# Chemometric Modeling of Trace Element Data for Origin Determination of Demantoid Garnets

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}Fe

_{2}(SiO

_{4})

_{3}) garnets. Demantoids have first been found in the Russian Urals in the river Bobrovka near the village Elizavetinskoye, close to Niznij Tagil (or often just called Tagil) in the middle of the 19th century [5,6]. Further deposits in the Urals were later found in Poldnevskoye (Poldnevaya), Korkodin, and Ufaley—all within a range of 100 km of Ekaterinburg. Only in the 1990s a second major deposit was found in the Erongo region of Namibia [7], which together with Russia represents most of the cut gems available in today’s market (Figure 1) [8]. Smaller deposits of gem quality material can also be found in Val Malenco in Italy [9], Antetezambato in Madagascar [10,11], Khuzdar, Baluchistan in Pakistan [12,13], as well as in Belqeys and Kerman, both located in Iran [14,15]. Certainly, there are many more deposits of demantoid garnets on the globe which are not mentioned here as to this date they do not play a major role in the gem market.

## 2. Materials and Methods

^{24}Mg,

^{27}Al,

^{47}Ti,

^{51}V,

^{52}Cr, and

^{55}Mn were measured by Laser ablation—inductively coupled plasma mass spectrometry (LA-ICP-MS) using a Cetac LSX-213 G2+ laser ablation system (Teledyne Cetac Technologies, Omaha, NE, USA) connected to a Thermo Fisher Scientific X-Series II ICP MS system (Thermo Fisher Scientific, Waltham, MA, USA).

^{43}Ca,

^{28}Si, and

^{56}Fe were also measured and utilized to monitor the smoothness of the Laser ablation process. Other elements (

^{7}Li,

^{59}Co,

^{60}Ni,

^{63}Cu,

^{64}Zn,

^{71}Ga,

^{74}Ge, and

^{88}Sr) were measured as well for representative samples but did not show any significant concentrations. Tuning and calibration were carried out using NIST 610, 612, and 614 glass standards; reference values were taken from Jochum et al. [17]. The laser ablation was operated with a 100 µm spot size, a laser fluence of 9.2 J cm

^{−2}and a repetition rate of 10 Hz. 130–230 shots were made on each spot and the ablated products transferred to the ICP-MS with a flow rate of 500 mL.min

^{−1}He gas. Each sample was measured at three different spots except for two pieces from Madagascar that have been sampled 5 times each.

^{n}(publicly available in-house written software), which provides a MatLab-based workflow for robust chemometric modelling [18]. Specifically, a table with sample IDs, meta data on the geographic origin of the sample as class information, and the respective element concentration profile for each sample were loaded in AI(OMICS)

^{n}, which was operated in MatLab (version R2018b, The Mathworks, Natick, MA, USA). Different strategies were applied, either considering all classes or considering just pair-wise comparison of classes. As a first step, native element concentrations were subjected to the data expansion module of AI(OMICS)

^{n}, which automatically generates, e.g., combinations of pair-wise ratios of parameters, logarithms of single parameters as well as parameter combinations for improved screening of non-linear events contributing variance between groups. To calculate logarithms in this context, concentrations below the limit of detection (LOD) were set to a fixed value of 0.1 ppm since negative and zero values cannot be logarithmized. The resulting variables were then subjected to a Kruskal-Wallis H-test in order to detect class-related variance and ranked according to their relevance [19]. A maximum of 50 of the most relevant variables got selected for further dimensionality reduction by principal component analysis (PCA). Subsequently, linear discriminant analysis (LDA) was applied to the extracted principal components (PCs). The number of PCs to be taken into account for LDA was determined by a cut-off at a threshold of 95% explained variance. This is a precaution in order to avoid overfitting and thus the consideration of random error (noise) rather than relationships between variables. Models were validated using the Monte-Carlo cross-validation (MCCV) method [20]. Specifically, 100 random models were generated with 90% of the data set as training data and the remaining 10% as test set. The overall quality of the models, i.e., the correctness of data predicted during MCCV, was judged using the multivariate correlation coefficient R

^{2}, the multivariate correlation coefficient after cross validation Q

^{2}, the accuracy (ACC), the Matthews correlation coefficient (MCC), the prevalence threshold as well as the confusion matrix. In particular the MCC has been found to be a very reliable metric for judging the quality of chemometric models. In contrast to the ACC, which constitutes the ratio of the sum of true positive and negative classifications and the total population, MCC also takes into account the false positive and false negative classifications. It can also be applied well in cases where group sizes differ significantly. The prevalence threshold is applied to test whether a particular class is represented with enough samples under consideration of the accuracy and precision of the respective model. To assess model robustness averaged class accuracies and their standard deviations were calculated based on the 100-fold MCCV. The importance of each element for the respective model was judged by calculating the variable importance in projection (VIP). Based on this information box plots were generated for these elements in R version 3.5.4 using the ‘ggplot2’ package [21,22,23,24,25,26,27].

