Enhancing Laser-Induced Breakdown Spectroscopy

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1. Introduction

With the rapid advancement of remote sensing technology and planetary exploration, laser-induced breakdown spectroscopy (LIBS) has emerged as an indispensable tool for analyzing the elemental composition of distant planetary surfaces [1]. LIBS is favored for its non-contact operation, rapid analysis capabilities, and high sensitivity, making it widely applicable across various fields such as geological exploration [2], environmental monitoring [3], and materials science [4]. In the context of Mars exploration, the successful deployment of instruments like ChemCam [5], SuperCam [6], and MarSCoDe [7,8] has underscored LIBS’s pivotal role in acquiring elemental composition data from the Martian surface.

However, LIBS-generated spectral data often encompasses thousands to tens of thousands of dimensions, many of which are redundant or noisy. Such redundancy not only amplifies the complexity of data processing, but also poses a risk of model overfitting, thereby jeopardizing the accuracy and reliability of analytical outcomes. Consequently, extracting the most representative and informative features from this high-dimensional data are essential for enhancing model performance and curtailing computational costs.

To surmount this challenge, a variety of feature selection methods have been proposed, including competitive adaptive reweighted sampling (CARS) [9], the ReliefF algorithm [10], least absolute shrinkage and selection operator (LASSO) [11], and variable importance in projection (VIP) [12]. While these methods have incrementally improved the efficiency and accuracy of LIBS data analysis, their dependence on specific model configurations or parameters often limits their versatility and adaptability.

Feature selection method based on mutual information (MI) has been applied in LIBS data analysis. For example, Zhao et al. used a feature selection method based on MI for melanoma screening, focusing primarily on improving the accuracy of early screening and staging, rather than optimizing the redundancy of feature sets [13]. Additionally, Lyu et al. combined empirical knowledge with data-driven variable selection methods in coal ash content analysis. Although they considered the relevance of features, the issue of feature redundancy was not systematically addressed [14]. These applications mainly focus on calculating a single MI value, without considering redundancy. In contrast, Ruan et al. proposed a hybrid method that combines MI filtering with the bidirectional selection (DBS) algorithm for feature selection in archeological ceramic classification. They utilized a weighting factor α to balance the relevance and redundancy of feature subsets [15]. However, the mRMR method employs mutual information quotient (MIQ), which not only considers the relevance between features and target variables, but also optimizes the feature set by minimizing the redundancy between features [16]. This approach improves model prediction performance while reducing the number of features and enhancing computational efficiency. The combined consideration of correlation and redundancy gives mRMR unique advantages in LIBS data analysis, enabling the more efficient extraction of the most valuable features for classification and prediction. The mRMR method is not dependent on a specific model, making it versatile and adaptable for various data analysis tasks. The mRMR algorithm has demonstrated remarkable effectiveness in fields such as bioinformatics, data classification, and image processing [17,18,19,20]. Although the application of MI in LIBS feature selection has been extensively studied and mRMR has shown remarkable effectiveness in various fields, there are relatively few studies in the literature specifically focusing on the use of mRMR in LIBS data analysis. Hence, further research is urgently needed to explore and develop this area.

This study aims to evaluate the efficacy of the mRMR method in feature selection for LIBS spectral data and to systematically compare it with other prevalent methods such as CARS, Regressional ReliefF (RReliefF), and neighborhood component analysis (NCA). Furthermore, this research delves into the synergistic application of mRMR with various regression models, including random forest (RF), support vector regression (SVR), back propagation neural network (BPNN), and partial least squares regression (PLSR), for the quantitative analysis of oxide content of major Martian elements (e.g., Si, Ti, Al, Fe, Mg, Ca, Na, and K) within the ChemCam dataset. The primary innovations of this study are:

A comprehensive comparison of mRMR with other feature selection methods in the context of LIBS data analysis;
An assessment of the quantitative analysis capabilities when mRMR is integrated with different regression models;
The proposal of an effective LIBS data analysis strategy to enhance the precision of quantitative analysis of Martian surface elemental composition.

Through this research, we aspire to contribute a novel methodological perspective to LIBS data analysis and provide scientific validation for the quantitative analysis of elemental composition on Mars and other planetary surfaces. Additionally, this study is poised to furnish robust data support and analytical tools for future planetary exploration missions.

2. Materials and Methods

2.1. ChemCam Dataset

The dataset used in this study is sourced from the Planetary Data System (PDS) (https://pds-geosciences.wustl.edu/msl/msl-m-chemcam-libs-4_5-rdr-v1/mslccm_1xxx/calib/ (assessed on 30 September 2024)) [21]. It was acquired under operational conditions similar to those of the Mars ChemCam, using the ChemCam laboratory test platform at Los Alamos National Laboratory (LANL). A total of 408 pressed powder samples were analyzed, and their LIBS spectra were recorded. For each sample, the oxide contents of Si, Ti, Al, Fe, Mg, Ca, Na, and K were determined using standard geochemical methods. This study aims to perform quantitative analysis of the oxide contents based on LIBS spectra, and thus, only those samples with well-defined oxide content measurements were included in the analysis.

Each spectral dataset consists of 6144 wavelength variables, covering three wavelength ranges: 240.8110–340.7970 nm, 382.1381–469.0898 nm, and 473.1842–905.5735 nm. Drawing on the findings elaborated in the literature [21], it is evident that under real-world experimental conditions, the spectral signals display substantial noise at the edge positions within the three channel ranges. Without a rigorous screening procedure for these data, there is a high likelihood of adverse impacts on subsequent feature selection and model training phases. Moreover, such an oversight could potentially undermine the accuracy and reliability of the analytical results. To minimize noise interference, wavelength intervals that did not contain key diagnostic peaks for the major elements were excluded. Specifically, the excluded ranges were: 240.8110–246.6350 nm, 338.4570–340.7970 nm, 382.1380–387.8590 nm, 473.1842–492.4265 nm, and 849.0956–905.5735 nm. The preprocessing approach employed in this study bears resemblance to the methodology utilized by Li et al. Specifically, this involves the masking of the initial 100 pixels and the final 50 pixels of the spectral data channel to exclude invalid data points [22]. After preprocessing, each spectral dataset retained 5484 valid wavelength variables for subsequent analysis. The LIBS spectra of 408 samples are presented in Figure 1, with Figure 1a displaying the averaged spectra for each sample across the entire dataset. To enhance the identification and annotation of elemental characteristic peaks, Figure 1b–d showcase enlarged segments of Figure 1a. Upon examination of Figure 1a, it is evident that the LIBS spectra contain numerous distinct characteristic peaks, which are pivotal for the estimation of oxide concentrations. Drawing from the Martian environmental LIBS library [23], we have marked the characteristic peaks for Si, Ti, Al, Fe, Mg, Ca, Na, and K in Figure 1b–d. To avoid congestion in the figures, we have annotated only two characteristic peaks per element. These peaks are sequentially ordered by increasing wavelength: Fe 275.01 nm and 275.65 nm, Mg 279.63 nm and 280.35 nm, Si 288.24 nm, Ti 307.95 nm and 308.89 nm, Si 390.66 nm, Ca 393.48 nm, Al 394.56 nm and 396.21 nm, Ca 396.96 nm, Na 589.16 nm, K 766.70 nm and 770.11 nm, and Na 819.71 nm. Considering the inherent challenges in visually assessing which characteristic peaks are more conducive to quantitative elemental analysis and which are not, the feature selection process for LIBS data is of paramount importance.

During the data processing stage, the SPXY algorithm [24] was employed to partition the spectral data, with 70% allocated for training and 30% for testing, ensuring the model’s ability to make effective predictions on unseen data. The detailed results of the data partitioning for the training and test sets of the eight oxides are provided in Table 1. Among the 408 samples, some have incomplete oxide content data, leading to a discrepancy between the total number of samples in the oxide training and test sets (as shown in Table 1) and the initial count of 408. Most samples contain 5 average spectra; however, samples MIX5B and JSC1413 each contain only 4 average spectra. For sample NAU2HIS, despite having 5 average spectra, one spectrum is abnormal. Therefore, for samples MIX5B, JSC1413, and NAU2HIS, we utilized only 4 spectra for analysis. Taking SiO₂ as an example, the SiO₂ content of sample MIX10 is missing, resulting in a total of 407 samples for SiO₂ analysis. In the training set, there are 285 samples, 282 with 5 spectra each and 3 with 4 spectra each, totaling 1422 spectra. The test set contains 122 samples, each with 5 spectra, totaling 610 spectra. The distribution of samples and spectra for other oxides follows a similar pattern to that of SiO₂ and will not be detailed further.

