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Article Basalt Tectonic Discrimination Using Combined Machine Learning Approach

Qiubing Ren 1 , Mingchao Li 1,* , Shuai Han 1, Ye Zhang 1, Qi Zhang 2 and Jonathan Shi 3

1 State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300354, China; [email protected] (Q.R.); [email protected] (S.H.); [email protected] (Y.Z.) 2 Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China; [email protected] 3 College of Engineering, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected] * Correspondence: [email protected]



Received: 20 March 2019; Accepted: 18 June 2019; Published: 22 June 2019


Abstract: Geochemical discrimination of basaltic magmatism from different tectonic settings remains an essential part of recognizing the magma generation process within the Earth’s mantle. Discriminating among mid-ocean ridge basalt (MORB), ocean island basalt (OIB) and island arc basalt (IAB) is that matters to geologists because they are the three most concerned basalts. Being a supplement to conventional discrimination diagrams, we attempt to utilize the machine learning algorithm (MLA) for basalt tectonic discrimination. A combined MLA termed swarm optimized neural fuzzy inference system (SONFIS) was presented based on neural fuzzy inference system and particle swarm optimization. Two geochemical datasets of basalts from GEOROC and PetDB served as to test the classification performance of SONFIS. Several typical discrimination diagrams and well-established MLAs were also used for performance comparisons with SONFIS. Results indicated that the classification accuracy of SONFIS for MORB, OIB and IAB in both datasets could reach over 90%, superior to other methods. It also turns out that MLAs had certain advantages in making full use of geochemical characteristics and dealing with datasets containing missing data. Therefore, MLAs provide new research tools other than discrimination diagrams for geologists, and the MLA-based technique is worth extending to tectonic discrimination of other volcanic rocks.

Keywords: basalt; tectonic setting; geochemical discrimination; machine learning; neural fuzzy inference system; particle swarm optimization

1. Introduction Magmatic rocks (e.g., basalt and granite) form in a wide variety of tectonic settings, which primarily include mid-ocean ridges, ocean islands and island arcs. Many scholars have expressed concern regarding the restoration of the original tectonic setting of ancient magmatic rocks [1,2]. The establishment of the plate tectonics hypothesis has brought the study of a tectonic setting discrimination of magmatic rocks to the forefront of geological research [3]. Previous studies have shown that it is difficult to achieve this goal through only the macro geological setting, and understanding the chemical composition of these rocks is crucial for accurate tectonic setting discrimination [1,4–6]. Magmatic rocks formed in different tectonic settings have unique geochemical characteristics, which are mainly reflected in the differences in composition [7–10]. The contents of major elements, trace elements and isotopic composition can be obtained by whole-rock geochemical analysis. Thus, it is feasible to discriminate the tectonic settings where magma formed and the chemical properties of magmatic source areas through elemental composition [11]. Basalt, as a typical magmatic rock, is widely distributed in the earth’s crust, and its chemical element content can provide sufficient information about mantle sources, mantle partial melting, magma crystallization processes and mantle metasomatism [12,13].

Minerals 2019, 9, 376; doi:10.3390/min9060376 www.mdpi.com/journal/minerals Minerals 2019, 9, 376 2 of 19

What is more, basalts are mafic rocks, which are closer to the composition of the parental magma that was extracted from its mantle source [14,15]. For these reasons, basalt is usually the research object of tectonic setting discrimination [16–18]. Mid-ocean ridge basalt (MORB), ocean island basalt (OIB) and island arc basalt (IAB) are the three main types of basalts most concerned by academia. MORBs erupt at mid-ocean ridge without subduction. MORBs are derived from the depleted upper mantle, and have a high degree of partial melting. MORBs generally have the characteristics of low K2O, TiO2 and depleted incompatible elements. OIBs are distributed as islands far away from the subduction zone, and within plate hotspot activity. OIBs come from the LILE-enriched regions of the lower mantle, whose origin is related to the mantle plume. IABs are typical products of subduction zone magmatism, which are derived from the depleted upper mantle and components from the subducted slab [19–23]. Thus, how to discriminate the tectonic settings of basalts has become an essential issue in geochemistry [24–26]. The basalt discrimination diagram method is an important scientific discovery after the rise of plate tectonics. In the 1970s, a group of scholars, represented by Pearce, devoted themselves to the geochemical tectonic discrimination of basalts, and established the discrimination diagram method, which opened up a new approach to the basalt geodynamics research [10,27–29]. According to different source compositions of MORB, OIB and IAB, they explored the influence of the depth of magma formed, the degree of partial melting, magma evolution on the basalt geochemical properties [30–32]. Starting from the statistical theory and combining with typical regional case studies, they put forward many basalt tectonic discrimination diagrams, and achieved great results [33–39]. Subsequently, the discrimination diagram method was rapidly extended, which promoted the development of basalt research and plate tectonics. However, with the accumulation of geochemical data, it is found that the common discrimination diagrams are not always applicable to new datasets [1,8,12,13,40–42]. Some scholars are still trying to explore other methods to establish new discrimination diagrams. Linear discriminant analysis [4], primordial mantle-normalized diagrams [43] and other geochemical discrimination tools [44] have been proposed sequentially as alternatives. Despite some progress, there has been no widespread academic response. Since then, the discrimination diagram method has fallen into a bottleneck. Therefore, we attempt to utilize the artificial intelligence method to discriminate among MORB, OIB and IAB. The main aim of this paper is to propose a new hybrid artificial intelligence method for tectonic setting discrimination of basalts, which we have termed swarm optimized neural fuzzy inference system (SONFIS). A set of scientific and reasonable model evaluation criteria were established, which included the quantitative classification accuracy and confusion matrix visualization. Two geochemical datasets of basalts were compiled to validate SONFIS. The classification performance of SONFIS was evaluated using the same datasets through comparisons with other well-established machine learning (ML) models, such as support vector machine (SVM), random forest (RF) and naive Bayes (NB). Moreover, the logistic regression classifier (LRC) and multilayer perceptron (MLP), which have not been applied in tectonic discrimination, have also been included for comparison. This paper is organized as follows. A literature review is introduced in Section2. Section3 depicts the limitations of conventional discrimination diagrams and how to overcome these problems. In Section4, the mathematical principles of the main methods used, including the neural fuzzy inference system (NFIS) and particle swarm optimization (PSO), are briefly described. Section5 presents the proposed hybrid SONFIS method and its implementation procedure. Section6 illustrates and discusses the results of comparative experiments and the applicability of ML modeling in tectonic discrimination. Conclusions and future work are finally provided in Section7.

2. Literature Review Data sharing has dramatically expanded with the establishment of some open-access and comprehensive global geochemical databases, such as GEOROC [45] and PetDB [46], which provide reliable data support for big data analysis [47,48]. Big data makes it possible to reevaluate the Minerals 2019, 9, 376 3 of 19 conventional discrimination diagrams that are based on small samples and traditional statistics [49–51]. Li et al. [43], Wang et al. [13] and Di et al. [7,18] adopted discrimination diagrams to conduct big data mining research on global data samples respectively, which again showed the inapplicability of the statistical method. In an attempt to make a simplification, tectonic setting discrimination was transformed into a multi-classification task, and ML techniques were introduced into geochemical tectonic discrimination due to their dominant classification performance and innate adaptability to big data [1,4,8,40,52]. At present, this technique is still scarcely applied for tectonic discrimination, despite its extensive application in other fields [3,53–55]. Only several commonly used ML algorithms (MLAs) have been utilized in a few cases beyond basalts, such as classification trees (CT), SVM, decision tree (DT), RF, sparse multinomial regression (SMR), NB and k-nearest neighbors (KNN), as summarized in Table1. These cases indicate that the ML technique, which can utilize more elements and have excellent generalization capability, has a good potential for tectonic discrimination, and it is expected to become another common method in addition to conventional discrimination diagrams.

Table 1. A summary of ML models and application scenarios in a few cases.