## 3. Results and Discussion

^{n}, which aims at automation of certain steps of data pre-processing, was applied for the remaining six elements Al, Cr, Mg, Mn, Ti, and V. Specifically, inverse concentrations, products and ratios of all element concentrations measured as well as the calculated dependencies are generated to allow a systematic screening. Especially ratios, e.g., plotted in 2D graphs, have been shown in the past to contain important information as compared to using pure concentrations [8]. Similarly, as concentrations of the various elements span several orders of magnitude and are heavily skewed, transformation into their logarithm—as pointed out above—may improve chemometric modeling based on linear approaches, including PCA, PCA-LDA, partial least squares-discriminant analysis (PLS-DA), orthogonal PLS-DA (OPLS-DA) and others. Here, the natural logarithm (base-e log) has been chosen for model building with the PCA-LDA approach [28,29]. For this specific operation it was necessary to convert measurements numbers smaller or equal to zero to a small value but not zero. Considering the concentrations detected for the individual elements a fixed value of 0.1 was chosen empirically, which—being in the order of the value of the smallest concentrations of all elements measured—prevents unnaturally small logarithms. Several rounds of data elimination were performed considering the ranks of variables especially in the Kruskal-Wallis H-test as well as in the VIP-scores in PCA-LDA models. In addition, model quality was judged by the R

^{2}and Q

^{2}values as a function of the number of principal components to exclude overfitting. The Matthews correlation coefficient (MCC), which in contrast to R

^{2}and Q

^{2}also takes into consideration the effect of true and false positive/negative predictions as well, is also considered as a relative quality criterion between different models. The prevalence threshold, which takes into account the quality of discriminating power under consideration of the size of the groups, was judged, too. All models were generated as PCA-LDA for optimal comparison. Finally, it was found that logarithmizing of the pure element concentration as single data transformation was sufficient to achieve over 95% explained variance with no more than 3 to 5 principal components for pair-wise comparisons of locations. As logarithmizing of variables is a common way of converting non-linear problems into linear ones, it is not unexpected that LDA performs better with the logarithms of the element concentrations. Variables, such as ratios of concentrations or their logarithms were excluded to reduce multi-collinearity, which may significantly impact modeling by overfitting. Of note, PCA-LDA itself was chosen because it is known to be less prone to overfitting due to multi-collinear variables. Additionally, in contrast to, e.g., a PLS-DA (where dimensionality reduction is supervised while it is un-supervised in the PCA approach), it allows better assessment of samples that do not belong to groups used to generate models through judgment of the Mahalanobis distance [30] of a sample relative to the group center in multidimensional space [31]. Note, too, that LDA in principle requires normality of the data, which is often not given in real-life data sets, in particular when only small sample numbers are available. Using logarithms of element concentrations is an important step towards normality, which, however, is not met for every element/country combination. Nevertheless, the LDA classification has been shown in the past to perform very robust, despite the assumption of normality is violated [32,33]. If the data were perfectly normally distributed LDA would determine the optimal solution for each hyperplane. In absence of normality several solutions for a hyperplane may exist, each describing only one local optimum. This in turn would lead to an increased number of mismatches in the cross-validation, which was not observed here. We wish to note that there are many other classification algorithms which in principle may perform similarly provided proper verification. In this study, however, the focus is not to find the best performing statistical model but to gain insights into whether trace element concentration allows classification of the origin at all and which elements play a role in this approach.

^{2}and Q

^{2}as well as the averaged group accuracy (mean ± standard deviation) from the 100-fold MCCV are shown to allow judgement of the model robustness. Diagonal elements in the confusion matrix represent the number (fraction) of correctly assigned cases. Vertical off-diagonal elements correspond to false negative predictions (i.e., wrong origin assigned; e.g., from the 73 true Namibian Demantoids two are mis-assigned to Russia and three to Italy (first row)), while horizontal off-diagonal elements are false negative results (i.e., wrong origin is assumed; e.g., five samples predicted to be from Namibia are indeed from Russia (first line)). The asterisk indicates that the prevalence threshold expected for the accuracy and precision of the model is slightly missed for the origin Iran.

^{n}package, where Mahalanobis distance of all samples used for model building and the sample under investigation is plotted versus their respective Euclidean distance at a 95% confidence level (α = 5%). In general, if an unknown sample is initially run in the multi-origin data set, one can obtain a first estimate of the samples’ country of origin. In a subsequent step this can be verified with the finer pair-wise models. In any case it is necessary to closely investigate the Mahalanobis distance for the sample in all models and to compare it with the typical confidence intervals obtained for true in-class samples.