2.2. Feature Selection Methods

2.2.1. mRMR

The mRMR was initially proposed by Ding et al. in 2003 [25]. The method aims to optimize the correlation between the feature set and the target variable while minimizing redundancy among features. The redundancy and correlation between features in mRMR are quantified by assessing the mutual information among variables, including both pairwise feature–feature mutual information and feature–response variable mutual information [26]. For two discrete random variables, their mutual information is defined as their probability density function:

(1) $I (X, Y) = \sum_{x_{i}} \sum_{y_{i}} P (X = x_{i}, Y = y_{i}) \log \frac{P (X = x_{i}, Y = y_{i})}{P (X = x_{i}) P (Y = y_{i})}$

The objective of the mRMR algorithm is to identify an optimal subset $S$ of features that maximizes the relevance $V_{S}$ between $S$ and the response variable $Y$ , while minimizing the redundancy $W_{S}$ of $S$ . The relevance $V_{S}$ and redundancy $W_{S}$ are quantified using mutual information:

(2) $V_{S} = \frac{1}{|S|} \sum_{X_{i} \in S} I (X_{i}, Y)$

(3) $W_{S} = \frac{1}{{|S|}^{2}} \sum_{X_{i}, X_{j} \in S} I (X_{i}, X_{j})$

here,

|S|

is the number of features in

S

The ranking of features in the mRMR algorithm is determined by employing the mutual information quotient (MIQ) value within a sequential forward search scheme.

(4) ${MIQ}_{X_{i}} = \frac{V_{X_{i}}}{W_{X_{i}}}$

The $V_{X_{i}}$ and $W_{X_{i}}$ represent the relevance and redundancy of the feature $X_{i}$ , respectively.

(5) $V_{X_{i}} = I (X_{i}, Y)$

(6) $W_{X_{i}} = \frac{1}{|S|} \sum_{X_{j} \in S} I (X_{i}, X_{j})$

The mRMR algorithm is implemented in this study using the ‘fsrmrmr’ function in MATLAB. This function provides feature indices and their corresponding score values, which are sorted based on their importance. Higher scores indicate greater significance of the features. The steps for sorting features using the ‘fsrmrmr’ function are as follows:

Identify the feature with the highest relevance from the given feature set $Ω$ and incorporate it into the empty set $S$ .
In the complement set $S^{C}$ of feature subset $S$ , search for features with non-zero relevance and zero redundancy. If there are no features in $S^{C}$ with non-zero relevance and zero redundancy, please proceed to step 4. Otherwise, select the feature with the highest relevance and add it to subset $S$ .
Repeat step 2 until the redundancy of all features in $S^{C}$ is not zero.
Identify the feature in $S^{C}$ that exhibits both non-zero relevance and redundancy, and include this selected feature in set $S$ based on its maximum MIQ value.
(7) $\begin{matrix} m a x \\ X_{i} \in S^{C} \end{matrix} {MIQ}_{X_{i}} = \begin{matrix} m a x \\ X_{i} \in S^{C} \end{matrix} \frac{I (X_{i}, Y)}{\frac{1}{|S|} \sum_{X_{j} \in S} I (X_{i}, X_{j})}$
Reiterate step 4 until the relevance of all features in $S^{C}$ is zero.
Incorporate zero-relevance features into $S$ in a random sequence.

We take the SiO₂ data in the dataset as an example to illustrate the specific application of the mRMR algorithm in LIBS. In this application example, the feature $X$ of SiO₂ is derived from LIBS spectral data, and its dimension is 1422 × 5484, indicating that we have 1422 spectral samples, and each spectral sample contains 5484 feature variables. The corresponding response variable $Y$ has a dimension of 1422 × 1, representing the SiO₂ content corresponding to the 1422 spectral samples.

When the mRMR algorithm is applied, first, the relevance and redundancy of feature $X$ are evaluated by calculating mutual information. Specifically, the calculated relevance $V_{X}$ between feature $X$ and response variable $Y$ has a dimension of 5484 × 1, while the redundancy $W_{X}$ between features has a dimension of 5484 × 5484. We calculate that the feature with the highest relevance is the 2669th feature, denoted as $X_{2669}$ , and then $X_{2669}$ is added to the initially empty set $S$ .

Next, we search for features with non-zero relevance and zero redundancy in the set $S^{C}$ composed of other features except $X_{2669}$ . After a careful search, we found no features meeting this condition. Therefore, according to the flow of the mRMR algorithm, we moved the operation to step 4.

In step 4, we calculated the MIQ values of each feature in the feature set $S^{C}$ according to Formula (7). In this case, the set $S$ in Equation (7) contains only one feature, $X_{2669}$ , that is, $|S| = 1$ , and $X_{j} = X_{2669}$ . After calculation, we found that the 838th feature $X_{838}$ has the largest MIQ value, so it is selected into the set $S$ .

After that, we followed the process of step 5 and step 6 to complete the sorting operation of all features step by step. In this process, we strictly followed the principle of the mRMR algorithm to ensure that each step of calculation and decision is in line with its logical requirements, thus achieving an effective feature ordering, which provides an important basis for subsequent data analysis and processing.

2.2.2. Overview of Other Feature Selection Methods

To comprehensively evaluate the performance of the mRMR algorithm for feature selection in LIBS data, this study compares it with the full spectrum (i.e., spectral data consisting of 5484 wavelength variables) and three widely used feature selection methods: CARS, RReliefF algorithm, and NCA. A brief overview of the basic principles of these methods is provided below.

CARS algorithm is inspired by the principle of “survival of the fittest” from Darwin’s theory of evolution. It aims to select the most informative wavelengths for predictive modeling from multi-component spectral data. Initially, the method employs the absolute values of the regression coefficients derived from a partial least squares model to evaluate the importance of each wavelength. In each iteration of CARS, a random subset of the sample set is selected using the Monte Carlo sampling method to construct a calibration model. Subsequently, a two-step procedure—combining the exponentially decreasing function (EDF) and adaptive reweighted sampling (ARS)—is applied to select key wavelengths. The EDF is used to efficiently eliminate wavelengths with smaller absolute regression coefficients, while ARS mimics natural selection by assigning higher selection probabilities to more important wavelengths. This competitive mechanism further refines the wavelength selection process. The subset of wavelengths that minimizes the root mean square error of cross-validation (RMSECV) is then identified as the optimal solution through cross-validation (CV). The CARS method effectively reduces redundancy in highly collinear spectral data, thereby improving the predictive accuracy of the model. Additionally, by selecting wavelengths that are closely related to chemical properties, CARS enhances model interpretability, making the results more insightful and meaningful [27].

RReliefF is a feature selection method specifically designed for regression tasks. It is an extension of the original ReliefF algorithm, adapted to handle cases where the target variable is continuous. Unlike traditional feature selection methods, RReliefF is capable of effectively addressing strong dependencies between features, offering a more precise evaluation of feature importance. By leveraging local information, the algorithm captures the global significance of features, making it particularly suitable for datasets with complex interdependencies among features [28].

NCA is a distance metric learning-based method for feature selection and dimensionality reduction. Its primary goal is to learn an optimal distance metric that minimizes the distance between similar samples in the transformed feature space. One key advantage of NCA is that it imposes no assumptions on the data distribution, making it highly versatile and applicable to a wide range of learning tasks. This characteristic makes NCA especially well-suited for handling nonlinear and complex data distributions [29,30].

2.3. Overview of Regression Models

2.3.1. RF

RF is an ensemble learning technique that enhances the overall model performance by constructing multiple decision trees and aggregating their predictions [31]. Recognized for its high robustness to outliers and noise, as well as its capacity to manage high-dimensional data, RF has been extensively applied in various fields [32]. In this study, we developed an RF model consisting of 150 decision trees, with each tree trained on a randomly selected subset of features to enhance the model’s generalization capabilities. Additionally, to manage the model complexity, we specified the minimum number of samples per leaf node to be 1.

2.3.2. SVR

SVR is a regression analysis technique predicated on the support vector machine (SVM) framework, which forecasts the target variable’s value by identifying the hyperplane that maximizes the margin [33]. A pivotal advantage of SVR is its employment of kernel functions, which facilitate the mapping of input data into a high-dimensional space, thereby enabling the execution of nonlinear regression analyses. Within the scope of this study, we opted for the radial basis function (RBF) as the kernel function for our SVR model. This selection was informed by the RBF kernel’s efficacy in managing nonlinearities within the data and its robust adaptability across diverse data distributions. In the SVR model, the regularization parameter and the kernel function parameter can be optimized using the grid search method. In this study, based on preliminary experimental exploration, the regularization parameter C was set to 4, and the kernel function parameter γ was set to 0.1.

2.3.3. BPNN

BPNN is a multi-layer feedforward neural network trained via the backpropagation algorithm [34]. The BPNN minimizes prediction errors by adjusting the weights and biases within the network.

In this study, our BPNN architecture comprises an input layer, a hidden layer, and an output layer. The input layer has neurons that correspond one-to-one with the selected features, ensuring the network’s effective capture of the data’s intrinsic characteristics. The hidden layer, containing 12 neurons, is configured to provide sufficient nonlinear mapping capabilities for managing complex input–output relationships. In this study, we utilized 5484 spectral features of SiO₂ as examples to investigate the performance of the BPNN model under different numbers of hidden layer neurons. As clearly depicted in Figure S1, when the number of hidden layer neurons ranges from 4 to 28, the BPNN model exhibits relatively minor variations in prediction performance, indicating that its impact on the prediction results is not substantial. Nevertheless, when the number of hidden layer neurons reaches 30 or more, the training and running of the model become challenging with typical computer configurations, due to the drastic increase in memory requirements. Taking into account multiple factors, including model performance, computing resource limitations, and practical operability, we ultimately set the number of hidden layer neurons to 12. This setting not only ensures that the model maintains a certain level of prediction ability but also guarantees its smooth operation under ordinary computing conditions.