Authors MLAs Rock Types Tectonic Settings Vermeesch [1] CT Basalt MOR, OI and IA Petrelli and Perugini [40] SVM Volcanic rocks CA, IA, IOA, BAB, CF, MOR, OP and OI Liu and Liu [3] SVM and DT Basalt CP and PF Ueki et al. [8] SVM, RF and SMR Volcanic rocks CA, IA, IOA, BAB, CF, MOR, OP and OI Han et al. [53] NB, KNN, SVM and RF Basalt MOR, OI and IA Jiao et al. [54] SVM, KNN and RF Gabbro CF, CM, IV and OI Abbreviations: MOR—mid-ocean ridge; OI—ocean island; IA—island arc; CA—continental arc; IOA—intra-oceanic arc; BAB—back-arc basin; CF—continental flood; OP—oceanic plateau; CP—continental plateau; PF—plateau flood; CM—convergent margin; IV—intraplate volcanic.

2.1. Neural Fuzzy Inference System (NFIS) To provide options for geochemists, MLAs applied in tectonic discrimination still need to be explored [10,56]. This study introduces a robust and quantitative MLA called NFIS. Developed since 1993, NFIS integrates the concise form of fuzzy logic with the learning algorithm of neural networks. It is a hierarchical topology based on neural networks to provide the fuzzy function of the fuzzy system [57]. The unified model not only has the self-learning ability of neural networks, but also makes up for the uninterpretability of neural networks using fuzzy logic [58,59]. Additionally, the unified model utilizes a hybrid algorithm of back-propagation and least squares estimation to adjust the antecedent and consequent parameters, and can automatically generate if-then rules [60]. Structurally, NFIS is an adaptive multi-layer feedforward neural network, composed by individual layers for fuzzification, product, normalization, defuzzification and output [61]. The algorithm has been widely used in different fields (e.g., financial industry [58], hydraulic engineering [59,60], and mining industry [61]) due to its excellent performance. Even so, literature surveys show that there is no research regarding the application of NFIS to geochemical tectonic discrimination, thus leading us to provide such applications in our current study.

2.2. Particle Swarm Optimization (PSO) On the other hand, model parameter selection has a significant influence on classification performance in ML modeling [60]. However, in previous studies, model parameters were not specially optimized when different MLAs were used for tectonic setting discrimination, but instead manual adjustment and grid searches have been common methods of parameter tuning, which makes it difficult to explore the full potential of model performance. Swarm intelligence optimization has always been an area of high priority research as a burgeoning evolutionary computation technique. The advantages of metaheuristic optimization techniques provide solutions to the parameter optimization problem using soft computing [62]. A number of well-known metaheuristic algorithms, such as genetic algorithm [63], ant colony optimization [64], differential evolution [65] and PSO [66], have been put forward as possible Minerals 2019, 9, x FOR PEER REVIEW 4 of 18

optimization has always been an area of high priority research as a burgeoning evolutionary Mineralscomputation2019, 9, 376technique. The advantages of metaheuristic optimization techniques provide solutions4 of 19 to the parameter optimization problem using soft computing [62]. A number of well-known metaheuristic algorithms, such as genetic algorithm [63], ant colony optimization [64], differential optimization techniques to address the parameter optimization weaknesses. Among them, PSO is evolution [65] and PSO [66], have been put forward as possible optimization techniques to address considered as a promising and powerful optimization technique, thus it was selected to search for the parameter optimization weaknesses. Among them, PSO is considered as a promising and optimal parameters with NFIS [67]. PSO, which was originally attributed to Kennedy and Eberhart [66], powerful optimization technique, thus it was selected to search for optimal parameters with NFIS is an advanced evolutionary algorithm for solving the continuous global optimization problem using a [67]. PSO, which was originally attributed to Kennedy and Eberhart [66], is an advanced non-linear technique. The idea was initially inspired by the bird flock patterns, and then a simplified evolutionary algorithm for solving the continuous global optimization problem using a non-linear model was established using swarm intelligence [68,69]. The PSO-based algorithm is well suited to deal technique. The idea was initially inspired by the bird flock patterns, and then a simplified model was with non-linear and non-convex design spaces with discontinuities on account of easy implementation, established using swarm intelligence [68,69]. The PSO-based algorithm is well suited to deal with high precision, good robustness and fast convergence [70]. The algorithm has attracted extensive non-linear and non-convex design spaces with discontinuities on account of easy implementation, attention in academic circles due to its strong ability to solve practical problems [58–61]. Therefore, high precision, good robustness and fast convergence [70]. The algorithm has attracted extensive PSO can automatically optimize the antecedent and consequent parameters, which subsequently attention in academic circles due to its strong ability to solve practical problems [58–61]. Therefore, improves the adaptability of NFIS. PSO can automatically optimize the antecedent and consequent parameters, which subsequently 3.improves Problem the Description adaptability and of ResearchNFIS. Contribution

3. ProblemSome problemsDescription with and existing Research discrimination Contribution diagrams are described. With the outstanding classification nature of MLAs, SONFIS and other MLAs can overcome the above problems to some extent,Some that problems is the main with contribution existing ofdiscrimination this paper. diagrams are described. With the outstanding classification nature of MLAs, SONFIS and other MLAs can overcome the above problems to some 3.1.extent, Limitations that is the of Conventionalmain contribution Discrimination of this paper. Diagrams

3.1. LimitationsThough basalt of Convention discriminational Discrimination diagrams areDiagrams widely used, they have some inherent limitations. RestrictedThough basalt by thediscrimination earlier data diagrams processing are technique, widely used, the they sampling have some method inherent was adoptedlimitations. and • • theRestricted typical by region the earlier was takendata processing as a research technique, example, the thus sampling the discrimination method was adopted diagrams and were the obtained.typical region Although was taken the discrimination as a research diagrams example, obtained thus the have discrimination achieved great diagrams results, were with theobtained. accumulation Although of the massive discrimina geochemicaltion diagrams data, the obtained earlier discriminationhave achieved great diagrams results, may with not the be applicable.accumulation Plotting of massive compiled geochemical global data data, on some the classicalearlier discrimination diagramsdiagrams illustrates may not thebe problem,applicable. with Plotting a significant compiled amount global of data these on data some being classical misclassified discrimination (see Figure diagrams1)[ 53 ].illustrates There is alsothe problem, significant with overlap a significant among MORB, amount OIB of and these IAB data samples. being misclassified (see Figure 1) [53]. BinaryThere is or also ternary significant discrimination overlap among diagrams MORB, are OIB the and most IAB commonly samples. used for basalt tectonic • • discrimination.Binary or ternary In otherdiscrimination words, only diagrams a few elements are the ormost element commonly ratios areused utilized, for basalt which tectonic could adiscrimination.ffect the discrimination In other words, effect. only In addition,a few elemen whents or information element ratios about are related utilized, elements which could in a discriminationaffect the discrimination diagram is missing,effect. In the addition, diagram when is not information available. about related elements in a discrimination diagram is missing, the diagram is not available. As such,such, the expanded known compositionalcompositional ranges of basalts that form in didifferentfferent tectonictectonic settings meanmean thatthat the the earlier earlier developed developed discrimination discrimination diagrams diagrams have have significant significant limitations limitations that castthat doubtcast doubt on the on validity the validity of results of results obtained obtained using using these these diagrams diagrams [42]. [42].

Ti/100 32700

16600 MORB ) -6 8600 IAB WPB WPB Ti (×10 Ti

2670 CAB 1860 IAB MORB IAB 1000 CAB 10 15 25 35 45 95 105 215 542 800 -6 Zr Y×3 Zr (×10 ) (a) (b)

Figure 1. Cont.

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FeOT

10.8 WPB

5.2 4 3.4 IAB Zr/Y OIB CB SCIB 1.4 MORB 6 IAB 0 OB MORB 16 36 60 77 112 137 226 268 596 -6 Zr (×10 ) MgO Al2O3 (c) (d)

TiO2

OIB MORB IAB OIB MORB IAB

OIAB

CAB

MnO×10 P2O5×10 (e)

FigureFigure 1. 1.Classical Classical basalt basalt tectonic tectonic discrimination discrimination diagrams: diagrams: (a )(a diagram) diagram of of Ti-Zr-Y, Ti-Zr-Y, (b )(b diagram) diagram of of Ti-Zr, (c) diagram of Zr/Y-Zr, (d) diagram of FeOT T-MgO-Al2O3 and (e) diagram of TiO2-MnO-P2O5. Ti-Zr, (c) diagram of Zr/Y-Zr, (d) diagram of FeO -MgO-Al2O3 and (e) diagram of TiO2-MnO-P2O5. Abbreviations:Abbreviations: WPB—withinWPB—within plateplate basalt; basalt; CAB—calc-alkaliCAB—calc-alkali basalt; basalt; CB—continentalCB—continental basalt; basalt; SCIB—spreadSCIB—spread center center island island basalt; basalt; OB—orogenic OB—orogenic basalt; basalt; OIAB—oceanic OIAB—oceanic island island alkali alkali basalt. basalt.

