1. Introduction

Laser-Induced Breakdown Spectroscopy (LIBS) has emerged as a popular elemental analysis technique due to its speed and simplicity [1]. It enables real-time, online detection of elements in solids, liquids, and gases without the need for complex sample pretreatment [2]. As a result, LIBS finds extensive applications in diverse fields such as aerospace [3,4], environmental monitoring [5–7], mineral extraction [8,9], metallurgical analysis [10], and atmospheric monitoring [11]. By employing high-energy laser pulses to ablate the sample surface and generate plasma, LIBS facilitates qualitative and quantitative analysis of sample composition through the collection and processing of emitted spectral signals [12]. The accuracy of classification in LIBS heavily relies on the number of samples and the information extracted from the collected spectra. In general, a larger and more representative sample dataset enhances model classification accuracy and improves the generalization capabilities of established models. Adequate LIBS spectral line information provides crucial insights into sample characteristics. However, the challenges associated with large-scale sample data and high-dimensional spectral information contribute to data redundancy. As the number of collected samples increases, the exponential growth in required computational resources and time poses difficulties in model building for researchers. While existing computational resources enable researchers to effectively handle high-dimensional data and computers can process datasets with large sample sizes, challenges arise when simultaneously dealing with large-scale sample sizes and high-dimensional data in classification tasks. These challenges include the following. (1) Enormous computational burden during the training and classification stages. (2) Substantial storage requirements to accommodate a large volume of training data. (3) Difficulty in determining decision rules within high-dimensional data. In data-driven learning tasks, larger sample sizes often lead to more accurate and realistic prediction results. In the case of learning tasks based on LIBS, to enhance prediction accuracy, richer sample data is often required, which further exacerbates the challenges associated with large-scale and high-dimensional data.Furthermore, LIBS spectral lines are susceptible to chemical and physical matrix effects and backgrounds, resulting in spectral signal uncertainty [13]. In many cases, obtaining sufficient sample data for certain detection objects is particularly challenging. Therefore, it becomes crucial to extract effective information from the original spectra to develop prediction models with enhanced reliability. Addressing this challenge is imperative for advancing LIBS research and applications.

With the advancement of LIBS technology, many researchers have started using chemometric methods such as partial least squares (PLS) and principal component analysis (PCA) to extract the main features of spectra and reduce noise and redundant information. These methods are used for simultaneous quantitative and qualitative analysis of various samples, including metals [14], rocks [15], soils [16], drugs [17], and biological [18] samples. To obtain more spectral features and establish nonlinear or complex regression relationships to improve the sensitivity and stability of LIBS analysis, more and more scholars have combined different chemometric methods with PLS and PCA. Cao et al. [19] combined PCA with k-nearest neighbors (KNN) to investigate the influence of different numbers of principal components on the classification accuracy of KNN in coal qualitative analysis. Yang et al. [20] compared the qualitative analysis results of iron ore using spectra after PCA dimensionality reduction combined with three machine learning models: KNN, artificial neural network (ANN), and support vector machine (SVM). They explored methods to achieve rapid and accurate classification of iron ore. Rao et al. [21]utilized PCA to select the wavelength range with the maximum principal component loadings and trained machine learning models (ensemble regression, support vector machine regression, gaussian kernel regression, and artificial neural network) in a simplified feature space. They extensively studied the necessity of using various machine learning models for quantitative analysis in spectra of lanthanide and actinide elements. Furthermore, Rao et al. [22]conducted a comprehensive review of multivariate regression techniques in various spectra, extensively discussing how to use PCA and PLS for multivariate analysis and combining them with various machine learning models in nuclear science research. Huang et al. [23] described the various challenges encountered in soil analysis using LIBS technology, with a focus on the importance of using machine learning models for LIBS data analysis. They provided detailed explanations of machine learning methods for quantitative/qualitative analysis using spectral feature extraction and feature selection.

The methods for quantitative and qualitative analysis in LIBS are continuously being updated, and the detection accuracy is becoming more precise. Data-driven learning tasks tend to make predictions that approach real situations as the sample size increases. However, research on situations with limited or insufficient sample sizes is rarely mentioned. In the case of small sample data, the limited number of samples reduces the computational burden, but it also hampers the acquisition of valid information due to the diminished representation of the overall dataset. Data-driven models that rely on excessive feature information often encounter the risk of overfitting, while the intricate nature of LIBS spectral data, combined with the small sample size in learning tasks, exacerbates the problem of model underfitting. Consequently, achieving accurate results with a limited amount of data remains a challenging issue in the context of LIBS. To address this challenge, Li et al. [24] employed a resampling technique, repeatedly extracting data randomly from the original spectra, in order to construct an extensive training set and mitigate the risk of overfitting when the sample size is less than 20. They further integrated Principal Component Analysis-Partial Least Squares (PCA-PLS) to develop a model for precise quantitative analysis of coal. Additionally, Li et al. [25] employed a statistical correction strategy to select effective spectra, compensating for the unreliability inherent in small sample estimation. They extracted features related to emission peak intensity and emission shape to reduce the influence of noise and enhance the informational content of the spectra. By combining these enhanced features with PCA-PLS, they achieved accurate determination of the blast heat of energetic materials. Therefore, investigating the extraction of effective features from a limited amount of LIBS sample data and developing a robust model warrant further exploration.

