normally measured by Mössbauer spectroscopy (Lacalamitaet al., 2011) or micro-XANES (Scordari et al., 2010), and the data mayhave different levels of accuracy. The accuracy of biotite Fe3+/ΣFe ana-lyzed by micro-XANES is believed to be ±10–20% (Righter et al.,

Fig. 5. Upper panels: correlation between predicted and reference total Fe3+ (IVFe3++VIFe3+

(predicted - reference) and FeOtot content of reference biotite.

2002). The accuracy of Mössbauer spectroscopy for the determinationof Fe oxidation state depends on several factors: the FeO concentrationof the sample, the recoil-free fraction, the thickness of the sample, andthe employed fitting method. The optimized accuracy of Mössbauerspectroscopy is accepted to be similar to that of wet chemistry (±0.05

) for the working dataset. Lower panels: correlation between discrepancy in Fe3+/ΣFe

8 X. Li et al. / Lithos 356–357 (2020) 105371

in Fe3+/ΣFe; McCammon et al., 1998; Sobolev et al., 1999; Dyar, 2002;Schuessler et al., 2008). Therefore, the relatively low prediction qualityfor total Fe3+ (Fig. 5a) is considered partly due to analytical uncer-tainties of the original data. In addition, another major source of errorcomes from the assumption of our model that tetrahedral sites arefully occupied by tetrahedral Fe3+ in addition to Si and Al (Eq. (6)).However, a Fe3+ doublet related to the tetrahedral coordination is notalways observed by Mössbauer spectroscopy, especially for biotiteswith low Fe contents (Lacalamita et al., 2011). The final propagated un-certainty is clearly displayed by the blowing up of error on Fe3+/ΣFevalues towards the low concentration of FeOT (Fig. 5d). The absoluteerror of predicted Fe3+/ΣFe generally shows a funnel-shaped trendalongwith the variation of FeOT content, with themajority of predictionerrors falling into the interval of ±0.4 if FeOT ≥ 5 wt% and further intothe interval of ±0.2 if FeOT ≥ 10 wt% (Fig. 5d).

5. Discussion

5.1. Improvements in predicting biotite formula compared to othermethods

Fig. 4 compares the result for WO2− on O(4) site occupancy pre-dicted by our model with the results predicted by the empirical equa-tion of WO2− = 2*Ti (Henry and Daigle, 2018) and the empiricalequation WO2− = VIAl + VIFe3+ + Ti + Cr (Righter et al., 2002). Thepredicted trends from this study (Fig. 4a) and from Henry and Daigle(2018)’smethod (Fig. 4b) are similar, but ourmodel shows a better con-sistency of predicted and measured data along the 1:1 line. In contrast,WO2− calculated by the amount of specific octahedral cations (Righteret al., 2002), which are based on the combinations of several Ti-Oxy sub-stitution mechanisms (Eqs. (1) and (2)), displays a rather bad predic-tion quality (Fig. 4c). This relationship also implies that Eq. (1) of Ti-oxy substitution (rather than Eq. (2)) is the major mechanism that ac-commodates the dehydrogenation at the O(4) site.

Fig. 5 illustrates the prediction qualities for total Fe3+ and Fe3+/ΣFeobtained using our method, the 7 cations method, and the iterative cat-ion normalization method proposed by Dymek (1983). It is apparentthat the prediction quality of the 7 cations method for the total Fe3+ ispoor (Fig. 5b), while the Dymek (1983)’s method tends to overestimateFe3+ (Fig. 5c). For the 7 cations method, the relationship between theprediction errors in Fe3+/ΣFe and FeOT content shows no systematiccorrelation (Fig. 5e), suggesting that this method should be abandonedeven if FeOT content of biotite is high. In addition, the Dymek (1983)’smethod can predict biotite Fe3+/ΣFe with an uncertainty ≤0.2 only ifFeOT content ≥20 wt%, and the uncertainty may increase remarkablyto ±0.6 if FeOT content is ≤15 wt% (Fig. 5f).

5.2. Application for Fe oxidation state

Our new method established based on the multi-variable statisticalmethod of PCR considers the chemical and structural constraints of bio-tite simultaneously,which implies that the calculated formula is built onthe multiple possible substitutions during the formation of biotite.These substitutions are governed by the physical-chemical conditionsof the systems in which biotite crystallized and in which biotite equili-brated with the surrounding melts, fluids or solid phases. However,after the formation of biotite, if there is any chemical modificationwhich do not follow the substitution laws, involving only an individualelement and/or individual crystallographic site, biotite compositionwould be modified without significant structure distortion and ourmethod would not be applicable. Typical cases of biotite for which ourproposedmethodwould not be applicable include post-formationmod-ifications at disequilibrium conditions, such as diffusion of mobile ionsor biotite oxidation or reduction processes under atmosphere or at ex-perimental conditions.