3.2.3.2. Feasibility Feasibility of of MLAs MLAs for for Tectonic Tectonic Discrimination Discrimination MLAsMLAs can can overcome overcome the twothe limitationstwo limitations of conventional of conventional discrimination discriminati diagrams.on diagrams. Taking SONFIS Taking asSONFIS an example, as an the example, advantages the advantages of MLAs over of MLAs discrimination over discrimination diagrams are diagrams described. are described. • MLAs originated from the era of big data, with strong adaptability to all kinds of data. As a MLAs originated from the era of big data, with strong adaptability to all kinds of data. As a • classifier, SONFIS can be trained based on the geochemical data of a large number of basalt classifier, SONFIS can be trained based on the geochemical data of a large number of basalt samples with known tectonic settings. For the samples with unknown tectonic settings, the samples with known tectonic settings. For the samples with unknown tectonic settings, the geochemical data measured can be directly input into the trained SONFIS, then the geochemical data measured can be directly input into the trained SONFIS, then the corresponding corresponding tectonic setting can be easily acquired. When different data serve as to train the tectonic setting can be easily acquired. When different data serve as to train the SONFIS, the model SONFIS, the model parameters update adaptively to get a new different classifier. In summary, parameters update adaptively to get a new different classifier. In summary, the performance of the performance of the classifier is related to the quantity and quality of geochemical data the classifier is related to the quantity and quality of geochemical data trained. trained. • MLAs have no limit on the amount of input geochemical data. Theoretically, the more the effective • MLAs have no limit on the amount of input geochemical data. Theoretically, the more the information,effective information, the better the the performance better the performanc of the classifier.e of the For classifier. samples For with samples unknown with tectonicunknown settings,tectonic the settings, classification the classification effect of SONFIS effect of is alsoSONF stillIS is satisfactory also still satisfactory even if some even input if some data input are missing.data are Therefore, missing. the Therefore, MLA-based the classifiers MLA-based have excellentclassifiers compatibility have excellent and robustness.compatibility and robustness. The advantages of MLAs can make up for the limitations of conventional discrimination The advantages of MLAs can make up for the limitations of conventional discrimination diagrams, that is, the MLA-based discrimination method can supplement the discrimination diagrams. diagrams, that is, the MLA-based discrimination method can supplement the discrimination This technique will be fully demonstrated through two datasets (in Section 5.2) and comparative diagrams. This technique will be fully demonstrated through two datasets (in Section 5.2) and experiments (in Section6). comparative experiments (in Section 6).

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4. Mathematical Mathematical Principles Principles of of Main Algorithms

4.1. Neural Neural Fuzzy Fuzzy Inference Inference System System (NFIS) In order order to to provide provide a asimplified simplified visual visual process, process, the the NFIS NFIS architecture architecture with with two twoinputs inputs and andone outputone output is illustrated is illustrated in Figure in Figure 2, 2where, where circle circless and and squares squares represent represent fixed fixed and and adaptive nodes,nodes, respectively [71]. [71]. The The mathematical mathematical principle principle of NFIS of NFIS is explained is explained within within the diagram the diagram of this general of this structure.general structure.

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 x y

A1 ω ω Π 1 1 ω x N 11f A2 f Σ

B1 Π ω f y ω N ω 22 2 2 B2 x y

Input Fuzzifi- Product Normali- Defuzzifi- Output cation zation cation Figure 2. GeneralGeneral architecture architecture of the neural fuzzy inference system (NFIS) model.

For a first-order first-order Takagi-Sugeno fuzzy model, a common rule is set with with two two fuzzy fuzzy if-then if-then rules, rules, which is presented as follows:

RuleRule(i) (i) →If Ifx xis isA Aiand andy yis isB B,i, thenthen f𝑓== p𝑝i ++ p𝑝ix𝑥++ p iy𝑝 𝑦,, ii = 1,2,1, 2, → i i i 0 1 2 where A and B are specified as the membership functions for inputs x, y, and 𝑝, 𝑝, 𝑝 are function parameterswhere A and ofB outputare specified f. It should as the also membership be noted that functions the node for in inputs the i-thx, yposition, and p0 ,ofp1 ,thep2 arek-th function layer is parameters of output f. It should also be noted that the node in the i-th position of the k-th layer denoted as 𝑂 , and the node functions in the same layer are of the same function family as described Ok below.is denoted as i , and the node functions in the same layer are of the same function family as described below. 4.1.1. Layer 1: Fuzzification Layer 4.1.1. Layer 1: Fuzzification Layer The fuzzy membership values for input factors are generated using a membership function in The fuzzy membership values for input factors are generated using a membership function in this this layer, as seen in Equation (1). The Gaussian membership function, namely Equation (2), is layer, as seen in Equation (1). The Gaussian membership function, namely Equation (2), is considered considered as the input membership function. as the input membership function. 𝑂 =𝜇(),𝑂 =𝜇(),𝑖 =1,2, (1) 1 = 1 = = Oi µAi(x), Oi µBi(y), i 1, 2, (1) ( ) ( ) (2) 𝜇() =𝑒 (x c )2 ,𝜇() =𝑒(y c )2 ,𝑖 =1,2, − i − i 2 2 = − 2σi = − 2σi = µAi(x) e , µBi(y) e , i 1, 2, (2) where A and B are the linguistic variables, 𝜇() and 𝜇() are the Gaussian membership functions, and 𝜎 and c are the antecedent parameters that respectively control the shapes of 𝜇 () where A and B are the linguistic variables, µAi(x) and µBi(y) are the Gaussian membership functions, and 𝜇 . and σ and() c are the antecedent parameters that respectively control the shapes of µAi(x) and µBi(y).

4.1.2. Layer Layer 2: 2: Product Product Layer Layer

Every node in Layer 2 computes the firingfiring strengthstrength ω𝜔i of of the the rule-based rule-based on on Equation Equation (3), (3), which which is isthe the product product of of all all incoming incoming signals. signals.

𝑂2 =𝜔 =𝜇 ⋅𝜇 ,𝑖 =1,2. (3) O = ω = µ () µ (), i = 1, 2. (3) i i Ai(x) · Bi(y)

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4.1.3. Layer 3: Normalization Layer Normalization in this layer is done in accordance with Equation (4). The output, called normalized firing strength ωi, is calculated based on the ratio of the i-th rule’s firing strength to the sum of all rules’ firing strengths. 3 ωi Oi = ωi = , i = 1, 2. (4) ω1 + ω2

4.1.4. Layer 4: Defuzzification Layer The layer is used for defuzzification where the value of the rule consequence for each node is calculated as follows. 4  i i i  Oi = ωi fi = ωi p0 + p1x + p2 y , i = 1, 2, (5) i i i where p0, p1 and p2 are the consequent parameters of the output function fi.

4.1.5. Layer 5: Output Layer In this layer, the overall output is generated through the summation of all incoming signals.

