# Factor Analysis of XRF and XRPD Data on the Example of the Rocks of the Kontozero Carbonatite Complex (NW Russia). Part I: Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

**X**variables of size (N × M), where N is the number of observations (rows) and M is the number of independent variables (columns). FA can be carried out for both variables (R-technique) and observations (Q-technique). The analysis involves examining either the correlation matrix or the covariance matrix of

**X**. The former approach is most often used. The

**X**matrix must be converted to a standardized

**X**

^{S}matrix, which is then decomposed into several latent variables (factors). These are calculated as eigenvectors of the correlation matrix of the standardized data. The magnitude of the corresponding eigenvalues represents the variance of the data by the eigenvector directions [15]. Decomposition of an

**X**

^{S}data matrix implies data separation into two parts—a structure part and an error part:

**X**

^{S}=

**AB**

^{T}+

**E**= Structure + Errors

**A**is the matrix of “factor scores” (of size N × n),

**B**is the matrix of “factor loadings” (of size M × n), the apex T means transpose, and n ≤ min(N, M) (where N is the number of samples, M is the number of independent variables, and n is the number of factors). The above inequality stipulates the transition to a space of lower dimension. The optimal n can be calculated through the sum of the eigenvalues of the used factors, which represents the data dispersion explained by these factors. The residuals (errors) are collected in an

**E**matrix in such a way that an

**A**matrix of factor scores describes the position of the samples in the new coordinate system. The

**B**matrix of factor loadings describes the new axis, which is built on the original one. The factor score (FS) values describe the magnitude of a factor. FSs characterize the observations. The factor loadings (FLs) are the coefficients of the correlation between the factors and the original variables. They characterize the entire dataset and not a specific observation. In particular cases, the FA procedure becomes more complicated (for example, due to the application of the singular value decomposition algorithm [16]). However, the general FA principle remains unified.

## 3. Materials and Methods

#### 3.1. Sample Description

_{2}, (2) silicocarbonatites (essentially carbonate rocks of endogenous origin containing > 20 wt% SiO

_{2}), and (3) a variety of carbonate-bearing silicate and aluminosilicate rocks (from normal to alkali content, with both Na and K alkalinity types).

#### 3.2. Analytical Techniques

#### 3.2.1. XRPD

#### 3.2.2. XRF

#### 3.3. Data Processing

_{max}-I

_{min}), where I

_{max}and I

_{min}represent the maximum and minimum values in the corresponding “raw” diffractogram. Since most diffractograms showed subhorizontal baselines, this simplified approach for estimating the intensity of the principal peak satisfied the correctness. The set of diffractograms, thus transformed, was supplemented with the contents of the chemical elements in the corresponding samples. Factor analysis was performed in the modification of the principal component method using IBM SPSS Statistics v. 23 (IBM Corp., Armonk, NY, USA; [14]). An R-technique of FA (by variables) was used. The VARIMAX rotation [31], which is the most commonly used orthogonal rotation in FA, was also applied. Factors were identified using the online American Mineralogist Crystal Structure Database (AMSCD) [32], the QualX v. 2.24 program with the indexed XRPD database of open-access POW_COD [33], and the commercial PDF2 [34]. The calculated factor loadings and factor scores are listed in Supplementary Tables S4 and S5, respectively.

## 4. Results and Discussions

#### 4.1. Types of the Extracted Factors

#### 4.2. Evaluation of the Stability of the Factor Solution

- Raw X-ray diffraction patterns not subjected to any spectral operations;
- Raw diffraction patterns subjected to baseline correction;
- Smoothed diffraction patterns subjected to baseline correction (the data used in the method of [13]).

- The subset of carbonatites sensu stricto selected according to the formal principle “SiO
_{2}< 20 wt %”, commonly used for the classification of carbonatites [36]—99 observations; - The cumulative subset of all other rocks of the collection (silicocarbonatites and carbonate-bearing silicate rocks)—99 observations;
- The entire set of samples from the Kontozero collection—198 observations.

#### 4.3. Interpretation of the Results of Factor Analysis

_{3}(Sr,Ba,Ce)

_{3}(CO

_{3})

_{5}(Figure 12C) and baryte (Figure 12D). Their diffractograms do not contain pronounced peaks due to the low content of these minerals in the rocks (up to several wt%). However, the proposed technique turned out to be sensitive even to these accessory phases.