The output layer’s neuron count matches the dimension of the prediction target, which is utilized to forecast the concentration of the target elements. In terms of activation functions, the hidden layer uses the hyperbolic tangent function (‘tansig’), while the output layer employs a pure linear activation function (‘purelin’) to produce continuous predictive outputs.

The network training process employs the Levenberg–Marquardt algorithm, an efficient optimization algorithm that integrates the stability of gradient descent with the rapid convergence of the Newton method. Regarding the training parameter configuration, we set the maximum number of training epochs to 1000 and the target minimum error threshold to 1 × 10⁻⁶, ensuring that the model achieves high-precision predictions during the training process. To further enhance training stability and accelerate convergence, we set the minimum performance gradient (i.e., the threshold for weight updates) and the maximum number of consecutive iterations without performance improvement (i.e., the patience parameter) to 1 × 10⁻⁶ and 6, respectively. Additionally, the learning rate was set to 0.01, and the momentum factor was also 0.01; these meticulously chosen parameters helped to balance the step length and update direction during training, thereby optimizing the network’s overall performance.

2.3.4. PLSR

PLSR is a multivariate analytical technique designed to establish linear relationships between predictor variables and response variables for predictive purposes. Particularly adept at handling high-dimensional datasets, PLSR effectively mitigates the issue of multicollinearity among predictor variables [35]. In this study, Monte Carlo cross-validation (MCCV) was employed to optimize the number of latent variables in the PLSR model. Specifically, PLSR modeling was conducted under varying numbers of latent variables through multiple random divisions of the dataset into calibration sets and test sets. Notably, the maximum number of latent variables set for cross-validation was 100, the data preprocessing method was ‘autoscaling’, the number of Monte Carlo simulations was set to 1000, and the calibration samples constituted 80% of the total sample size. For SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O, the results of MCCV are depicted in Figure S2. As evident from the figure, with an increase in the number of latent variables, RMSECV values for each component exhibit a trend of initially decreasing and subsequently stabilizing. Through detailed analysis, when RMSECV reaches its minimum, the corresponding numbers of latent variables for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O are 67, 73, 57, 76, 61, 70, 81, and 67, respectively. These findings provide a crucial parameter foundation for the subsequent establishment of an accurate PLSR model, thereby facilitating the enhancement of the accuracy and reliability of the model’s predictions for each component.

3. Results

3.1. Results of Feature Selection

3.1.1. mRMR Feature Selection

In the present study, we utilized the mRMR algorithm for feature selection from LIBS data to evaluate the relative importance of various wavelength variables. Figure 2 illustrates the mRMR algorithm’s ranking of feature importance for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O, showcasing only the top 500 wavelength variables according to their importance scores. Analysis of Figure 2 indicates that the higher-ranked wavelength variables possess higher mRMR scores, which are strongly associated with the concentrations of the respective oxides. Figure 2a presents the mRMR feature selection outcomes for SiO₂, where the top 10 wavelength variables are observed to have considerably higher scores than others. A similar pattern is observed in Figure 2b–d, as well as in Figure 2f–h, where the top eight wavelength variables demonstrate significantly elevated scores relative to the remaining variables. In Figure 2e–g, the mRMR feature selection analysis for MgO and Na₂O reveals that the top seven wavelength variables exhibit significantly higher scores compared to the rest, with a pronounced distinction between the seventh and eighth ranked variables. Beyond the high-scoring top-ranked wavelength variables, the scores for other variables descend gradually and asymptotically approach a stable value.

We have conducted an in-depth analysis of the relationship between the features selected by the mRMR algorithm and the target oxide elements. The specific findings are illustrated in Figure 3 and Table 2. Figure 3 depicts the correlation between the top 50 features ranked by importance using mRMR and the characteristic lines of the target oxide elements. In Figure 3, the black spectrum represents the average spectrum derived from 408 samples, the red stars mark the peak positions of the element characteristic lines, and the brown dotted vertical lines indicate the 50 features selected by mRMR, with the height of these lines reflecting the significance of each feature. The correspondence between the top 50 mRMR-selected features and the characteristic lines of the target oxide elements is detailed in Table 2.

Among the top 50 features identified by mRMR, 8 are located at or near the characteristic line of Si, 21 are located at or near the characteristic line of Ti, 8 are located at or near the characteristic line of Al, 27 are located at or near the characteristic line of Fe, 10 are located at or near the characteristic line of Mg, 9 are located at or near the characteristic line of Ca, 7 are located at or near the characteristic line of Na, and 9 are located at or near the characteristic line of K.

Further analysis reveals that two features selected by mRMR are located at or near the characteristic lines of Fe 516.37 nm, Mg 285.25 nm, Mg 293.74 nm, Ca 393.48 nm, Ca 612.39 nm, and Na 819.70 nm. Additionally, three features are identified at or near the characteristic lines of Fe 438.48 nm and Na 589.16 nm, and four features are found at or near the characteristic line of K 766.70 nm. Moreover, with the exception of Si, the most important feature selected by mRMR for each of the other seven elements is located at or near the characteristic line of the corresponding element.

By employing the mRMR feature selection method, we identified the feature variables from the original dataset that exhibit the highest correlation with the target variables and the lowest redundancy among themselves. This study systematically evaluated the impact of these mRMR-selected features on the performance of four regression models, RF, SVR, BP, and PLSR, with the aim of determining the optimal set of feature variables. Figure 4 illustrates the relationship between the number of feature variables and the predictive performance of these four regression models on the test set in the context of quantitative analysis of eight oxides. In Figure 4, each subplot corresponds to a specific oxide, with the x-axis representing the number of feature variables, and the left and right y-axes representing the coefficient of determination (R²) and the root mean square error (RMSE) of the test set predictions, respectively.

As observed in Figure 4, except for the PLSR models, the R² values generally show an increasing trend followed by a stabilizing trend as the number of features increases. Meanwhile, the RMSE values exhibit a rapid decrease initially, followed by gradual stabilization. This indicates that a moderate increase in the number of features can improve the predictive accuracy of the model. However, increasing the number of features does not always result in a continuous reduction in RMSE. Once the number of features reaches a certain threshold, the RMSE may stabilize or even slightly increase, suggesting that too many features may introduce noise and reduce model performance.

In the PLSR models for SiO₂ (Figure 4a), Al₂O₃ (Figure 4c), and FeO_T (Figure 4d), as the number of features increases, although the detailed variations of R² and RMSE fluctuated, they generally exhibited a trend where R² increased first and then decreased, while RMSE decreased first and then increased. In the PLSR models for TiO₂ (Figure 4b), MgO, CaO, Na₂O, and K₂O (Figure 4e–h), the R² and RMSE demonstrated irregular changes. However, their trends were consistently opposite, with the maximum R² corresponding to the minimum of RMSE. Based on these observations, we selected the number of features that minimizes the RMSE as the optimal feature set for constructing the quantitative analysis model.

3.1.2. Results of Other Feature Selection Methods

We also optimized the number of selected feature variables using the RReliefF and NCA algorithms, with the corresponding results presented in Figures S3 and S4, respectively. For models other than the PLSR model, similar to the situation in Figure 4, the trends of R² and RMSE generally align with those of the feature variables selected by the mRMR algorithm as the number of features increases. In the PLSR model presented in Figure S3, for SiO₂, TiO₂, Al₂O₃ (Figure S3a–c), MgO (Figure S3e), and K₂O (Figure S3h), the R² initially increases, then decreases with the increasing number of features, and eventually stabilizes. Conversely, the RMSE exhibits an opposite trend: it first decreases, then increases, and finally reaches a stable state. For the PLSR model of FeO_T (Figure S3d), both R² and RMSE vary irregularly, yet their changing states are in opposition. Moreover, in the PLSR models of CaO (Figure S3f) and Na₂O (Figure S3g), the overall R² shows a downward tendency, while the corresponding RMSE demonstrates an upward trend.

By employing the NCA feature selection method and examining the relationship between feature weights and the corresponding number of features, we extensively explore the impact of varying numbers of features on the predictive performance of the regression model. Figure S4 illustrates the model performance for eight oxides with feature weights ranging from 22 to 43, thus presenting R² and RMSE values for 22 to 43 data points. While R² and RMSE exhibit some irregular fluctuations as the number of features increases, an inverse relationship between these two metrics is consistently observed. Based on this trend, we selected the feature set that minimizes RMSE, as indicated in Figures S3 and S4, as the optimal set for constructing the quantitative analysis model.

In determining the number of features selected by the CARS method, we followed the fundamental principles of the CARS approach. Specifically, CARS assesses the importance of each wavelength by using the absolute values of the regression coefficients from the PLSR model, and selects wavelengths based on this criterion. The process of CARS feature selection is illustrated in Figure S5. Specifically, each subgraph (a), (b), (c), (d), (e), (f), (g), and (h) in Figure S5, respectively, represents the feature selection process of CARS for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O. Each subgraph consists of three sections: (i), (ii), and (iii).