X P 5 i ωi fi O = ωi fi = P , i = 1, 2. (6) i ω i i i

4.2. Particle Swarm Optimization (PSO) PSO is initialized with a set of random particles (solutions) that have random positions and velocities. Seeking an optimal solution through an iterative process is the next step of the algorithm, in which particle positions are adjusted based on experimentation with the particle of interest and that of other particles. To find a potential solution, the best position of each particle tracked is called the personal best pbest. The global best gbest is also the global best value achieved by other particles in the swarm. Each particle can be trained to accelerate from its own pbest and gbest positions during the learning process. This can be achieved by calculating the new velocity term for every particle based on its distance from the pbest and gbest positions. The pbest and gbest velocities are then randomly weighted to produce a new velocity value for the particle, which then affects the next position of the particle in the next iteration. Compared with other optimization algorithms, the main advantage of PSO is its simplicity. The velocity update equation (see Equation (7)) and movement equation (see Equation (8)) are the only equations required in PSO. Equation (7) is used to adjust the velocity vectors given to pbest and gbest, while Equation (8) determines the actual motion of particles through their specific velocity vectors. Furthermore, Shi and Eberhart [72] introduced the inertia weight w into PSO in order to improve the convergence speed. The inertia weight determines the contribution rate of the particle’s previous velocity to its velocity at the current step.     vnew→ = w →v + r C p→ →p + r C g→ →p , (7) · 1 1 · best − 2 2 · best −

pnew→ = →p + vnew→ , (8) where vnew→ , →v , pnew→ and →p are the new velocity, current velocity, new position and current position, respectively. C1 and C2 are the acceleration constants, p→best is the personal best position and g→best is the global best position among all particles. r1 and r2 are random values in the range (0,1) sampled from a uniform distribution, and w is the inertia weight. MineralsMinerals2019 2019,,9 9,, 376 x FOR PEER REVIEW 88 ofof 1918

5.5. MethodologyMethodology

5.1.5.1. OverallOverall Methodology:Methodology: The ProposedProposed Hybrid SONFISSONFIS MethodMethod AA newnew hybrid hybrid artificial artificial intelligence intelligence method method for tectonic for tectonic discrimination discrimination of basalts, of calledbasalts, SONFIS, called wasSONFIS, put forward was put based forward on NFIS based and on PSO NFIS methods. and PSO PSO methods. was introduced PSO was to introduced improve the to flexibility improve and the generalizationflexibility and abilitygeneralization of NFIS byability seeking of NFIS the optimal by seeking parameters the optimal corresponding parameters to dicorrespondingfferent datasets. to Thedifferent architecture datasets. of The the proposedarchitecture hybrid of the SONFIS proposed method hybrid is SONFIS shownin method Figure 3is. shown The procedure in Figure of 3. SONFISThe procedure is outlined of SONFIS below. is outlined below. StepStep 1:1: Dataset Dataset preparation. preparation. The trainingThe training and test an setsd weretest determinedsets were determined by percentage by segmentation. percentage Thesegmentation. independent The and independent dependent and variables dependent were alsovariables specified. were also specified. StepStep 2:2: ModelModel initialization.initialization. An initial SONFIS model waswas generatedgenerated afterafter allall antecedentantecedent andand consequentconsequent parametersparameters werewere confirmedconfirmed byby trialtrial and and error. error. StepStep 3:3: ParameterParameter optimization.optimization. PSO was employed to optimizeoptimize thethe modelmodel byby searchingsearching forfor thethe optimaloptimal valuesvalues ofof thethe antecedentantecedent andand consequentconsequent parameters.parameters. Cross-validationCross-validation playedplayed anan importantimportant rolerole therein.therein. StepStep 4:4: FitnessFitness evaluation.evaluation. The overalloverall classificationclassification accuracy was taken as thethe fitnessfitness function,function, whichwhich waswas thethe bridgebridge betweenbetween NFISNFIS andand PSO.PSO. ModelModel tuningtuning waswas basedbased onon fitnessfitness evaluationevaluation results.results. StepStep 5:5: ModelModel validation.validation. The maximummaximum numbernumber ofof iterationsiterations oror certaincertain classificationclassification precisionprecision waswas consideredconsidered asas thethe stoppingstopping criterion.criterion. OnceOnce thethe optimaloptimal parametersparameters werewere found,found, thethe SONFISSONFIS obtainedobtained waswas validated.validated.

Training set Values of trial and error

Dataset preparation PSO implementation Initialization of SONFIS (1) Independent variables • Swarm initialization model and its parameters • Major elements (wt%) • Initialize the particle's • Minor and trace position and velocity elements (ppm) • • Update the particle's Isotopic ratios SONFIS model parameter position and velocity (2) Dependent variables optimization • Find the global optimal • Mid-ocean ridge (MOR) solution • Ocean island (OI) • Island arc (IA) No Fitness evaluation using Stopping criterion classification accuracy Yes Test set Optimal SONFIS model Model validation

Figure 3. Architecture description of the proposed hybrid swarm optimized neural fuzzy Figure 3. Architecture description of the proposed hybrid swarm optimized neural fuzzy inference inference system (SONFIS) method for tectonic discrimination of basalts. system (SONFIS) method for tectonic discrimination of basalts.

5.2.5.2. MethodologyMethodology ImplementationImplementation ProcedureProcedure

5.2.1.5.2.1. DataData AcquisitionAcquisition andand PreprocessingPreprocessing TheThe twotwo geochemicalgeochemical datasetsdatasets ofof basaltsbasalts fromfrom GEOROCGEOROC andand PetDBPetDB werewere usedused toto testtest thethe classificationclassification performanceperformance ofof thethe proposedproposed SONFISSONFISmodel. model. OneOne waswas aa high-dimensionalhigh-dimensional dataset,dataset, andand thethe otherother waswas low-dimensional.low-dimensional. The The usability usability of of SONFIS SONFIS could could be be validated validated by by using using both both types types of ofdatasets. datasets.

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Dataset 1 with High-Dimensional Features An extensive dataset of 938 basalt samples was obtained after datadata cleaningcleaning [[1].1]. The dataset includedincluded 296 296 MORB MORB samples samples from from the the East Pacific Pacific Rise, Mid Atlantic Ridge, Indian Ocean and Juan de Fuca Ridge. Three Three hundred hundred and and nineteen nineteen OIB OIB samples were from St. Helena, Helena, the the Canary, Canary, Cape Verde, Caroline,Caroline, Crozet, Crozet, Hawaii-Emperor, Hawaii-Emperor, Juan Juan Fernandez, Fernandez, Marquesas, Marquesas, Mascarene, Mascarene, Samoan Samoan and Society and Societyislands. islands. The remaining The remaining 323 IAB samples323 IAB came samples from thecame Aeolian, from the Izu-Bonin, Aeolian, Kermadec, Izu-Bonin, Kurile, Kermadec, Lesser Kurile,Antilles, Lesser Mariana, Antilles, Scotia Mariana, and Tonga Scotia arcs. Theand globalTonga distribution arcs. The global map of distribution basalt samples map is shownof basalt in T samplesFigure4. is Each shown basalt in Figure sample 4. was Each assigned basalt sample with 11 was major assigned elements with (SiO 11 2major, TiO2 ,elements Al2O3, Fe (SiO2O32,, FeOTiO2,, AlCaO,2O3, MgO, Fe2O3 MnO,, FeOT K, CaO,2O, Na MgO,2O and MnO, P2O K5),2O, 35 Na minor2O and and P trace2O5), 35 elements minor (La,and Ce,trace Pr, elements Nd, Sm, (La, Eu, Gd,Ce, Pr, Tb, Nd,Dy, Ho,Sm, Er,Eu, Tm, Gd, Yb, Tb, Lu, Dy, Sc, Ho, V, Cr,Er, Co,Tm, Ni, Yb, Cu, Lu, Zn, Sc, Ga,V, Cr, Rb, Co, Sr, Y,Ni, Zr, Cu, Nb, Zn, Sn, Ga, Cs, Rb, Ba, Sr, Hf, Y, Ta, Zr, Pb, Nb, Th Sn, and Cs, U) 143 144 87 86 206 204 207 204 208 204 Ba,and Hf, five Ta, isotopic Pb, Th ratios and (U)Nd an/d fiveNd, isotopicSr/ Sr, ratiosPb (/143Nd/Pb,144Nd,Pb/ 87Sr/Pb86 andSr, 206Pb/Pb204/ Pb,Pb). 207Pb/ Accordingly,204Pb and 208everyPb/204 samplePb). Accordingly, was represented every by sample a 51-dimensional was represen vector.ted Theby basica 51-dimensional statistics for eachvector. dimension The basic are statisticsshown in for Table each2. dimension are shown in Table 2.