#### 4.4. The Algorithm for the Selection of Representative Samples

- XRPD and XRF analysis and primary data processing, including baseline corrections and removal of zero values. The output of this step is a database suitable for FA (an example is shown in the supplementary Table S3);
- Factor analysis of the obtained results. The outputs are (a) tables of FL values for XRPD and XRF variables (see Supplementary Table S4), (b) graphs of factor loadings for XRPD variables (e.g., Figure 5), and (c) factor scores for each sample (see Supplementary Table S5);
- Compilation and examination of all FL graphs on a single chart. The output is the rejection of all non-interpreted noise factors (see Figure 3);
- Interpretation of the easily interpretable factors by combining the two proposed techniques (Figure 10). The output is a highly confident mineralogical explanation of these factors;
- Interpretation of difficult-to-interpret factors by searching for the mineral (s) according to the position of the most intense peaks in the AMCSD database. The output is an assumption about the nature of these factors;
- Routine mineralogical (optical microscopy, SEM + EPMA, Raman) examination of the samples with the highest FSs. The output is a verification of the FA results, a collection of the most representative samples, an idea of the mineral composition of all studied rocks at the level of main, minor, and most accessory minerals, all within a reasonably short time.

## 5. Conclusions and Future Perspectives

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Examples of FL graphs of the most informative factors (red line—factor #5 “alkaline amphibole”; blue line—factor #9 “orthoclase”; green line—factor #12 “monticellite”).

**Figure 6.**(

**A**) An example of a robust linear relationship between the factor score (FS) values of analogous factors resulting from factor analysis of three XRPD datasets: “Raw”—raw data without spectral operations; “Raw + BL”—raw data with baseline correction; “Smooth + BL”—smoothed data with baseline correction. (

**B**) Comparison of factor loadings graphs of the same factors.

**Figure 7.**Comparison of the FL graphs of a specific factor, containing information about the baseline in the raw dataset (green line), and its closest analog from the result of applying FA to the “raw data vs. baseline correction” dataset (blue line).

**Figure 8.**Comparison of the FL graphs of three analogous factors obtained by processing the XRPD datasets of the entire set of rocks (green line), the set of carbonatites sensu stricto (blue line), and the cumulative set of silicocarbonatites and carbonate-bearing silicate rocks of Kontozero (red line).

**Figure 9.**Comparison of the fluorapatite diffraction pattern from the RRUFF database (ID R050122, blue line) with the FL graph of the fluorapatite factor (red line): (

**A**) for Kontozero rocks (this study); (

**B**) for the rocks of the Petyayan-Vara occurrence (Vuoriyarvi massif, NW Russia) from [4].

**Figure 10.**(

**A**) Comparison of the full FL graph of fluorite factor #14 (red line) with the diffractogram of a sample with the maximum FS of this factor (blue line); (

**B**) Ranked range of FS values of the fluorite factor; (

**C**) Comparison of the most significant peaks of the FL graph of fluorite factor (red line) with the diffractogram of a sample with the maximum FS of this factor (blue line); (

**D**) Comparison of the full FL graph of fluorite factor with fluorite peaks from the AMCSD online XRPD database; (

**E**) QualX v. 2.24 dialog box with a diffractogram of the sample with the maximum FS of fluorite factor (red peaks correspond to PDF2 fluorite card).

**Figure 11.**(

**A**–

**F**) The minerals of the studied rocks of the Kontozero complex: Ab—albite; Adr—andradite; Anc—Analcime; Ant—anatase; Ap—apatite; Brt—baryte; Bt—biotite; Cal—calcite; Chl—chlorite; Di—diopside; Dol—dolomite; Fl—fluorite; Ilm—ilmenite; Mag—magnetite; Mtc—Monticellite; Ntr—natrolite; Or—orthoclase; Py—pyrite; Qz—quartz; Sd—siderite; Srp—serpentine; and Str—strontianite. All images are backscattered electron (BSE) photos. Figure 11 is taken from [35].

**Figure 12.**Comparison of the FL graphs (red lines) of (

**A**) burbankite, factor #17, and (

**B**) baryte, factor #30, with diffraction patterns of the samples showing the maximum FSs of the corresponding factors (blue lines). (

**C**) Burbankite (Bur) and (

**D**) baryte (Btr) in the same samples in BSE photos.

**Table 1.**The correlation coefficients between scores of similar factors resulting from factor analysis (FA) of the X-ray powder diffraction (XRPD) datasets subjected to different spectral operations.

Raw data (factors #): | 1 * | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Raw data + baseline (factors #): | 1 | 2 | 3 | 4 | 7 | 5 | 6 | 8 | 9 |

Correlation coefficient between FSs: | 0.91 | 0.94 | 0.43 | 0.97 | 0.97 | 0.97 | 0.94 | 0.97 | 0.97 |

Raw data (factors #): | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Smoothed data + baseline (factors #): | 1 | 2 | 11 | 5 | 6 | 4 | 7 | 9 | 8 |

Correlation coefficient between FSs: | 0.76 | 0.89 | 0.20 | 0.94 | 0.96 | 0.95 | 0.87 | 0.94 | 0.95 |

1 * | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 17 | 22 | 24 | 26 | 30 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cal ** | Adr + Srp | Dol | Ab | Amp | Anl | Di + Mag | Ap | Or | Bt | Bur | Py | Ilm | Str | Brt | Ant | |