Section (i) is used to depict the variation trend of the number of sampled variables with the increase in the number of sampling runs, where the abscissa denotes the number of sampling runs and the ordinate represents the number of sampled variables. It can be clearly observed that as the number of sampling runs increases, the number of sampled variables exhibits a decreasing tendency.

Section (ii), the abscissa remains the number of sampling runs, while the ordinate is the fivefold cross-validation RMSECV value. With the increment of sampling runs, the fivefold RMSECV value changes gradually and reaches a relatively stable or minimum value at a specific point. This variation trend reflects the performance alteration of the model during the feature selection process. Generally, a smaller RMSECV value indicates a stronger predictive ability of the model.

Section (iii) presents the regression coefficients of each variable, where the abscissa is still the number of sampling runs, and the ordinate is the regression coefficient for each variable. In this section, lines with different colors illustrate the variation paths of the regression coefficients of each variable as the number of sampling runs increases. The interweaving and changes in these lines vividly demonstrate the dynamic variation in the contribution degree of different variables to the model during the feature selection process, suggesting that the CARS algorithm is continuously adjusting the weights of the variables to screen out the most valuable features for the model.

By analyzing the subgraphs in Figure S5, we selected the features corresponding to the minimum point of the 5-fold RMSECV. Specifically, CARS selected 466, 525, 217, 412, 473, 352, 478, and 506 features for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O, respectively. These selected feature numbers will be utilized in the subsequent model construction and analysis to enhance the accuracy and efficiency of the model. The CARS feature selection was grounded in the PLSR model. As reported in references [36,37], the numbers of features utilized in RF, SVR, and BP regression models after CARS feature selection are all determined by the number of features selected in the PLSR model.

In summary, Table S1 presents the optimal number of features selected by various feature selection methods for the regression models employed in the quantitative analysis of eight oxides. These findings offer valuable insights for developing efficient and accurate quantitative analysis models.

3.2. Regression Models

In this study, we investigated the impact of various feature selection methods combined with regression models on the predictive performance of spectral data for several oxides, including SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O. The feature selection techniques employed were the mRMR algorithm, CARS, the RReliefF algorithm, and NCA. These methods were coupled with four regression models, RF, SVR, BPNN, and PLSR, to predict the target variables. Table S1 summarizes the comparative performance metrics of these combined approaches, focusing on R² for both the training and test datasets, as well as RMSE. These metrics were used to evaluate the accuracy and generalizability of the different methodologies in predicting the regression outcomes.

3.2.1. Reducing Overfitting Through Feature Selection

According to the data presented in Table S1, we have computed the discrepancy in R² between the training and test sets for each regression model. These calculated differences are presented in Table 3, which serves as a crucial reference for assessing the performance consistency of different regression models. In the context of model evaluation, for both the training and test sets, an R² value closer to one and a minimal difference between them typically signify the superior generalization capabilities of the model. A large difference in R² values between the training and test sets may indicate potential overfitting issues. To highlight instances where models might exhibit a propensity for overfitting, in Table 3, we have bold-faced those entries where the R² discrepancy exceeds 0.1.

We observed that directly using all 5484 wavelength variables for regression modeling led to significant overfitting. This was characterized by a high R² in the training set, but a relatively low R² in the test set. Specifically, in our study, the RF model for Al₂O₃, the SVR model for all oxides, the BPNN model for SiO₂, TiO₂, and Al₂O₃, and the PLSR model for TiO₂, Al₂O₃, Fe₂O₃, and Na₂O all exhibited varying degrees of overfitting. Specifically, the difference between the R² values of the training set and the test set for these models was above 0.1008, with the maximum difference reaching 0.4272. However, after applying the mRMR feature selection, only the SVR model for TiO₂ showed slight overfitting, with an R² difference between the training and test sets of 0.1020. The R² differences for the other models ranged from 0.0082 to 0.0920, indicating that mRMR feature selection effectively mitigated overfitting.

Following the CARS feature selection, some regression models for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, and Na₂O still exhibited overfitting, with the R² difference between the training and test sets ranging from 0.1002 to 0.2683. The R² differences for the remaining models ranged from 0.0141 to 0.0975. Among the features selected by the RReliefF algorithm, only the RF model for TiO₂ and the PLSR model for FeO_T showed overfitting, with R² differences of 0.1188 and 0.1233, respectively. The R² differences for the remaining models ranged from 0.0079 to 0.0948. For the NCA algorithm, the SVR model for TiO₂, and the PLSR models for FeO_T, MgO, and Na₂O exhibited R² differences exceeding 0.1000, with values of 0.1096, 0.1003 0.1281, and 0.1006, respectively. The R² differences for the other models ranged from 0.0106 to 0.0997.

3.2.2. Analysis of Number of Features

We analyzed the number of features extracted by four feature selection methods—mRMR, CARS, RReliefF, and NCA—in the prediction of various oxides. In Table S1, we categorized the quantitative analysis models of eight oxides based on feature selection methods and calculated the average number of features for four regression models under each feature selection method. This allowed us to determine the number of features for each oxide under the corresponding feature selection method, as shown in Table 4. The results indicate that the mRMR method extracted the fewest features in the prediction of MgO, and K₂O, with a relatively small number of features extracted for FeO_T, CaO, and Na₂O predictions, and a relatively large number of features extracted for SiO₂, TiO₂, and Al₂O₃ predictions. The CARS method extracted the fewest features for Al₂O₃ prediction, and the highest number of features for predictions of other oxides. As for the RReliefF method, it extracted the largest number of features in the prediction of Al₂O₃. For SiO₂ and TiO₂, a small number of features were extracted, and for other oxides, a relatively large number of features were obtained. The NCA method extracted the fewest features for SiO₂, TiO₂, FeO_T, CaO, and Na₂O predictions, with a medium number of features for other oxides. In summary, there are significant differences in the number of features extracted by different feature selection methods for the prediction of various oxides, which may affect the predictive performance and generalization capability of the models.

3.2.3. Comparison of Prediction Results

The integration of feature selection methods—mRMR, CARS, RReliefF, and NCA—with regression models, such as RF, SVR, BPNN, and PLSR, for predicting the oxide contents has shown promising performance across all methods, as shown in Table S1. Table 5 provides a comprehensive summary of models exhibiting remarkable predictive performance for diverse oxides. Through a meticulous comparison of the R² and RMSE values across the models, it becomes evident that for each oxide, the specific model demonstrates exceptional proficiency in feature selection and regression prediction. The outstanding performance of these models is manifested not only by the elevated R² values on the training set, which signify a strong model fit, but also by the diminished RMSE values on the test set, thereby confirming their efficacy and robustness in practical applications within the realm of oxide analysis.

In the analysis of SiO₂, the mRMR-SVR model achieved the highest R² value in the test set, demonstrating its superior efficacy in feature extraction and the precision of predictive modeling. For TiO₂, the full-spectrum RF approach outperformed alternative models, with the mRMR-RF combination also exhibiting robust performance. The mRMR-SVR model again demonstrated exceptional predictive capabilities for Al₂O₃, underscoring its strength in feature selection and regression analysis. In the case of FeO_T, the NCA-RF model emerged as the top performer, highlighting its advanced feature selection and regression modeling prowess.

Regarding MgO, both the mRMR-RF and NCA-RF models demonstrated commendable performance, with the NCA-RF model showing a marginal advantage in terms of RMSE on the training set. The mRMR-RF method also excelled in the regression analysis for CaO, achieving a notable enhancement in predictive performance concurrent with a significant reduction in the number of features. For Na₂O, the RReliefF-BP model displayed strong predictive capabilities, closely trailed by the mRMR-BP model. Lastly, in the regression analysis for K₂O, the mRMR-SVR model outperformed its counterparts, while the NCA-SVR model also presented admirable predictive performances.

We further investigated the prediction performance of four feature selection methods on the contents of eight oxides under the condition of identical feature numbers and the application of RF regression. The results are presented in Table S2. It was found that the features selected by the mRMR method attained the maximum R² values and the minimum RMSE values in the test set predictions for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, and Na₂O. Nevertheless, when the number of features was 506, the prediction results of mRMR-selected features were slightly inferior to those of the RReliefF and NCA methods for K₂O prediction.

3.2.4. mRMR-RF Prediction Results

In this study, we conducted a comprehensive evaluation of the mRMR-RF method’s performance in predicting the content of eight oxides, with results detailed in Table S1 and depicted in Figure 5. The predictive outcomes reveal that, with the exception of TiO₂ which exhibited test set R² values below 0.9000, the R² values for the other oxides surpassed 0.9000. This indicates the mRMR-RF method’s efficacy in predicting the content of these oxides. The detailed results are as follows: For SiO₂, the mRMR-RF method achieved a test set R² of 0.9445 and an RMSE of 2.39; for TiO₂, the R² was 0.8985 with an RMSE of 0.22; for Al₂O₃, the R² was 0.9022 with an RMSE of 1.52; for FeO_T, the R² was 0.9476 with an RMSE of 1.14; for MgO, the R² was 0.9769 with an RMSE of 0.43; for CaO, the R² was 0.9903 with an RMSE of 0.44; for Na₂O, the R² was 0.9575 with an RMSE of 0.35; and for K₂O, the R² was 0.9342 with an RMSE of 0.46.