MORB samples OIB samples IAB samples 90°N

60°N

30°N

0°

30°S

60°S

90°S 60°W 30°W 0° 30°E 60°E 90°E 120°E 150°E 180° 150°W 120°W 90°W 60°W Figure 4. GlobalGlobal distribution distribution map map of basalt sample collection sites.

Table 2. Basic statistics for dataset 1 with high-dimensional features. Table 2. Basic statistics for dataset 1 with high-dimensional features.

No.No. Elements Elements Min. Min. Max. Mean Mean No. Elements Min.Min. Max. Max. Mean Mean 1 SiO 36.55 54.30 49.15 27 V 27.00 622.00 284.65 1 SiO2 2 36.55 54.30 49.15 27 V 27.00 622.00 284.65 2 TiO2 0.18 4.91 1.81 28 Cr 0.00 3700.00 246.80 2 TiO2 0.18 4.91 1.81 28 Cr 0.00 3700.00 246.80 3 Al2O3 8.52 26.16 15.71 29 Co 10.00 460.00 45.83 3 4Al Fe2O23O 3 8.520.00 26.16 19.33 4.2515.71 3029 Ni Co 0.00 10.00 900.00460.00 120.2745.83 T 4 5 Fe2FeOO3 0.001.24 19.33 15.11 7.614.25 3130 Cu Ni 5.00 0.00 6001.00900.00 112.14120.27 6 CaOT 0.35 14.81 10.54 32 Zn 28.70 441.00 95.77 5 7FeO MgO 1.24 1.57 15.11 22.60 7.397.61 3331 Ga Cu 9.00 5.00 6001.00 48.00 18.96112.14 6 8CaO MnO 0.35 0.01 14.81 19.00 0.2010.54 3432 Rb Zn 0.0028.70 116.73441.00 15.1495.77 7 9MgO K2O 1.57 0.02 22.60 9.68 0.707.39 3533 SrGa 6.68 9.00 1590.0048.00 385.8818.96 8 10MnO Na2 O0.01 0.74 19.00 5.95 2.700.20 3634 RbY 7.00 0.00 296.00116.73 30.8615.14 11 P2O5 0.01 2.35 0.29 37 Zr 0.00 988.30 145.80 9 12K2O La 0.02 0.00 317.009.68 17.090.70 3835 Nb Sr 0.00 6.68 130.001590.00 19.17385.88 10 13Na2 CeO 0.74 0.00 420.005.95 36.852.70 3936 Sn Y 0.45 7.00 12.00296.00 2.0030.86 14 Pr 0.36 26.20 5.86 40 Cs 0.00 5.00 0.32 11 P2O5 0.01 2.35 0.29 37 Zr 0.00 988.30 145.80 15 Nd 0.00 780.00 21.36 41 Ba 1.30 1088.00 219.33 12 16La Sm 0.00 0.52 317.00 923.00 6.8217.09 4238 HfNb 0.110.00 21.90130.00 3.6919.17 13 17Ce Eu 0.00 0.20 420.00 288.00 2.2336.85 4339 Ta Sn 0.010.45 6.8012.00 1.232.00 14 Pr 0.36 26.20 5.86 40 Cs 0.00 5.00 0.32 15 Nd 0.00 780.00 21.36 41 Ba 1.30 1088.00 219.33 16 Sm 0.52 923.00 6.82 42 Hf 0.11 21.90 3.69 17 Eu 0.20 288.00 2.23 43 Ta 0.01 6.80 1.23

Minerals 2019, 9, 376 10 of 19

Table 2. Cont.

No. Elements Min. Max. Mean No. Elements Min. Max. Mean 18 Gd 1.03 43.60 5.52 44 Pb 0.00 47.00 3.89 19 Tb 0.10 28.30 0.97 45 Th 0.00 27.00 2.47 20 Dy 0.00 594.00 7.93 46 U 0.00 6.20 0.74 21 Ho 0.29 6.10 1.01 47 143Nd/144Nd 0.50 0.52 0.51 22 Er 0.80 259.00 3.58 48 87Sr/86Sr 0.70 0.71 0.70 23 Tm 0.11 2.80 0.48 49 206Pb/204Pb 17.08 38.46 18.93 24 Yb 0.26 193.00 3.38 50 207Pb/204Pb 15.39 15.83 15.56 25 Lu 0.04 243.00 0.94 51 208Pb/204Pb 18.83 40.17 38.29 26 Sc 0.00 88.00 34.95 The unit of the major element concentration is wt%, and so is the minimum, maximum and mean. Moreover, the unit of the minor and trace element concentration is ppm, and so is the minimum, maximum and mean.

Dataset 2 with Low-Dimensional Features The data volume of the low-dimensional dataset was larger than that of dataset 1, containing 1582 basalt samples [10]. Five hundred and thirty-nine MORB samples came from the Atlantic and Pacific mid-ocean ridges, 463 OIB samples were mainly distributed in the Atlantic and Pacific regions and 580 IAB samples were from the west and east coast of the Pacific Rim. Unlike dataset 1, each basalt sample in dataset 2 contained only 12 elements, which were K2O, CaO, SiO2, MgO, NiO, Na2O, T FeO , TiO2, Al2O3, MnO, Cr2O3 and P2O5. The basic statistics for each element corresponding to three different tectonic settings are presented in Table3.

Table 3. Basic statistics for dataset 2 with low-dimensional features.

MORB OIB IAB Elements Min. Max. Mean Min. Max. Mean Min. Max. Mean

K2O 0.01 0.05 0.02 0.00 0.38 0.03 0.00 0.09 0.01 CaO 0.10 0.54 0.31 0.02 0.58 0.31 0.01 0.66 0.21 SiO2 29.93 41.85 40.12 34.71 42.43 39.71 27.32 42.93 39.01 MgO 36.43 51.30 46.64 23.64 51.71 45.47 25.24 52.66 42.65 NiO 0.01 0.44 0.20 0.01 0.43 0.22 0.00 0.55 0.16 Na2O 0.01 0.10 0.02 0.00 0.53 0.14 0.00 0.21 0.02 FeOT 7.72 23.90 12.59 7.58 42.33 14.28 7.11 37.35 17.18 TiO2 0.01 0.11 0.03 0.00 0.19 0.03 0.00 0.21 0.03 Al2O3 0.01 0.55 0.06 0.00 0.95 0.09 0.00 0.94 0.06 MnO 0.01 0.53 0.20 0.03 0.73 0.21 0.08 0.76 0.27 Cr2O3 0.01 0.21 0.06 0.00 0.23 0.05 0.00 0.29 0.04 P2O5 0.01 0.14 0.07 0.01 0.10 0.03 0.00 0.12 0.02 The unit of the element concentration is wt%, and so is the minimum, maximum and mean.

It should be noted that both datasets contained missing data [1,10]. Nothing was done with the missing data in this study, once again illustrating the superiority of the proposed SONFIS method. What was more, each geochemical feature in both datasets was treated as an input attribute, while the output classes were unified into three basalt tectonic settings, that is, MORB, OIB and IAB. Since the fuzzy membership values of the SONFIS model were between 0 and 1, the values of the input features were adjusted to fit this range. Three output classes, MORB, OIB and IAB, were symbolized by 0, 1 and 2, respectively. Percentage segmentation is a conventional method used for model validation in ML modeling. Both datasets were then randomly split into two subsets, one as a training set (80%) and the other as a test set (20%) [10]. The segmentation results of the two datasets are shown in Tables4 and5, separately. MineralsMinerals2019 2019, 9,, 9 376, x FOR PEER REVIEW 11 11of of18 19

TableTable 4. 4.Classification Classification accuracy accuracy ofof tectonictectonic discriminationdiscrimination of basalts basalts in in dataset dataset 11 based based on on six six MLAs. MLAs.

