Abstract
Machine-learning methods are evaluated to study the intriguing and debated topic of discrimination among different tectonic environments using geochemical and isotopic data. Volcanic rocks characterized by a whole geochemical signature of major elements (SiO2, TiO2, Al2O3, Fe2O3T, CaO, MgO, Na2O, K2O), selected trace elements (Sr, Ba, Rb, Zr, Nb, La, Ce, Nd, Hf, Sm, Gd, Y, Yb, Lu, Ta, Th) and isotopes (206Pb/204Pb, 207Pb/204Pb, 208Pb/204Pb, 87Sr/86Sr and 143Nd/144Nd) have been extracted from open-access and comprehensive petrological databases (i.e., PetDB and GEOROC). The obtained dataset has been analyzed using support vector machines, a set of supervised machine-learning methods, which are considered particularly powerful in classification problems. Results from the application of the machine-learning methods show that the combined use of major, trace elements and isotopes allows associating the geochemical composition of rocks to the relative tectonic setting with high classification scores (93 %, on average). The lowest scores are recorded from volcanic rocks deriving from back-arc basins (65 %). All the other tectonic settings display higher classification scores, with oceanic islands reaching values up to 99 %. Results of this study could have a significant impact in other petrological studies potentially opening new perspectives for petrologists and geochemists. Other examples of applications include the development of more robust geothermometers and geobarometers and the recognition of volcanic sources for tephra layers in tephro-chronological studies.
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Acknowledgments
We thank the editor (Prof. O. Müntener) and two unknown reviewers for valuable comments and suggestions that contributed to increase the quality our manuscript. We also acknowledge Rebecca Astbury for the proofreading of the final version of the manuscript. This project was supported by the ERC Consolidator “CHRONOS” project (Grant No. 612776) and by the Microsoft Research Azure Award Program (Maurizio Petrelli: Azure Machine Learning Award).
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Appendices
Appendix A: mathematical principles of support vector machines
An extensive introduction to the mathematical principles of support vector machines is reported in Abedi et al. (2012) and Cortes and Vapnik (1995). To introduce the formulation of support vector machines, we first discuss a two-class problem.
Consider a training dataset of S dimensional samples (e.g., S chemical elements as input) x i with i = 1, 2, 3,…, n where n is the number of samples. To each sample, a label y i is assigned. The label y i is equal to 1 for the first class and −1 for the second class.
In the case the two classes are linearly separable, then there exists a group of linear separators that satisfy the following equation (Kavzoglu and Colkesen 2009):
As a consequence, the separating hyper-plane can be formalized as a decision function:
with sgn(x) defined as follows:
The parameters of w and b can be obtained by solving the optimization function:
subject to:
An example of two-dimensional problem where two different populations can be divided by a linear function is reported in Fig. 8A. However, there are problems where a nonlinear trend can separate the different populations more efficiently (Fig. 8B).
In these cases, a projection function \(\phi \left( x \right)\) can be utilized to map the training data form the original space x to a Hilbert space X. This means that a nonlinear function is learned by a linear learning machine in a high-dimensional feature space while the capacity of the system is controlled by a parameter that does not depend on the dimensionality of the space (Cristianini and Shawe-Taylor 2000). This is called “kernel trick” and means that the kernel function transforms the data into a higher dimensional feature space allowing for performing a linear separation (Cortes and Vapnik 1995).
As reported by Abedi et al. (2012), the training algorithm in the Hilbert space only depends on data in this space through a dot product (i.e., a function with the form \(\phi \left( {x_{i} } \right) \times \phi \left( {x_{j} } \right)\)). As a consequence, a kernel function K can be formalized as follows:
The two-class problem can be also solved as follows (El-Khoribi 2008):
subject to:
The decision function can be now rewritten as (Yang et al. 2008):
Many potential functions can be utilized as \(K\left( {x_{i} ,x_{j} } \right)\)(Zuo and Carranza 2011). Among these, the radial basis function (RBF) utilized in our work is defined as follows:
As reported by Cortes and Vapnik (1995), support vector machines were originally developed for the solution of two-class problems, but many of the potential applications are characterized by more than two classes (multiclass problems). In order to solve multiclass problems, the two most popular approaches are the One Vs One (OVO) and the One Vs Rest (OVR) approach (Fig. 9). In OVO, one SVM classifier is built for all possible pairs of classes (Fig. 9B, C) (Knerr et al. 1990; Dorffner et al. 2001). The output from each classifier is obtained in the form of a class label. The class label with the highest frequency is assigned to that point in the data vector (Hsu and Lin 2002). Since the number of SVMs required in this approach is M(M − 1)/2, it is not suitable for those datasets characterized by a large number of classes (Dorffner et al. 2001).
On the contrary, in OVR, one SVM is built for each of the M classes. The SVM for a particular class is constructed using the training examples from that class as positive examples and the training examples of the rest of (M-1) class as negative examples (Fig. 9D).
In other words, in the OVO (One Vs One) approach, each population is compared with each other population, separately. In the OVR (One Vs Rest) approach, each population is compared with all the other populations mixed together, simultaneously.
Appendix B: the logic behind classification
Figure 10 reports a flowchart showing the steps to be implemented to determine the tectonic environment of igneous. The first step consists in verifying whether the learning process has been already performed. In the case the learning is missing, a new learning process is required. To complete this task, the reference dataset has to be normalized and split into two portions: the learning and test dataset. The role of the learning dataset is to train the system and develop a provisional model. The role of the test dataset is to check the goodness of the provisional model developed using the learning dataset. To complete this task, the samples belonging to the test dataset are evaluated as unknowns using the provisional model. If the validation process is completed successfully, the provisional model is converted to a final model. On the contrary, the whole classification process is aborted and more detailed studies are required.
When the final model is ready, the samples belonging to the unknown population are processed by the system. Results are then cross-validated using conventional techniques such as petrographic inspections, classical geochemical investigations and field observations.
Finally, if the provisional results are confirmed, the unknown samples can be safely assigned to a specific tectonic setting. On the contrary, further investigations are needed.
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Petrelli, M., Perugini, D. Solving petrological problems through machine learning: the study case of tectonic discrimination using geochemical and isotopic data. Contrib Mineral Petrol 171, 81 (2016). https://doi.org/10.1007/s00410-016-1292-2
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DOI: https://doi.org/10.1007/s00410-016-1292-2