Furthermore, we conducted a comparative analysis of the mRMR-RF method against AOMB-CNN, elastic net, and two variants of PLS1-SM, with the results detailed in Table 6. In the comparison of RMSE values, the mRMR-RF method exhibited the lowest RMSE in predicting SiO₂, TiO₂, FeO_T, MgO, CaO, Na₂O, and K₂O, with RMSE values of 2.39, 0.22, 1.14, 0.43, 0.44, 0.35, and 0.46, respectively. These values were significantly lower than those obtained using other methods. Regarding the prediction of Al₂O₃, the mRMR-RF method’s RMSE value was 1.52, which, although not the lowest, remained lower than those of elastic net and the two PLS1-SM methods.

4. Discussion

4.1. Feature Selection

4.1.1. The Physical Significance of the mRMR-Selected Features

Utilizing the mRMR algorithm for feature selection, we successfully identified the wavelength variables most highly correlated with the concentrations of the target oxides, thereby validating the efficacy of the mRMR algorithm in the context of LIBS data. The mRMR feature selection results depicted in Figure 2 highlight the importance of the top-ranked wavelength variables in the construction of predictive models, as evidenced by their higher scores. The trend in the scores of the wavelength variables suggests that those ranked higher contribute more significantly to the model, while the impact of other variables, though less pronounced, is not negligible. This provides a scientific basis for the selection of an optimal feature set.

Through analysis of the correspondence between the features selected by mRMR and the target oxide elements (as shown in Figure 3 and Table 2), it can be demonstrated that the features selected by mRMR possess a clear physical significance. Figure 3 clearly illustrates the close relationship between the top 50 features selected by mRMR and the characteristic lines of the target oxide elements. Specifically, a significant number of these features are located near the characteristic lines of various elements. Notably, for Ti, there are 21 mRMR features located at or near its characteristic lines, and for Fe, there are 27, indicating a distinct directional preference for these elements.

Further analysis of different elements reveals specific correlations. For instance, Fe has two mRMR-selected features near the characteristic line at 516.37 nm and three near 438.48 nm; K has four mRMR-selected features at or near 766.70 nm. These findings further substantiate the correlation between the features selected by mRMR and the characteristic lines of the elements. For Si, Al, Mg, Ca, Na, and K, there are 7 to 10 mRMR-selected features located at or near the characteristic lines of the corresponding elements. Although the number is relatively smaller compared to Ti and Fe, it still indicates that the features selected by mRMR effectively capture the characteristics related to the peaks of these elements. Moreover, for the other seven elements except Si, the features ranked first in mRMR importance are located at or near the corresponding element characteristic lines. This fact fully demonstrates that the features selected by the mRMR algorithm are scientifically reasonable and meaningful, and can reflect the key physical characteristics of the target oxide elements.

4.1.2. Influence of the Number of Features on Regression Models

Employing the mRMR feature selection method, we successfully identified from the original dataset the feature variables that exhibit the highest correlation with the target variables and the lowest redundancy among themselves. This process significantly influences subsequent model performance. The results depicted in Figure 4 suggest that a moderate increase in the number of feature variables can enhance the model’s predictive accuracy. However, an increase in the number of features does not invariably lead to a reduction in RMSE values. Once the number of feature variables surpasses a certain threshold, the RMSE may stabilize or even marginally increase, implying that an overabundance of feature variables might introduce noise, thereby degrading model performance. This observation highlights the necessity of balancing the quantity and quality of features during selection to mitigate overfitting and noise. In the PLSR models for SiO₂ (Figure 4a), Al₂O₃ (Figure 4c), and FeO_T (Figure 4d), as the number of features increases, the R² exhibits a trend of first increasing and then decreasing. In contrast, RMSE shows an opposite trend; specifically, RMSE decreases first and then increases. This phenomenon indicates that in these specific instances, an increase in the number of features may initially ameliorate model performance, but eventually results in a deterioration. This could be due to an excess of features introducing unnecessary complexity and noise, adversely affecting the model’s predictive capabilities.

Furthermore, the feature selection results obtained using the RReliefF and NCA algorithms further confirmed the relationship between the number of features and model performance. Although the specific features selected by each algorithm varied, both the R² and RMSE demonstrated opposite trends, providing additional evidence of the impact of feature selection on the performance of the regression model.

In certain oxide quantitative analysis models, the number of features selected by the mRMR method is not the smallest. Nevertheless, the minimum redundancy advantage of the mRMR algorithm cannot be evaluated merely based on the number of features. During the feature selection process, the mRMR algorithm does not solely consider the quantity of features. Instead, it achieves feature selection by comprehensively calculating the correlation between features and target variables, as well as the redundancy among features. Taking the quantitative analysis model of SiO₂ in Table S1 as an instance, the mRMR-RF approach selected 492 features, while CARS-RF, RReliefF-RF, and NCA-RF selected 466, 490, and 304 features, respectively. Although the number of features selected by these methods is smaller than the 492 features determined by mRMR, the 492 features selected by mRMR are chosen after a comprehensive consideration of their high correlation with the target variable and their low mutual redundancy. This implies that these features can provide more efficient information regarding the target variable with less overlap, thereby resulting in superior performance in subsequent model predictions.

Furthermore, under the condition of an identical number of features and an RF regression model, we compared the prediction performances of the four feature—selection methods. The results are presented in Table S2. The data demonstrated that the features selected by mRMR achieved the maximum R² values and the minimum RMSE values in the test set predictions for SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, and Na₂O. This further verifies that the features selected by mRMR have the strongest correlation with the target variables and the lowest redundancy among them. Notably, when the number of features is 506, for the prediction of K₂O, the prediction results of mRMR-selected features are slightly inferior to those of RReliefF and NCA. This indicates that there is a certain amount of redundant information when the 506 features selected by mRMR are applied to RF regression. As can be observed from Table S1, when mRMR selects 212 features for RF prediction, its prediction outcome is better than that of using 506 features in Table S2. In conclusion, during the feature selection process, the mRMR algorithm can fully take into account the correlation and redundancy of features, thus providing robust support for the construction of efficient prediction models.

In addition, taking CaO as an example, we investigated the impact of features selected manually and those selected by the mRMR method on the regression model. Specifically, we meticulously selected 30 characteristic wavelengths of Ca manually, with the relevant details presented in Table S3. Subsequently, these 30 characteristic spectral lines were employed in four regression models, namely the RF, SVR, BP, and PLSR models. The results obtained were meticulously compared with those derived from using the features selected by mRMR, as depicted in Table S4. Among the RF, SVR, and BP regression models, the prediction results based on mRMR features are markedly superior to those based on manually selected features, as evidenced by their better performance in various evaluation metrics. However, in the PLSR model, the manually selected features yielded relatively favorable prediction results. Based on the comparison results, it is evident that the mRMR feature selection algorithm is effective in most scenarios, capable of providing more advantageous features to the regression model and thereby enhancing the model’s prediction performance. Nevertheless, in the specific case of the PLSR model, its efficacy may be inferior to that of manual feature selection. This finding offers a valuable reference for the subsequent feature selection strategy, suggesting that in practical applications, we should comprehensively consider the feature selection method in accordance with the specific type of regression model to achieve the optimal prediction effect.

4.1.3. The Necessity of Feature Selection

In scientific research, the necessity of feature selection depends on the specific application context and characteristics of the data. For complex datasets that contain a significant amount of redundant or irrelevant information, feature selection can effectively reduce the computational burden, enhance model training efficiency, and potentially improve the generalization ability of the model. However, in some cases, particularly when CNNs are employed, feature selection may not be required. CNNs are capable of automatically learning useful features from raw data, thus obviating the need for explicit feature selection.

For example, Li et al. adopted a full-spectrum approach in their study, bypassing feature selection [22]. This was because LIBS inherently contains distinctive “fingerprint” information about elemental composition, which CNNs can leverage without relying on a preselected subset of features. In such contexts, the convolutional and pooling layers of CNNs are designed to automatically extract and identify relevant features directly from the data, making traditional feature selection steps unnecessary in certain applications.

Although feature selection is commonly used in traditional machine learning to reduce data dimensionality and computational complexity, CNNs typically do not face significant computational challenges due to their inherent properties of sparse connectivity and weight sharing, even when dealing with large datasets. In certain cases, however, when the input data plays a critical role in the analysis performance, feature selection can be guided by physical significance and prior knowledge. For example, Xing et al. developed a dual Li-line DP-CNN to effectively address the matrix effect encountered in LIBS when measuring lithium in saline samples [41]. In CNN models, even without explicit feature extraction, the network can optimize the feature selection and extraction process during training, thereby enhancing performance. Gou et al. proposed an improved CNN model, SFEMARNet, which incorporates a spectral feature extraction (SFE) module and a multiple attention residual (MAR) module [42]. This model is designed to extract detailed features and highlight relevant information from LIBS spectral data. In some cases, feature selection can further improve model performance by removing irrelevant or noisy features. For instance, Ding et al. employed feature selection methods based on the RF algorithm, specifically mean decrease accuracy (MDA) and mean decrease impurity (MDI), to filter the data [43]. Their results showed that the MDA-CNN model outperformed other models, effectively identifying soil origins. These findings suggest that while feature selection may not always be essential in CNNs, it can enhance model performance and interpretability in specific applications.