Tectonic Training ClassificationClassification Accuracy Accuracy (%) (%) Tectonic Settings TrainingTest Set Set Test Set Settings Set LRC LRCNB NB MLP MLPSVM SVM RF RF SONFIS SONFIS MORB 239 57 91.23 82.46 85.96 91.23 91.23 92.98 MORB 239 57 91.23 82.46 85.96 91.23 91.23 92.98 OIB OIB262 262 57 87.72 57 87.7284.21 84.2198.25 98.25 91.2391.23 96.49 96.49 98.2598.25 IAB IAB 249 249 74 89.19 74 89.1993.24 93.2489.19 89.1993.24 93.24 95.9595.95 95.95 95.95 TotalTotal 750 750 188 18889.36 89.3687.23 87.2390.96 90.9692.02 92.02 94.68 94.68 95.7495.74 MarkedMarked in italicsin italics and and bold denotesbold denotes the classification the classificati accuracyon ofaccuracy the best classifierof the best for diclassifierfferent tectonic for different settings. tectonic settings.

TableTable 5. 5.Classification Classification accuracy accuracy ofof tectonictectonic discriminationdiscrimination of basalts basalts in in dataset dataset 22 based based on on six six MLAs. MLAs.

Tectonic Training ClassificationClassification Accuracy Accuracy (%) (%) Tectonic Settings TrainingTest Set Set Test Set Settings Set LRC LRCNB NB MLP MLPSVM SVM RF RF SONFIS SONFIS MORBMORB 424 424 115 11574.78 74.7890.43 90.4385.22 85.2284.35 84.35 82.61 82.61 96.5296.52 OIBOIB 374 374 89 55.06 89 55.0641.57 41.5788.76 88.7629.21 29.21 85.39 85.39 94.3894.38 IABIAB 468 468 112 11285.71 85.7175.89 75.8990.18 90.1887.50 87.50 92.86 92.86 97.3297.32 Total 1266 316 73.10 71.52 87.97 69.94 87.28 96.20 Total 1266 316 73.10 71.52 87.97 69.94 87.28 96.20 Marked in italics and bold denotes the classification accuracy of the best classifier for different tectonic settings. Marked in italics and bold denotes the classification accuracy of the best classifier for different tectonic settings.

5.2.2.5.2.2. Model Model Parameter Parameter Configuration Configuration ForFor parameter parameter learning learning and and tuning, tuning, the theK-fold K-fold cross-validation cross-validation technique technique was was adopted adopted for model for optimizationmodel optimization in order in to order verify to theverify classification the classifica performancetion performance of the of trainingthe training model model and and adjust adjust the parameters,the parameters, as shown as shown in Figure in 5Figure[ 1,8,10 5 ].[1,8,10]. As a general As a general rule for rule providing for providing empirical empirical evidence, evidence,K = 10 is generally𝐾=10 preferred.is generally The preferred. raw dataset The wasraw first dataset divided was into first 10 divided subsets. into The 10 hold-out subsets. method The hold-out was then repeatedmethod 10 was times, then such repeated that each10 times, time, such one ofthat the each 10 subsets time, one was of used the 10 as subsets the test was set and used the as remaining the test nineset subsetsand the were remaining grouped nine together subsets to formwere agrouped training set.together The errorto form estimation a training was finallyset. The averaged error overestimation all 10 trials was tofinally get the averaged total eff overectiveness all 10 tria ofls the tomodel. get the total effectiveness of the model.

Iteration 1 Iteration 2 Iteration 3 Iteration K Fold 1 Fold 1 Fold 1 Fold 1 Fold 2 Fold 2 Fold 2 Fold 2 Fold 3 Fold 3 Fold 3 Fold 3

Fold K Fold K Fold K Fold K

Training set Test set FigureFigure 5.5. KK-fold-fold cross-validation technique. technique.

TheThe fuzzy fuzzyc -meansc-means clustering clustering [[60]60] waswas employedemployed to to transfer transfer all all training training samples samples into into 𝑐=10c = 10 clusters,clusters, which which was was determined determined by trialby trial and and error. error. The fuzzyThe fuzzy if-then if-then rules wererules generatedwere generated by clusters, by andclusters, the optimal and the parameters optimal parameters of the rule of were the acquiredrule werethrough acquired the through PSO algorithm, the PSO algorithm, as shown as in shown Figure 3. Duringin Figure the 3. optimization During the optimization process, diff erentprocess, combinations different combinations of antecedent of an andtecedent consequent and consequent parameters wereparameters generated, were and generated, each combination and each was combinat tested withion was 10-fold tested cross-validation, with 10-fold resultingcross-validation, in a total ofresulting 100 iterations. in a total The of variant 100 iterations. of classification The variant accuracy of classification was used for accuracy fitness evaluation.was used for When fitness the evaluation. When the optimization was completed, the optimal position 𝑔 of the swarm was optimization was completed, the optimal position gbest of the swarm was obtained, providing the obtained, providing the highest classification accuracy once the stopping criterion was satisfied. The highest classification accuracy once the stopping criterion was satisfied. The optimized values of the antecedent and consequent parameters of SONFIS were then determined, and the corresponding Minerals 2019, 9, 376 12 of 19 model was finally validated using the test set prior to being used for discriminating the tectonic settings of unknown samples.

5.2.3. Model Performance Evaluation Evaluation metrics are fundamental to evaluating the proposed SONFIS method with existing datasets. Using specific metrics, model performance evaluation results can be reasonably given and accurately expressed. A comprehensive evaluation system, including the classification accuracy and confusion matrix, was used to assess the classification effect of each MLA. The classification accuracy Ci of an individual program i depended on the number of samples correctly classified and was quantitatively evaluated by Equation (9) [53].

T C = 100% (9) i N × where T is the number of sample cases correctly classified, and N is the total number of sample cases. As a visual tool, the confusion matrix is often used to describe the performance of a classifier on the test set for which the actual values are known, and intuitively exhibits classification precision [10]. It can also be described as a specific type of contingency table with two dimensions and identical sets of classes in each dimension. The horizontal dimension denotes the actual class, and the vertical dimension denotes the predicted class. The number of correct and incorrect predictions were summarized with count values, and consequently broken down by each class. Some derived indexes (e.g., precision and recall) could also be derived from the confusion matrix. In summary, a confusion matrix provided insight not only into identifying the errors being made by a classifier, but more importantly, the types of errors being made.

5.2.4. Model Validation Scheme Design To assess the classification performance of the proposed SONFIS method when processing geochemical data, several experiments were carried out using the two datasets described in Section 5.2.1. It is imperative to verify the effectiveness of the PSO algorithm using optimizing model parameters, and then compare it with two other common optimization techniques, manual adjustment and grid search. Furthermore, three additional common MLAs (SVM, RF and NB) were also compared with the SONFIS model. The new LRC and MLP models were also additionally used for performance comparison. Finally, the proposed SONFIS method and the other five MLAs were all implemented in MATLAB® R2016b.

6. Results and Discussion

6.1. Optimization Effect Verification The parameters of NFIS were tuned based on the same datasets (in Section 5.2.1) and computing platform (Intel® CoreTM i7-8700 CPU @ 3.20 GHz, 16 GB of RAM, and a 64-bit windows 10 OS), using manual adjustment, grid search and a PSO algorithm (i.e., SONFIS), sequentially. Each method was conducted ten times, and the average classification accuracy obtained was used as the comparison value. The results show that the classification accuracy of SONFIS was approximately five to ten percent higher than that of the other two optimized models when discriminating among the tectonic settings of basalts. The improvement of classification performance was consistent with literature [58]. The SONFIS model converged after 100 iterations, consequently taking less time, which suggested the PSO algorithm was superior in both performance and efficiency for parameter optimization.