The Fe-oxidation/deprotonation is a common post-formation pro-cess observed in Fe-bearing hydrous silicate minerals, especially for

amphibole and biotite groups that have a relatively incompact structurein favor of H2 diffusion. This dynamic process under heat-treated exper-iment condition has been investigated in detail for amphibole (e.g. DellaVentura et al., 2018; Oberti et al., 2018; Popp et al., 2006; Zema et al.,2012), biotite sensu stricto (e.g. Chon et al., 2003; Hogg and Meads,1975; Rancourt et al., 1993; Sanz et al., 1983) and phlogopite sensustricto (e.g. Chon et al., 2006; Virgo and Popp, 2000; Zema et al., 2010).

The whole process could be expressed as:

VIFe2þ þ WOH

− ¼ VIFe3þ þ WO

2− þ 1=2H2 gð Þ ð13Þ

which involves the oxidation of Fe2+ to Fe3+ and a simultaneous releaseof hydrogen from the linked hydroxyl group to retain the charge bal-ance. Rancourt et al., (1993) decomposed this whole process intomulti-ple steps: (1) firstly, local dissociation of hydroxyl groups (OH− =O2− + H+); (2) the associated octahedral Fe2+ is oxidized in situ intoFe3+ (Fe2+ = Fe3+ + e−); (3) when diffusing out from the crystal, thereleased hydrogen could catalyze with local hydroxyl groups(H+ + OH− = H2O) or react with environmental oxygen(2H+ + 1/2O2 = H2O) to form molecular water. It has been proposedthat this Fe-Oxy dehydrogenation mechanism (Eq. (13)) can also occurin some natural igneous biotites in volcanic systems, and the resultantH2O (formed by the release H2 and environmental O2)may have playeda role in triggering volcanic eruption (Feeley and Sharp, 1996).

As demonstrated by the heat-treated experiments under atmo-sphere, the Fe-oxidation/deprotonation process in biotite may start attemperatures as low as 400–500 °C and can process continue untilbeing broken down into α-Fe2O3 at 900–1000 °C (Hogg and Meads,1975; Vedder and Wilkins, 1969). Mössbauer spectra of samples afterheat-treated experiments have revealed the progressive change of coor-dination environment of the Fe3+ from Fe3+(O5OH) to Fe3+(O6) undergradual heating (Hogg and Meads, 1975). The rate of the oxidation andthe deprotonation is controlled by the oxygen pressure (Sanz et al.,1983), and the Fe at 2 M(2) sites is more easily oxidized than that atM(1) site (Ferrow, 1987). This whole process is also concomitant withchange of unit-cell parameters (Chon et al., 2003; Chon et al., 2006;Zema et al., 2010). In addition, the oxidation/deprotonation process(Eq. (13)) is demonstrated to be reversible via hydrothermal experi-ments with various oxygen buffers (Virgo and Popp, 2000).

In summary, the Fe-oxidation process involves the oxidation of Fe2+

to Fe3+, H+ and electron diffusion, and crystal structural unit rearrange-ment. However, onlyH+ and electron are generally open to the environ-ment. Thus, any multiple regression with major elements in the crystalwould not give a reasonable estimation for Fe3+ content that has beenmodified by post-formation oxidation. Therefore, the application ofour model in relevant cases must be in caution. We made a test onEMPA data of biotites that have been treated by heating in the presenceof oxygen (Virgo and Popp, 2000), and a low Fe3+/ΣFe ratio of 0.25 isobtained. This value is very close to the starting Fe3+/ΣFe ratio (0.19)prior to the heating experiments, but rather lower than the ratios(0.87–1.00) that were finally achieved afterwards.

Therefore, our method can only predict the initial Fe oxidation stateand the associated H2O content when biotite was formed in an equilib-rium state, in amagma chamber or a hydrothermal system. The effect ofpost-formation oxidation (Eq. (13)) or reduction (reverse of Eq. (13))processes on biotite composition, which usually exclusively involvesH2 diffusion, could not be detected by our method. On the other hand,formulae of such biotites predicted from our method, particularly theinformation of OH and Fe3+/ΣFe, actually reflect the physical-chemicalcondition of their host magmatic or hydrothermal system prior to anypost-formation process.

6. Concluding remarks

In order to establish a reliable method for calculating biotite formulafrom routine EMPA data, we developed a PCR-based machine learning

9X. Li et al. / Lithos 356–357 (2020) 105371

method built on data training and test using a large reference dataset ofwell-characterized biotites. This method can provide distribution of allcations and anions, including the tetrahedral Fe3+, octahedral Fe2+, oc-tahedral Fe3+, OH and WO2− at the O(4) site, and the prediction qualityfor major elements is robust with R2 N 0.95. In comparison to previousnormalization schemes, our model shows significant improvements inestimating OH and WO2− at O(4) site and Fe3+/ΣFe ratio.

Declaration of Competing Interest

The authors declare that there is no conflict of interest regarding thepublication of this article.

Acknowledgments

This study was supported by DFG (German Research Foundation)project BE 1720/40. We thank two anonymous reviewers and theircomments have greatly improved this paper. The Microsoft Excelspreadsheet for calculating biotite formula from EMPA data are pro-vided as a supplementary file online.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.lithos.2020.105371.

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