In summary, the application of feature selection in CNNs should be determined based on the specific task, data characteristics, and model requirements. While feature selection can improve model performance in certain applications, it is not always necessary, especially in cases where deep learning models are capable of automatically learning relevant features from the data. The decision to incorporate feature selection should thus be made carefully, considering whether manual feature engineering provides additional benefits over the inherent feature extraction abilities of CNNs.

4.1.4. The Computational Complexity of Feature Selection Algorithms

In terms of computational complexity, as depicted in Table 7, the mRMR algorithm exhibits a relatively balanced state in both time complexity and space complexity, thereby rendering it suitable for medium-sized datasets. The CARS algorithm manifests high computational complexity during the iterative process, particularly when handling large-scale datasets and high-dimensional data. The time complexity of the RReliefF algorithm is approximately equivalent to that of the mRMR algorithm; however, its space complexity is relatively low, making this algorithm potentially more advantageous in scenarios with large data volumes but low feature dimensions. When the NCA algorithm is employed to process high-dimensional data, both its time complexity and space complexity reach a high level. In practical application scenarios, owing to various factors such as the scale of specific datasets, feature dimensions, and available computing resources, appropriate algorithms should be meticulously selected to ensure the efficiency and feasibility of the algorithms.

4.2. Evaluation of Feature Selection Methods in Regression Models

4.2.1. The Role of Feature Selection in Reducing Overfitting of Regression Models

The results of this study demonstrate that combining various feature selection methods with regression models can significantly improve the predictive performance of spectral data while mitigating overfitting. In particular, the mRMR feature selection method shows exceptional effectiveness in reducing overfitting. This is likely due to the ability of the mRMR algorithm to select features that are highly relevant to the target variable while minimizing feature redundancy, thereby enhancing the model’s generalization capacity. In contrast, the CARS method, which selects a larger number of features, captures more information, but may also introduce irrelevant features, potentially increasing the risk of overfitting, particularly when predicting certain oxides. In addition, the overfitting phenomenon that occurs when CARS is applied to RF, SVR, and BP regression models might be attributed to the fact that the features selected by CARS are determined based on the PLSR model. As a result, these features may not be the most optimal for RF, SVR, and BP regression models. This further demonstrates that the reliance of feature selection algorithms on specific regression models harbors certain drawbacks. The RReliefF and NCA feature selection methods resulted in overfitting for a small number of models. In contrast, out of the 32 regression models constructed for eight oxides using the mRMR feature selection approach, only the mRMR-SVR model for TiO₂ exhibited a marginal degree of overfitting. This finding strongly endorses the efficacy of the mRMR feature selection method in mitigating overfitting in regression models.

From this point, it can be fully proved that the mRMR algorithm has significant advantages in reducing feature redundancy. This algorithm is designed to optimize the selection process by maximizing the correlation of features with the target variable (maximum relevance), while concurrently minimizing the redundancy among the features (minimum redundancy). The mRMR algorithm employs MI as a key metric, which is particularly adept at revealing nonlinear relationships that may exist between features. The comprehensive evaluation of both relevance and redundancy by the mRMR algorithm enables it to discerningly avoid the inclusion of features that, despite their correlation with the target variable, introduce excessive redundancy when combined with the existing feature set. This approach ensures that the resultant feature subset is not only simplified but also enhanced in its predictive efficacy. In comparison, the CARS algorithm, which is less focused on redundancy, is more dependent on the underlying model and the sampling methodology. The RReliefF algorithm, while robust in its feature weighting approach, does not inherently address redundancy and is susceptible to the influences of noise and the distribution of the sample data. The NCA algorithm, although it does account for redundancy in its feature selection process, does not prioritize it as a core objective, and it is associated with a higher computational complexity. Consequently, the mRMR algorithm is superior in managing feature redundancy, leading to more efficient and effective feature subsets that are better suited for predictive modeling.

4.2.2. Predictive Performance of Regression Models

We evaluated the predictive performance of four regression methods—RF, SVR, BPNN, and PLSR—in combination with various feature selection techniques. The results indicate that RF-based regression models, when coupled with different feature selection methods, generally exhibit superior predictive performance. In particular, the combination of the RF and mRMR feature selection method, denoted as mRMR-RF, stands out in the regression prediction of oxide contents. This approach is characterized by high R² values for both the training and test sets, as well as a low RMSEP for the test set, suggesting exceptional performance in both feature selection and regression modeling. SVR also demonstrates commendable performance in predicting the contents of most oxides, particularly excelling in the prediction of Al₂O₃ and K₂O.

In contrast, the BPNN regression method generally yields moderate results, often underperforming relative to RF and SVR, especially when combined with feature selection techniques for the regression of SiO₂, TiO₂, and FeO_T. In the BPNN model, the quantity of neurons constitutes a crucial factor that exerts a substantial influence on the model’s performance. A greater number of neurons will augment the count of parameters within the model, thereby endowing the model with enhanced fitting capabilities [44]. However, this augmented fitting ability may lead the model to overly capture noise and outliers present in the training data, consequently giving rise to overfitting phenomena. In LIBS, many scholars use BPNN. When the number of neurons in the input layer is set to 36-210, the number of neurons in the hidden layer is usually set to 5 or 10 [45,46,47,48]. In our experiment, 12 hidden layer neurons were chosen; nevertheless, overfitting still emerged in the CARS-BP model of SiO₂, TiO₂, Al₂O₃, MgO, and Na₂O, as well as in the full-spectrum BP model of SiO₂, TiO₂, and Al₂O₃. Specifically, the disparity in the R² between the training set and the test set exceeds 0.1. This outcome implies that the currently configured number of hidden layer neurons might still be excessive. Hence, it is viable to contemplate reducing the complexity of the model by diminishing the number of neurons, with the anticipation of alleviating the overfitting issue to a certain degree.

When the number of hidden layer neurons reaches over 1000, the expressive capacity of the neural network becomes extraordinarily potent, enabling it to fit extremely intricate data patterns. In such a scenario, the model may not necessarily rely on feature selection to enhance its performance since it possesses the ability to extract valid information from a vast array of features. For instance, Liu et al. successfully forecasted the MgO content by adopting a network structure with 4506 neuron nodes in the input layer and 1024 neuron nodes in the hidden layer [49].

Nonetheless, this does not signify that feature selection entirely forfeits its significance in neural networks with a high neuron count. For high-dimensional data, even if the neural network harbors a large number of neurons, it may still encounter the problem of data dimension catastrophe. Feature selection can effectively curtail the data dimension, thereby mitigating this issue. Even if the model performance is not conspicuously affected, feature selection remains conducive to enhancing the interpretability of the model, permitting us to discern which features are pivotal for the prediction results. Additionally, feature selection can fortify the stability of the model and diminish the fluctuations of the model induced by feature noise. In conclusion, when constructing a BP neural network model, the relationship between the number of neurons and feature selection ought to be comprehensively evaluated. The number of neurons should be rationally determined in accordance with the specific characteristics of the data and the research requirements, and the feature selection method should be duly applied to optimize the performance of the model.

PLSR tends to exhibit lower R² and higher RMSEP values in most cases, especially for TiO₂, Al₂O₃, and FeO_T, which suggests its limited applicability in these scenarios. This underperformance may be due to a mismatch between PLSR’s assumptions about the feature space and the actual structure of the data. In conclusion, the mRMR-RF combination demonstrates superior performance in predicting the content of various oxides, likely due to the robust capabilities of RF in handling high-dimensional data and the effectiveness of mRMR in selecting relevant features. These findings provide valuable insights for the selection of appropriate regression models and feature selection methods in similar tasks in future research.

This study compares the combined effects of different feature selection methods and quantitative analysis techniques, revealing the advantages of each approach and its applicability in specific scenarios. The results demonstrate that the combination of mRMR and RF exhibited significant performance advantages in the spectral analysis of multiple elements. On the other hand, the combination of NCA and SVR achieved high prediction accuracy using a reduced number of features. These findings underscore the importance of selecting appropriate feature selection and quantitative analysis methods in LIBS to enhance predictive performance. Future studies could explore the broader application of these methods to other types of spectral analysis and optimize relevant parameters to further improve predictive accuracy. Additionally, the potential benefits of combining multiple methods should be considered to develop integrated feature selection and quantitative analysis strategies, which could lead to enhanced accuracy and efficiency in LIBS spectral analysis. Such methodological advancements may contribute to broader scientific and technological progress in the field of spectral analysis.