6.2. MLA Performance Comparison For dataset 1, the basalt tectonic discrimination results obtained by six MLAs are shown in Table4 and Figure6. It can be seen from Table4 that all MLAs had good classification performance for the Minerals 2019, 9, 376 13 of 19 Minerals 2019, 9, x FOR PEER REVIEW 13 of 18 forthree the tectonic three tectonic settings. settings. The data The demonstrated data demonstr thatated SONFIS that SONFIS was the was most the capable, most capable, followed followed by RF, bySVM RF, and SVM MLP, and allMLP, of whichall of which had an had overall an overall classification classification accuracy accuracy of over of over 90%. 90%. However, However, LRC LRC and andNB didNB notdid performnot perform well well in comparison. in comparison. SONFIS SONFIS also also demonstrated demonstrated top top performance performance in terms in terms of the of theclassification classification accuracy accuracy of singleof single tectonic tectonic setting. setting. The The discrimination discrimination eff ectseffects of MLPof MLP on on OIB OIB and and RF RFon on IAB IAB were were also also outstanding. outstanding. In In Figure Figure6, 6, darker darker colored colored squares squares on on the the main main diagonal diagonal indicatedindicated higherhigher accuracy accuracy of of the the classificati classificationon results. In In the the confusion confusion matrix matrix for for each each MLA, MLA, the the squares squares on on the the mainmain diagonal werewere the the darkest, darkest, indicating indicating that that the the number number of correct of correct classifications classifications was large. was Again,large. Again,the SONFIS the SONFIS model performedmodel performed best in the best three in the classes. three Overall, classes. allOverall, MLAs all used MLAs could used take could advantage take advantageof the high-dimensional of the high-dimensional information information in dataset 1 in and dataset had a remarkable1 and had a discrimination remarkable discrimination ability, which ability,was not which achieved was in not the achieved discrimination in the discrimination diagrams [53]. diagrams These results [53]. were These additionally results were consistent additionally with consistentprevious results with previous obtained results by Ueki obtained et al. [8 ].by Ueki et al. [8].

60 52 23 60 47 2860 49 35 MORB MORB MORB 45 45 45 2 50 5 2 48 7 0 56 1 OIB

OIB 30 30 OIB 30 15 3566 15 2369 15 3566 IAB IAB IAB 0 MORB OIB IAB MORB OIB IAB MORB OIB IAB (a) (b) (c)

52 1460 52 3260 53 2260 MORB MORB MORB 45 45 45 4 52 1 1 55 1 0 56 1 OIB OIB 30 OIB 30 30

15 15 3269 1271 15 1271 IAB IAB IAB 0 MORB OIB IAB MORB OIB IAB MORB OIB IAB (d) (e) (f)

FigureFigure 6. ConfusionConfusion matrix matrix visualization visualization of of tectonic tectonic discrimination discrimination of of basalts basalts in in dataset dataset 1 1 based based on on differentdifferent MLAs:MLAs: (a()a logistic) logistic regression regression classifier classifier (LRC); (LRC); (b) naïve (b) Bayesnaïve (NB); Bayes (c) multilayer(NB); (c) multilayer perceptron perceptron(MLP); (d) support(MLP); ( vectord) support machine vector (SVM); machine (e) random(SVM); ( foreste) random (RF) andforest (f) (RF) SONFIS. and (f) SONFIS.

ForFor dataset 2,2, thethe basalt basalt tectonic tectonic discrimination discrimination results results acquired acquired by six by MLAs six MLAs are shown are shown in Table in5 Tableand Figure 5 and7 .Figure In this 7. dataset, In this there dataset, were there only were a few only elements a few and elements no isotopes. and no As isotopes. noted by As Petrelli noted and by PetrelliPerugini and [40 ],Perugini a small [40], number a small of elements number can of aelementsffect the classificationcan affect the e ffclassificationect of MLAs, effect which of was MLAs, also whichconfirmed was byalso the confirmed classification by resultsthe classification below. For example,results below. the three For models,example, MLP,SVM the three and models, RF, did MLP, well SVMin the and dataset RF, 1,did but well did notin the perform dataset well 1, withbut did dataset not 2.perform However, well the with SONFIS dataset model 2. However, still classified the SONFISbest, with model the single still classified and overall best, classification with the single accuracy and overall reaching classification about 95%, accuracy demonstrating reaching that about the 95%,integration demonstrating of the PSO that algorithm the integration improved of the the PSO adaptability algorithm of SONFIS.improved The the e ffadaptabilityects of information of SONFIS. loss Thewere effects also visually of information illustrated loss in were Figure also7. Unlikevisually Figure illustrated6, not in all Figure of the darkest7. Unlike squares Figure were 6, not on all the of themain darkest diagonal squares within were every on subgraph. the main However,diagonal thewithin confusion every matrixsubgraph. corresponding However, tothe the confusion SONFIS matrixmodel wascorresponding consistent withto the Figure SONFIS6, which model demonstrated was consistent the advantageswith Figure of 6, using which the demonstrated combined model. the advantages of using the combined model.

100 100 104 29 86 17 12 80 80 98 98 80 MORB MORB MORB 60 60 60 27 49 13 27 37 25 6 79 4 OIB OIB OIB 40 40 40

11 5 96 20 26 1 85 20 65101 20 IAB IAB IAB

MORB OIB IAB MORB OIB IAB MORB OIB IAB

Minerals 2019, 9, x FOR PEER REVIEW 13 of 18 for the three tectonic settings. The data demonstrated that SONFIS was the most capable, followed by RF, SVM and MLP, all of which had an overall classification accuracy of over 90%. However, LRC and NB did not perform well in comparison. SONFIS also demonstrated top performance in terms of the classification accuracy of single tectonic setting. The discrimination effects of MLP on OIB and RF on IAB were also outstanding. In Figure 6, darker colored squares on the main diagonal indicated higher accuracy of the classification results. In the confusion matrix for each MLA, the squares on the main diagonal were the darkest, indicating that the number of correct classifications was large. Again, the SONFIS model performed best in the three classes. Overall, all MLAs used could take advantage of the high-dimensional information in dataset 1 and had a remarkable discrimination ability, which was not achieved in the discrimination diagrams [53]. These results were additionally consistent with previous results obtained by Ueki et al. [8].

60 52 23 60 47 2860 49 35 MORB MORB MORB 45 45 45 2 50 5 2 48 7 0 56 1 OIB

OIB 30 30 OIB 30 15 3566 15 2369 15 3566 IAB IAB IAB 0 MORB OIB IAB MORB OIB IAB MORB OIB IAB (a) (b) (c)

52 1460 52 3260 53 2260 MORB MORB MORB 45 45 45 4 52 1 1 55 1 0 56 1 OIB OIB 30 OIB 30 30

15 15 3269 1271 15 1271 IAB IAB IAB 0 MORB OIB IAB MORB OIB IAB MORB OIB IAB (d) (e) (f)

Figure 6. Confusion matrix visualization of tectonic discrimination of basalts in dataset 1 based on different MLAs: (a) logistic regression classifier (LRC); (b) naïve Bayes (NB); (c) multilayer perceptron (MLP); (d) support vector machine (SVM); (e) random forest (RF) and (f) SONFIS.

Petrelli and Perugini [40], a small number of elements can affect the classification effect of MLAs, which was also confirmed by the classification results below. For example, the three models, MLP, SVM and RF, did well in the dataset 1, but did not perform well with dataset 2. However, the SONFIS model still classified best, with the single and overall classification accuracy reaching about 95%, demonstrating that the integration of the PSO algorithm improved the adaptability of SONFIS. The effects of information loss were also visually illustrated in Figure 7. Unlike Figure 6, not all of the darkest squares were on the main diagonal within every subgraph. However, the confusion

Mineralsmatrix 2019corresponding, 9, 376 to the SONFIS model was consistent with Figure 6, which demonstrated14 of the 19 advantages of using the combined model.

100 100 104 29 86 17 12 80 80 98 98 80 MORB MORB MORB 60 60 60 27 49 13 27 37 25 6 79 4 OIB OIB OIB 40 40 40

11 5 96 20 26 1 85 20 65101 20 IAB IAB Minerals 2019, 9, x FOR PEER REVIEW IAB 14 of 18 MORB OIB IAB MORB OIB IAB MORB OIB IAB ((a)) ((b)) ((c)) 100 95 218 111 22 100 97 10 8 80 80

MORB MORB MORB 80 60 60 39 26 24 10 76 3 3 84 2 60 OIB OIB 40 OIB 40 40

10 4 98 20 80104 20 21109 20 IAB IAB IAB 0 MORB OIB IAB MORB OIB IAB MORB OIB IAB (d) (e) (f)

Figure 7. Confusion matrix visualization of tectonic discrimination of basalts in dataset 2 based on didifferentfferent MLAs:MLAs: ( a)(a logistic) logistic regression regression classifier classifier (LRC); (LRC); (b) naïve (b) Bayes naïve (NB); Bayes (c) multilayer(NB); (c) perceptronmultilayer (MLP);perceptron (d) support(MLP); ( vectord) support machine vector (SVM); machine (e) random (SVM); forest(e) random (RF) and forest (f) (RF) SONFIS. and (f) SONFIS.