_{0}, is Russia (we assume the sample is from Russia) there is a 93% chance the result is correct, and the gem is indeed from Russia. More than 9 out of 10 tests will deliver a correct result. However, in the case H

_{0}is Italy (assuming the gem is from Italy) chances for the specimen being indeed an Italian demantoid are as low as 75% meaning one out of four tests will be mis-assigned. In such a case it is particularly important to critically evaluate all other models by attempting to rule out other solutions. Ultimately, there will always be a certain risk that a new sample is not well represented in the model. In such instances it is indicated to utilize additional approaches, such as analysis of inclusions (if present) to gain more confidence in the results obtained.

^{2}and Q

^{2}as well as the averaged group accuracy (mean ± standard deviation) from the 100-fold MCCV are shown to allow judgement of the model robustness. Diagonal elements in the confusion matrix represent the number (fraction) of correctly assigned cases. Models highlighted in red and blue denote perfectly (exclusively correct classifications) respectively almost perfectly preforming models.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Demantoid gemstones (0.75–3.23 ct) from Russia (top row) and Namibia (bottom row) represent the majority of today’s market.

**Figure 2.**PCA-LDA score plot with all classes including the log-transformed concentrations of the six relevant elements Mg, Al, Ti, V, Cr and Mn. Each dot represents a single element profile from an individual measurement after MCCV. Axis shown represent the first two linear discriminants. Ellipses confer to the 95% confidence interval according to Hotellings T

^{2}, which is the multivariate analog to the t-test [34]. Note that graphics represent a projection from 6-dimensional space, i.e., overlap in this projection does not necessarily mean samples cannot be separated (cf. confusion matrix in Table 1).

**Figure 3.**Representative PCA-LDAs for the pair-wise models (

**a**) Italy-Namibia; (

**b**) Italy-Pakistan; (

**c**) Italy-Russia; (

**d**) Italy-Iran; (

**e**) Italy-Madagascar; (

**f)**Namibia-Pakistan; (

**g**) Namibia-Russia, (

**h**) Namibia-Iran; (

**i**) Namibia-Madagascar, (

**j**) Pakistan-Russia, (

**k**) Pakistan-Iran, (

**l**) Pakistan-Madagascar, (

**m**) Russia-Iran, (

**n**) Russia-Madagascar, (

**o**) Iran-Madagascar. Each dot represents a single element profile from an individual measurement after MC cross-validation. Axis shown represent the first two linear discriminants. Ellipses confer to the 95% confidence interval according to Hotellings T

^{2}.

**Figure 4.**Boxplot showing the natural logarithm of all measured element concentrations. Samples are grouped and colored by origin. The graph displays the median, the first quartile, the third quartile as well as the maximum and minimum of each group. Outliers are represented as black dots.

**Table 1.**Confusion matrix showing prediction accuracy of the method (for details refer to the text).

Model | ACC = 0.84 | MCC = 0.78 | R^{2}_{cum} = 1 | Q^{2}_{cum} = 0.85 | |||
---|---|---|---|---|---|---|---|

All Origins | |||||||

n(total) = 403 | True Namibia | True Pakistan | True Madagascar | True Russia | True Iran | True Italy | prevalence threshold (calc/observed) |

Predicted Namibia | 68 (93.2%) | 0 | 0 | 5 (4.3%) | 0 | 0 | 0.12/0.18 |

Predicted Pakistan | 0 | 91 (92.9%) | 1 (7.1%) | 7 (6.0%) | 0 | 7 (9.0%) | 0.12/0.18 |

Predicted Madagascar | 0 | 0 | 13 (92.9%) | 0 | 0 | 0 | 0.03/0.03 |

Predicted Russia | 2 (2.7%) | 1 (1.0%) | 0 | 96 (82.1%) | 3 (13.0%) | 18 (23.1%) | 0.24/0.29 |

Predicted Iran | 0 | 1 (1.0%) | 0 | 3 (2.6%) | 17 (73.9%) | 0 | 0.10/0.06 * |

Predicted Italy | 3 (4.1%) | 5 (5.1%) | 0 | 6 (5.1%) | 3 (13.0%) | 53 (67.9%) | 0.22/0.19 |

n(class) | 73 | 98 | 14 | 117 | 23 | 78 | |

false classifications (n (%)) | 65 (16.1%) | ||||||

correct classifications (n (%)) | 338 (83.9 %) | ||||||

n(latent variables) | 6 | ||||||

Averaged Group Accuracy from MC cross validation | 0.92 ± 0.10 | 0.91 ± 0.10 | 0.94 ± 0.15 | 0.81 ± 0.12 | 0.72 ± 0.31 | 0.68 ± 0.18 |

**Table 2.**Confusion matrix for a representative model for pair-wise testing of the country of origin (15 individual models).