4.2.3. Predictive Performance of the mRMR-RF Model

The mRMR-RF method demonstrated strong predictive performance in the estimation of eight oxides, with R² values exceeding 0.9000 for SiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O, indicating its ability to effectively capture the characteristic information of each oxide and enhance prediction accuracy. The R² values for TiO₂ in the test set were relatively lower, which may be attributed to the more complex chemical behavior of the oxide or the presence of sample-specific factors that affected prediction accuracy. When compared with AOMB-CNN, elastic net, and two PLS1-SM methods, mRMR-RF exhibited superior performance in most oxide predictions, particularly in terms of RMSE values. Although mRMR-RF did not yield the lowest RMSE values for Al₂O₃, its performance remained competitive, outperforming or matching the other methods, especially considering the relatively higher RMSE values of the elastic net and PLS1-SM methods. These results suggest that mRMR-RF can consistently provide high accuracy and low prediction error for complex oxide content prediction tasks. Overall, mRMR-RF demonstrated strong performance in predicting the content of the eight oxides, with its predictive accuracy significantly surpassing that of other methods in most cases. Therefore, mRMR-RF holds significant potential for application in geochemical analysis and offers a solid foundation for future research and method optimization.

5. Conclusions

In this study, four feature selection methods—mRMR, CARS, RReliefF, and NCA—were employed in combination with four regression models: RF, SVR, BP, and PLSR. The objective was to predict the content of eight oxides (SiO₂, TiO₂, Al₂O₃, FeO_T, MgO, CaO, Na₂O, and K₂O) in the ChemCam dataset. The following conclusions were drawn through comparative analysis:

Advantages of the mRMR algorithm: The mRMR algorithm demonstrated significant advantages in feature selection. It not only successfully identified features strongly correlated with the target variables but also minimized feature redundancy, thereby reducing the dimensionality of the dataset while maintaining its representativeness. Compared to CARS, RReliefF, and NCA, mRMR typically selected a moderate or low number of features, indicating its superior efficiency in feature selection.
Model performance comparison: When the mRMR algorithm was paired with the RF and SVR models, the prediction performance for oxide content was notably superior. This can be attributed to the fact that RF and SVR models are better equipped to handle nonlinear relationships and high-dimensional data. In contrast, BP and PLSR models exhibited poorer performance in terms of both prediction accuracy and stability, particularly in terms of prediction errors, as reflected by the RMSEC and RMSEP metrics.
Oxide-specific performance differences: Significant variations in predictive performance were observed across different oxides, which may be related to the distinct distribution characteristics and chemical behaviors of the oxides in the samples.

In conclusion, this study not only validates the effectiveness of the mRMR algorithm in feature extraction for LIBS data but also demonstrates the potential of feature selection algorithms in improving model performance. The findings provide valuable insights for the practical application of LIBS technology in predicting the elemental content of geological samples and offer robust methodological support for quantitative analysis in geological chemistry. Future research could explore the application of these methods to other types of samples and more complex geological settings, as well as investigate the development of new algorithms to further enhance prediction performance.

Author Contributions

Conceptualization, M.W. and M.L.; methodology, M.W.; software, F.C.; validation, F.C. and M.L.; formal analysis, M.W.; investigation, M.L.; resources, R.G.; data curation, M.W. and M.L.; writing—original draft preparation, M.W.; writing—review and editing, M.W., Y.L. and L.Y.; visualization, F.C.; supervision, F.W. and X.C.; project administration, Y.L. and L.Y.; funding acquisition, Y.L., R.G. and L.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The spectra used in this study are available to download from the NASA Planetary Data System at https://pds-geosciences.wustl.edu/msl/msl-m-chemcam-libs-4_5-rdr-v1/mslccm_1xxx/calib/ (accessed on 29 November 2024).

Acknowledgments

The authors express gratitude to all members of the Ultrafast Laser Precision Measurement and Processing Technology Group at China University of Petroleum (East China) for dedicating their time to discussions and providing valuable feedback.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

View Image - Figure 1. LIBS spectra of 408 samples, (a) 408 spectra composed of the average spectrum of each sample, (b), (c), and (d) are parts of (a), and the characteristic peaks of the elements are labeled in the figures.

Figure 1. LIBS spectra of 408 samples, (a) 408 spectra composed of the average spectrum of each sample, (b), (c), and (d) are parts of (a), and the characteristic peaks of the elements are labeled in the figures.

View Image - Figure 2. mRMR algorithm ranking of feature importance for different oxides, displaying only the top 500 wavelength variables by importance score, (a) SiO2, (b) TiO2, (c) Al2O3, (d) FeOT, (e) MgO, (f) CaO, (g) Na2O, and (h) K2O.

Figure 2. mRMR algorithm ranking of feature importance for different oxides, displaying only the top 500 wavelength variables by importance score, (a) SiO2, (b) TiO2, (c) Al2O3, (d) FeOT, (e) MgO, (f) CaO, (g) Na2O, and (h) K2O.

View Image - Figure 3. Correspondence between the top 50 features selected by mRMR and the wavelengths of LIBS spectra, where (a), (b), (c), (d), (e), (f), (g), (h) correspond to elements Si, Ti, Al, Fe, Mg, Ca, Na, and K, respectively.

Figure 3. Correspondence between the top 50 features selected by mRMR and the wavelengths of LIBS spectra, where (a), (b), (c), (d), (e), (f), (g), (h) correspond to elements Si, Ti, Al, Fe, Mg, Ca, Na, and K, respectively.

View Image - Figure 4. The influence of the number of mRMR-selected features on the predictive performance of RF, SVR, BP and PLSR, (a) SiO2, (b) TiO2, (c) Al2O3, (d) FeOT, (e) MgO, (f) CaO, (g) Na2O, and (h) K2O.

Figure 4. The influence of the number of mRMR-selected features on the predictive performance of RF, SVR, BP and PLSR, (a) SiO2, (b) TiO2, (c) Al2O3, (d) FeOT, (e) MgO, (f) CaO, (g) Na2O, and (h) K2O.

View Image - Figure 5. Prediction results for the contents of eight oxides obtained by the mRMR-RF model, (a) SiO2, (b) TiO2, (c) Al2O3, (d) FeOT, (e) MgO, (f) CaO, (g) Na2O, and (h) K2O.

Figure 5. Prediction results for the contents of eight oxides obtained by the mRMR-RF model, (a) SiO2, (b) TiO2, (c) Al2O3, (d) FeOT, (e) MgO, (f) CaO, (g) Na2O, and (h) K2O.

Table 1

Training and test set information for 8 oxides.

Oxide	Dataset	No. of Samples	No. of Spectra	Content (wt.%)
SiO₂	Training set	285	1422	0.02~98
SiO₂	Test set	122	610	2.0449~84.63
TiO₂	Training set	279	1392	0~5.81
TiO₂	Test set	119	595	0.01~3.5903
Al₂O₃	Training set	284	1417	0.01~38.79
Al₂O₃	Test set	122	610	0.1024~28.32
FeO_T	Training set	285	1423	0.02~86.31
FeO_T	Test set	122	609	0.25~50.0715
MgO	Training set	284	1417	0.01~56.14
MgO	Test set	121	605	0.04~20.3331
CaO	Training set	285	1422	0.01~56.42
CaO	Test set	122	610	0~25.18
Na₂O	Training set	280	1398	0~43.97
Na₂O	Test set	120	599	0.03~6.7779
K₂O	Training set	279	1392	0~12.05
K₂O	Test set	120	600	0~12.05

Table 2

Characteristic lines of target oxide elements corresponding to the top 50 features selected by mRMR.

Element	Characteristic Lines (nm)
Si	254.26, 263.21, 288.24, 390.66, 493.42, 505.74, 637.31, 780.22
Ti	264.74, 281.87, 283.30, 320.35, 321.52, 334.13, 334.77, 335.00, 338.13, 390.59, 416.48, 417.31, 439.52, 446.98, 451.40, 453.72, 455.68, 506.35, 590.497, 666.84, 842.67
Al	257.62, 281.70, 308.31, 390.18, 396.26, 537.33, 624.51,704.40
Fe	255.04, 256.11, 259.36, 260.02, 264.48, 266.55, 273.44, 278.89, 287.50, 291.30, 292.99, 299.35, 319.79, 321.49, 390.76, 393.14, 402.30, 410.19, 426.17, 438.48, 440.60, 495.90, 505.123, 516.37
Mg	280.35, 285.25, 293.74, 435.31, 448.24, 457.24, 517.41, 518.50
Ca	393.48, 422.79, 443.62, 445.60, 458.72, 612.39, 645.16
Na	312.61, 589.16, 818.55, 819.70
K	413.59, 438.94, 509.19, 694.07, 766.70, 770.11

Table 3

Differences in R² between the training set and the test set in regression models.