6.3. Contrast with Convention Conventionalal Discrimination Diagrams Some scholars [[26,53]26,53] have utilizedutilized conventionalconventional discrimination diagrams to classify the corresponding dataset in Section 5.2.1 into MORB MORB,, OIB and IAB respectively, whichwhich facilitatesfacilitates the performance contrast between MLAs and discriminationdiscrimination diagrams. For For dataset dataset 1, 1, Han Han et al. [[53]53] T adopted the TiOTiO22-MnO-P2OO55, ,FeO FeOT-MgO-Al-MgO-Al2O2O3, 3Ti-Zr-Y,, Ti-Zr-Y, Zr/Y-Zr Zr/Y-Zr and and Ti-Zr Ti-Zr five five common diagrams for tectonic discrimination (see Figure 11).). InIn thethe initialinitial state,state, thethe discriminationdiscrimination eeffectffect of the Ti-Zr diagram waswas outstanding,outstanding, whose whose classification classification accuracy accuracy was was about about 75%. 75%. When When the missingthe missing data data was processed,was processed, the classification the classification accuracy accuracy of the of Zr th/Y-Zre Zr/Y-Zr diagram diagram could reachcould 90%, reach but 90%, it was but still it was inferior still toinferior some to MLAs some (e.g., MLAs MLP, (e.g., SVM, MLP, RF SVM, and RF SONFIS). and SONFIS). For dataset For dataset 2, Li et 2, al. Li [ 26et] al. established [26] established the novel the T T FeOnovel/ NaFeO2O-FeOT/Na2O-FeO/CaOT/CaO diagram, diagram, which which had the had excellent the excellent classification classification effect for effect MORB, for OIBMORB, and IABOIB throughand IAB trialthrough and error,trial and while error, the discriminationwhile the discrimination effect between effect MORB between (or OIB)MORB and (or IAB OIB) was and still IAB not satisfactory.was still not Onsatisfactory. the contrary, On thethe proposed contrary, SONFISthe proposed method SONFIS performed method well performed for each tectonic well for setting. each tectonicThe setting. contrast may be incomplete because it did not cover all the basalt discrimination diagrams, but itThe still contrast showed may several be incomplete advantages because of MLAs: it did (1) not MLAs cover could all the take basalt advantage discrimination of all the diagrams, element information,but it still showed even if several the dataset advantages contained of missing MLAs: data,(1) MLAs that is, could MLAs take were advantage less selective of all about the samples.element (2)information, The classification even if accuracythe dataset of MLAs contained was oftenmissing higher, data, and that the is, classification MLAs were e fflessect ofselective singlesetting about wassamples. uniform. (2) The (3) MLAsclassification could be accuracy easily transplanted of MLAs was to tectonic often higher, discrimination and the ofclassification other volcanic effect rocks. of single setting was uniform. (3) MLAs could be easily transplanted to tectonic discrimination of 6.4. Discussion: Applicability and Deficiency of MLA-Based Discrimination Method other volcanic rocks. MLAs can extract valuable information from diverse data, which was taken into account in the6.4. selectionDiscussion: of Applicability basalt samples. and Deficienc The geochemicaly of MLA-Based data Discrimination used in this paper Method were global data of the Cenozoic era, with a certain tectonic environment background, excluding data before the Cenozoic MLAs can extract valuable information from diverse data, which was taken into account in the era. The amount of data was large and the source was numerous, including the fresh, altered and selection of basalt samples. The geochemical data used in this paper were global data of the metamorphic basalts. The richness and diversity of geochemical datasets made for a reliable SONFIS Cenozoic era, with a certain tectonic environment background, excluding data before the Cenozoic model with high classification accuracy. Nevertheless, it was insufficient that basalts that were era. The amount of data was large and the source was numerous, including the fresh, altered and influenced by crustal contamination had not been considered because only three types of tectonic metamorphic basalts. The richness and diversity of geochemical datasets made for a reliable settings (MORB, OIB and IAB) were studied. More tectonic settings also need to be classified accurately SONFIS model with high classification accuracy. Nevertheless, it was insufficient that basalts that by MLAs in the follow-up study. were influenced by crustal contamination had not been considered because only three types of tectonic settings (MORB, OIB and IAB) were studied. More tectonic settings also need to be classified accurately by MLAs in the follow-up study. Compared with binary or ternary discrimination diagrams, another deficiency of MLAs was that the high-dimensional data (more than three dimensions) were difficult to visualize. Although geologists could directly call on the trained model for tectonic discrimination and obtain satisfactory classification results, the visualization problem would limit its practical application to some extent. It is imperative to solve the visualization problem, and many high dimensional visualization (dimensionality reduction) methods are being tried, such as principal component analysis and t-SNE. In addition, some MLAs belong to the black box model, which makes it difficult

Minerals 2019, 9, 376 15 of 19

Compared with binary or ternary discrimination diagrams, another deficiency of MLAs was that the high-dimensional data (more than three dimensions) were difficult to visualize. Although geologists could directly call on the trained model for tectonic discrimination and obtain satisfactory classification results, the visualization problem would limit its practical application to some extent. It is imperative to solve the visualization problem, and many high dimensional visualization (dimensionality reduction) methods are being tried, such as principal component analysis and t-SNE. In addition, some MLAs belong to the black box model, which makes it difficult for geochemists to utilize the internal classification rules directly. The reasonable interpretation of classification results obtained by MLAs also needs to be resolved.

7. Conclusions and Future Work A combined technique based on fuzzy logic and neural networks, termed NFIS, was newly introduced into the research area of tectonic discrimination of basalts in this paper. A metaheuristic optimization algorithm PSO was also utilized to avoid model parameters restricting the classification performance of NFIS. As such, NFIS and PSO were combined to construct a novel intelligent discrimination method called SONFIS. The accuracy and generalizability of NFIS were compared with that of five well-established MLAs using high-dimensional and low-dimensional datasets. The model evaluation results were subsequently represented by classification accuracy calculations and confusion matrix visualization. The conclusions are summarized as follows:

It could be found from Section 6.1 that with the help of PSO, the overall classification accuracy of • SONFIS was about 5% higher than that of NFIS optimized by manual adjustment and grid search. Compared with grid search-optimized NFIS, SONFIS was more complicated, but it demonstrated better classification performance, indicating that the combined model was worth exploring. SONFIS had excellent generalization capacity, since PSO could automatically search for • optimal parameters for different datasets. SONFIS could also be accurately applied to both high-dimensional and low-dimensional datasets, which was valuable for the study of petrology and geochemistry. The comparative experiments show that SONFIS was competitive for the two datasets used, • demonstrating classification accuracy over 90% for both datasets. Furthermore, more elements could be utilized by SONFIS, giving it a superior ability to avoid the unreliability of the discrimination results. The other five well-established MLAs were also excellent methods for the tectonic discrimination • of basalts, showing that ML was a particularly useful and promising tool in geochemical research. The combination of large databases and ML techniques might yield unexpected results.

Although ML had outstanding potential in petrological and geochemical research, it inevitably had some disadvantages, such as difficulty in interpretation and expression. It is necessary to explore effective methods to solve these problems in future studies, so as to improve the usability of ML in Earth Science.

Author Contributions: Overall framework design, M.L.; Methodology and validation, Q.R.; Data collection and preliminary analysis, S.H.; Draft review and editing, Y.Z.; Investigation and conceptualization, Q.Z.; Data curation and formal analysis, J.S. All the authors discussed the results and commented on the manuscript. Funding: This research was jointly funded by the National Natural Science Foundation for Excellent Young Scientists of China (Grant No. 51622904), the Tianjin Science Foundation for Distinguished Young Scientists of China (Grant No. 17JCJQJC44000) and the National Natural Science Foundation for Innovative Research Groups of China (Grant No. 51621092). Conflicts of Interest: The authors declare no conflict of interest. Minerals 2019, 9, 376 16 of 19

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