Si | −0.71 | 0.19 | −0.15 | 0.29 | 0.14 | 0.24 | 0.22 | −0.15 | 0.24 | 0.11 | −0.05 | 0.05 | 0.08 | −0.05 | −0.05 | 0.03 |

Ti | −0.67 | 0.24 | −0.14 | 0.05 | −0.02 | 0.17 | 0.05 | −0.24 | 0.04 | 0.23 | −0.07 | 0.10 | 0.20 | −0.04 | −0.05 | 0.15 |

Zr | −0.30 | −0.09 | −0.09 | 0.29 | −0.11 | 0.48 | −0.05 | 0.19 | 0.34 | −0.01 | −0.05 | 0.15 | 0.13 | −0.07 | −0.04 | 0.08 |

Al | −0.53 | 0.00 | −0.15 | 0.53 | −0.11 | 0.38 | −0.04 | −0.15 | 0.35 | 0.07 | 0.00 | 0.05 | 0.08 | −0.05 | −0.04 | 0.06 |

Ca | 0.83 | −0.09 | 0.01 | −0.19 | −0.17 | −0.16 | −0.21 | 0.13 | −0.13 | −0.14 | 0.05 | −0.05 | −0.07 | 0.06 | 0.06 | −0.04 |

Sr | 0.59 | −0.23 | 0.01 | −0.01 | −0.24 | −0.10 | −0.29 | −0.05 | −0.08 | −0.18 | 0.22 | −0.08 | −0.03 | 0.36 | 0.06 | −0.08 |

Mg | −0.55 | 0.32 | 0.07 | −0.29 | 0.32 | −0.10 | 0.35 | −0.08 | −0.21 | 0.17 | −0.07 | 0.01 | 0.02 | −0.05 | −0.04 | −0.01 |

Fe | −0.78 | 0.19 | −0.13 | −0.04 | 0.08 | 0.06 | 0.41 | 0.04 | −0.06 | 0.10 | −0.07 | 0.03 | 0.04 | −0.12 | −0.08 | 0.02 |

Mn | −0.55 | 0.07 | −0.08 | −0.12 | 0.21 | −0.09 | 0.23 | 0.08 | −0.13 | 0.04 | 0.09 | −0.07 | −0.13 | 0.02 | −0.02 | −0.20 |

Ba | −0.19 | −0.14 | −0.10 | 0.03 | −0.13 | 0.04 | −0.25 | −0.12 | 0.15 | −0.03 | 0.08 | −0.04 | −0.03 | 0.20 | 0.25 | −0.04 |

K | −0.47 | −0.13 | −0.20 | 0.13 | −0.05 | 0.12 | −0.05 | −0.15 | 0.66 | 0.19 | −0.09 | 0.06 | 0.04 | −0.06 | −0.03 | 0.07 |

Na | −0.51 | −0.19 | 0.08 | 0.56 | 0.26 | 0.42 | −0.01 | −0.07 | 0.02 | 0.04 | 0.07 | 0.00 | 0.07 | −0.09 | −0.03 | 0.04 |

P | −0.07 | −0.08 | −0.08 | −0.13 | −0.07 | −0.02 | −0.06 | 0.95 | −0.07 | −0.02 | 0.06 | 0.09 | −0.01 | −0.05 | −0.02 | −0.01 |

S | −0.20 | −0.14 | 0.16 | 0.05 | −0.19 | −0.05 | −0.29 | 0.24 | 0.02 | −0.03 | −0.11 | 0.67 | −0.01 | −0.05 | −0.02 | 0.01 |

L.O.I. | 0.75 | −0.27 | 0.26 | −0.12 | −0.07 | −0.20 | −0.25 | −0.09 | −0.10 | −0.13 | 0.06 | −0.02 | −0.14 | −0.08 | 0.00 | 0.07 |

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**MDPI and ACS Style**

Fomina, E.; Kozlov, E.; Bazai, A.
Factor Analysis of XRF and XRPD Data on the Example of the Rocks of the Kontozero Carbonatite Complex (NW Russia). Part I: Algorithm. *Crystals* **2020**, *10*, 874.
https://doi.org/10.3390/cryst10100874

**AMA Style**

Fomina E, Kozlov E, Bazai A.
Factor Analysis of XRF and XRPD Data on the Example of the Rocks of the Kontozero Carbonatite Complex (NW Russia). Part I: Algorithm. *Crystals*. 2020; 10(10):874.
https://doi.org/10.3390/cryst10100874

**Chicago/Turabian Style**

Fomina, Ekaterina, Evgeniy Kozlov, and Ayya Bazai.
2020. "Factor Analysis of XRF and XRPD Data on the Example of the Rocks of the Kontozero Carbonatite Complex (NW Russia). Part I: Algorithm" *Crystals* 10, no. 10: 874.
https://doi.org/10.3390/cryst10100874