Method	SiO₂	TiO₂	Al₂O₃	FeO_T	MgO	CaO	Na₂O	K₂O
mRMR-RF	0.0508	0.0821	0.0920	0.0435	0.0211	0.0082	0.0374	0.0584
mRMR-SVR	0.0402	0.102 0	0.0548	0.0430	0.0349	0.0125	0.0358	0.0359
mRMR-BP	0.0390	0.0832	0.0587	0.0342	0.0270	0.0084	0.0224	0.0191
mRMR-PLSR	0.0650	0.0397	0.0611	0.0886	0.0636	0.0231	0.0415	0.0530
CARS-RF	0.0758	0.0876	0.1136	0.0554	0.0312	0.0141	0.1004	0.0734
CARS-SVR	0.1674	0.1025	0.0736	0.0912	0.0689	0.0208	0.0911	0.0473
CARS-BP	0.1236	0.1505	0.1027	0.0889	0.1148	0.0235	0.1002	0.0707
CARS-PLSR	0.16 20	0.2683	0.12 49	0.20 49	0.165 3	0.0743	0.154 8	0.0975
RReliefF-RF	0.0724	0.1188	0.0948	0.0624	0.0349	0.0122	0.0434	0.0665
RReliefF-SVR	0.0556	0.0310	0.0856	0.0510	0.0312	0.0122	0.0257	0.0435
RReliefF-BP	0.0574	0.0704	0.0648	0.0517	0.0273	0.0084	0.0229	0.0236
RReliefF-PLSR	0.0514	0.0715	0.0605	0.1233	0.0691	0.0079	0.0366	0.0694
NCA-RF	0.0695	0.0854	0.0937	0.0387	0.0226	0.0106	0.0552	0.0582
NCA-SVR	0.0505	0.1096	0.0812	0.0519	0.0530	0.0135	0.0483	0.0427
NCA-BP	0.0780	0.0568	0.0936	0.0421	0.0399	0.0141	0.0412	0.0490
NCA-PLSR	0.0364	0.0997	0.0852	0.1003	0.1281	0.0373	0.1006	0.0519
RF	0.0547	0.0800	0.1008	0.0438	0.0228	0.0086	0.0448	0.0654
SVR	0.4272	0.1641	0.2975	0.232 0	0.3458	0.1536	0.1794	0.1341
BP	0.1388	0.1728	0.1314	0.0793	0.0869	0.0308	0.0675	0.0554
PLSR	0.0905	0.194 5	0.120 6	0.1316	0.0973	0.0504	0.1029	0.0714

Table 4

The number of features extracted by four feature selection methods in the quantitative analysis models of eight oxides.

Method	SiO₂	TiO₂	Al₂O₃	FeO_T	MgO	CaO	Na₂O	K₂O
mRMR	332	370	224	194	284	249	257	332
CARS	466	525	217	412	473	352	478	506
RReliefF	269	335	348	299	345	299	273	463
NCA	130	246	224	174	326	108	155	415

Table 5

Models with excellent predictive performance of eight oxides.

Oxide	Method	Number	R²_C	R²_P	RMSEC	RMSEP
SiO₂	mRMR-SVR	230	0.9912	0.9510	1.7832	2.2465
SiO₂	mRMR-RF	492	0.9953	0.9445	1.3110	2.3903
TiO₂	RF	5484	0.9810	0.9010	0.1189	0.2181
TiO₂	mRMR-RF	386	0.9806	0.8985	0.1202	0.2208
Al₂O₃	mRMR-SVR	164	0.9827	0.9279	0.9795	1.3068
Al₂O₃	CARS-SVR	217	0.9889	0.9153	0.7868	1.4166
FeO_T	NCA-RF	56	0.9912	0.9525	0.8151	1.0883
FeO_T	mRMR-SVR	200	0.9915	0.9485	0.7991	1.1332
MgO	mRMR-RF	410	0.9980	0.9769	0.3475	0.4317
MgO	NCA-RF	286	0.9981	0.9755	0.3453	0.4437
CaO	mRMR-RF	492	0.9985	0.9903	0.3760	0.4356
CaO	RF	5484	0.9985	0.9899	0.3745	0.4439
Na₂O	RReliefF-BP	254	0.9880	0.9651	0.3622	0.3126
Na₂O	mRMR-BP	326	0.9874	0.9650	0.3701	0.3130
K₂O	mRMR-SVR	488	0.9958	0.9599	0.1244	0.3613
K₂O	NCA-SVR	247	0.9960	0.9533	0.1226	0.3900

Table 6

Comparison the RMSEPs of different methods.

Oxide	mRMR-RF	AOMB-CNN [38]	Elastic Net [39]	PLS1-SM [21]	PLS1-SM [40]
SiO₂	2.39	2.45	4.32	4.33	4.62
TiO₂	0.22	0.35	0.42	0.94	0.46
Al₂O₃	1.52	1.25	1.94	2.85	2.26
FeO_T	1.14	1.39	2.78	2.01	2.21
MgO	0.43	0.90	1.17	1.06	1.19
CaO	0.44	0.89	1.29	2.65	1.89
Na₂O	0.35	0.67	0.54	0.62	0.57
K₂O	0.46	0.55	0.66	0.72	0.72

Table 7

Computational complexity evaluation.

Algorithm	Time Complexity	Space Complexity	Remarks
mRMR	$O (n^{2} m)$ ¹	$O (n m)$	The time complexity is dominated by the calculation of mutual information and is affected by factors such as data sparsity; the space complexity depends on the storage of the feature matrix and intermediate results related to mutual information.
CARS	$O (k n t)$ ²	$O (n m + p)$ ³	It is more time-consuming when handling large datasets with a large number of samples, and it requires additional storage for model parameters.
RReliefF	$O (n^{2} m)$	$O (m)$	It has a similar time complexity order of magnitude to mRMR, with simple calculations. In some cases, it may be faster than mRMR due to data distribution. It only stores feature weights and a small amount of intermediate results, making it more memory-efficient.
NCA	$O (n^{2} d)$ ⁴	$O (n^{2})$	In high-dimensional data, the time complexity is high. The space is mainly used for storing the distance matrix. It may have advantages when the feature dimension is low, and the neighborhood structure is important.

¹ $n$ is the number of samples, and $m$ is the number of features. ² $k$ is the number of iterations, and $t$ is the time per iteration. ³ $p$ is the number of model parameters. ⁴ $d$ is the feature dimension.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/rs17030416/s1, Figure S1: The change in prediction performance of BPNN under different number of hidden layer neurons; Figure S2: Optimization of the number of latent variables in PLSR models, (a) SiO₂, (b) TiO₂, (c) Al₂O₃, (d) FeO_T, (e) MgO, (f) CaO, (g) Na₂O, and (h) K₂O; Figure S3: The influence of the number of RReliefF-selected features on the predictive performance of RF, SVR, BP and PLSR, (a) SiO₂, (b) TiO₂, (c) Al₂O₃, (d) FeO_T, (e) MgO, (f) CaO, (g) Na₂O, and (h) K₂O; Figure S4: The influence of the number of NCA-selected features on the predictive performance of RF, SVR, BP and PLSR, (a) SiO₂, (b) TiO₂, (c) Al₂O₃, (d) FeO_T, (e) MgO, (f) CaO, (g) Na₂O, and (h) K₂O; Figure S5: Feature selection process of CARS. As the number of sampling runs increases, plot (i) the trend of the number of sampled variables. The points highlighted in plot (i) indicate the number of variables selected when the fivefold RMSECV value reaches its minimum; plot (ii) the fivefold RMSECV values, and plot (iii) the regression coefficients of each variable. The asterisk line marks the point where the optimal fivefold RMSECV value reaches its minimum. (a) SiO₂, (b) TiO₂, (c) Al₂O₃, (d) FeO_T, (e) MgO, (f) CaO, (g) Na₂O, and (h) K₂O; Table S1: Evaluation of spectral data regression prediction performance based on different feature selection methods and regression model combinations; Table S2: Prediction results of four feature selection methods for the contents of eight oxides, using RF regression with the same number of features; Table S3: The thirty manually selected characteristic lines of Ca; Table S4: Comparison of regression model results between mRMR-selected features and manually selected features of Ca.

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Abstract

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Laser-induced breakdown spectroscopy (LIBS) is a rapid, non-contact analytical technique that is widely applied in various fields. However, the high dimensionality and information redundancy of LIBS spectral data present challenges for effective model development. This study aims to assess the effectiveness of the minimum redundancy and maximum relevance (mRMR) method for feature selection in LIBS spectral data and to explore its adaptability across different predictive modeling approaches. Using the ChemCam LIBS dataset, we constructed predictive models with four quantitative methods: random forest (RF), support vector regression (SVR), back propagation neural network (BPNN), and partial least squares regression (PLSR). We compared the performance of mRMR-based feature selection with that of full-spectrum data and three other feature selection methods: competitive adaptive re-weighted sampling (CARS), Regressional ReliefF (RReliefF), and neighborhood component analysis (NCA). Our results demonstrate that the mRMR method significantly reduces the number of selected features while improving model performance. This study validates the effectiveness of the mRMR algorithm for LIBS feature extraction and highlights the potential of feature selection techniques to enhance predictive accuracy. The findings provide a valuable strategy for feature selection in LIBS data analysis and offer significant implications for the practical application of LIBS in predicting elemental content in geological samples.

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Title

Enhancing Laser-Induced Breakdown Spectroscopy Quantification Through Minimum Redundancy and Maximum Relevance-Based Feature Selection

Author

Wang, Manping

; Lu, Yang; Liu, Man; Cui, Fuhui; Gao, Rongke

; Wang, Feifei; Chen, Xiaozhe; Yu, Liandong

First page

416

Publication year

2025

Publication date