Model | Type of Deposit | VIP1 | VIP2 | VIP3 | VIP4 | VIP5 | VIP6 |
---|---|---|---|---|---|---|---|

All origins | ln(Mg) | ln(Cr) | ln(Ti) | ln(Al) | ln(Mn) | ln(V) | |

Italy-Namibia | asbestos vs. skarn | ln(Al) | ln(Cr) | ln(Mn) | ln(Ti) | ln(Mg) | ln(V) |

Italy-Pakistan | asbestos vs. asbestos | ln(Mn) | ln(Mg) | ln(Al) | ln(Cr) | ln(Ti) | ln(V) |

Italy-Russia | asbestos vs. asbestos | ln(Mg) | ln(Al) | ln(V) | ln(Ti) | ln(Cr) | ln(Mn) |

Italy-Iran | asbestos vs. asbestos | ln(Al) | ln(Ti) | ln(Cr) | ln(Mn) | ln(Mg) | ln(V) |

Italy-Madagascar | asbestos vs. skarn | ln(Cr) | ln(Al) | ln(Mg) | ln(V) | ln(Ti) | ln(Mn) |

Namibia-Pakistan | asbestos vs. skarn | ln(Al) | ln(Cr) | ln(Mg) | ln(Mn) | ln(V) | ln(Ti) |

Namibia-Russia | asbestos vs. skarn | ln(Al) | ln(Mn) | ln(V) | ln(Ti) | ln(Cr) | ln(Mg) |

Namibia-Iran | asbestos vs. skarn | ln(Mn) | ln(Mg) | ln(Cr) | ln(Ti) | ln(Al) | ln(V) |

Namibia-Madagascar | skarn vs. skarn | ln(Cr) | ln(Mg) | ln(Mn) | ln(Al) | ln(Ti) | ln(V) |

Pakistan-Russia | asbestos vs. asbestos | ln(Mg) | ln(Al) | ln(Cr) | ln(Ti) | ln(V) | ln(Mn) |

Pakistan-Iran | asbestos vs. asbestos | ln(V) | ln(Cr) | ln(Al) | ln(Ti) | ln(Mg) | ln(Mn) |

Pakistan-Madagascar | asbestos vs. skarn | ln(Cr) | ln(Mn) | ln(Mg) | ln(Al) | ln(Ti) | ln(V) |

Russia-Iran | asbestos vs. asbestos | ln(Cr) | ln(Al) | ln(Mg) | ln(Mn) | ln(Ti) | ln(V) |

Russia-Madagascar | asbestos vs. skarn | ln(V) | ln(Cr) | ln(Al) | ln(Mg) | ln(Ti) | ln(Mn) |

Iran-Madagascar | asbestos vs. skarn | ln(Cr) | ln(V) | ln(Al) | ln(Ti) | ln(Mg) | ln(Mn) |

**Table 4.**Scores of the individual elements as a function of their VIP-rank. Although it appears that Ti and V are overall of less importance Table 3 clearly indicates their important role for particular models.

Rank VIP | Al | Cr | Mg | Mn | Ti | V |
---|---|---|---|---|---|---|

1 | 4 | 5 | 3 | 2 | 0 | 2 |

2 | 4 | 5 | 3 | 2 | 1 | 1 |

3 | 4 | 3 | 4 | 2 | 1 | 2 |

4 | 3 | 1 | 1 | 3 | 7 | 1 |

5 | 1 | 2 | 4 | 1 | 6 | 2 |

6 | 0 | 0 | 1 | 6 | 1 | 8 |

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**MDPI and ACS Style**

Bindereif, S.G.; Rüll, F.; Schwarzinger, S.; Schwarzinger, C.
Chemometric Modeling of Trace Element Data for Origin Determination of Demantoid Garnets. *Minerals* **2020**, *10*, 1046.
https://doi.org/10.3390/min10121046

**AMA Style**

Bindereif SG, Rüll F, Schwarzinger S, Schwarzinger C.
Chemometric Modeling of Trace Element Data for Origin Determination of Demantoid Garnets. *Minerals*. 2020; 10(12):1046.
https://doi.org/10.3390/min10121046

**Chicago/Turabian Style**

Bindereif, Stefan G., Felix Rüll, Stephan Schwarzinger, and Clemens Schwarzinger.
2020. "Chemometric Modeling of Trace Element Data for Origin Determination of Demantoid Garnets" *Minerals* 10, no. 12: 1046.
https://doi.org/10.3390/min10121046