In our study, we obtained aluminum alloy spectral data using femtosecond LIBS and analyzed this small sample spectral data utilizing a three-layer stacked machine learning model. The application of femtosecond laser offers several advantages, including high power density, precision, a small heat-affected zone, absence of thermal damage, and enhanced safety [26]. The shorter interaction time with the sample diminishes the formation of non-excited states and minimizes the thermal diffusion effect [27]. Compared to the long-duration continuous pulses of nanosecond lasers, femtosecond lasers have an extremely short interaction time with the sample. During the ablation process, there is no laser-plasma interaction (plasma shielding) and negligible heat transfer from the laser interaction zone to the surrounding lattice leading to improvement [28]. Moreover, the very short pulse width of the femtosecond laser results in a remarkably high peak power and energy density, thereby inducing nonlinear effects in the sample and enhancing the spectral signal [29]. The capability of femtosecond LIBS to simultaneously detect multiple elements has spurred its development and application in analyzing complex multi-element samples.

Overall, by utilizing a stacked machine learning approach, we aim to address the challenges associated with small sample sizes in LIBS analysis and achieve highly accurate qualitative analysis of aluminum alloys. The proposed model employs a stacked architecture to address the challenges associated with learning from small-sample LIBS data. Its primary objective is to extract effective features from aluminum alloy spectral data at different levels, facilitating the development of a robust qualitative analysis model while preserving the interpretability of spectral features. To overcome the limitations posed by small sample sizes in LIBS data, the first layer of the stacked model utilizes the random forest resampling mechanism to obtain reliable sample features. Additionally, an automatic peak search algorithm is employed to select independent spectral peaks, while the Voigt function is employed to fit and calculate the full-width at halfmax (FWHM) of the selected peaks. In the second layer, three heterogeneous learners, namely KNN, extreme gradient boosting (XGBoost), and SVM, are integrated to capture the diversity and complexity of features within different feature spaces of the samples. Cross-validation is applied within each base learner, further enhancing the reliability of the prediction results. The outputs of the three learners, along with the output of the first layer, form a new dataset that serves as the input for the third layer. In this layer, logistic regression is employed as the classifier to generate category probabilities for the final prediction.

Furthermore, in this study, a comparative analysis of the experimental results was conducted using various classification algorithms, including RF, SVM, and KNN, both individually and in combination with PCA. Performance metrics such as accuracy, precision, recall, and F-1 score were utilized for evaluating and comparing the classification performance of these models. Additionally, the impact of the spectral line spreading mechanism on the model was thoroughly investigated. Subsequently, the adequacy of the number of spectra required for small-sample learning based on the stacked model was examined and validated. The proposed model demonstrated effective extraction and utilization of feature information from spectra, thereby presenting a viable approach for model construction in scenarios with limited sample sizes.

2. Experimental setup and methods

2.1 Experimental setup and sample configuration

The experimental setup for LIBS is illustrated in Fig. 1. The setup comprises various components, including a femtosecond laser system (Libra, Coherent, USA), an energy attenuation system, a spectrometer (Mechelle 5000, Andor), a pulse delay trigger, a 3D translation stage, optical lenses, and a computer, as depicted in Fig. 1. The femtosecond laser operates at a wavelength of 800 nm with a pulse width of 50 fs, a repetition rate of 1.0 kHz, and an energy stability of 0.5%. Throughout the experiment, the laser energy was maintained at 1.8 mJ. The femtosecond laser-induced plasma emission spectra were collected and coupled into a fiber probe (core diameter 200 µm) of an intermediate dispersive grating spectrometer equipped with an intensified charge-coupled device (ICCD) detector (1024 × 1024 pixels, DH334T) using a fused silica lens L2 (f = 75 mm). The spectrometer possessed a acquisition wavelength range of 200-975 nm, with a precision of 0.05 nm and a resolution of λ/Δλ=5000. The ICCD detector exhibited a gate width of 5 µs and a delay time of 4 µs. To prevent excessive ablation of the target sample, the aluminum alloy samples were mounted on an XYZ-3D translation stage, ensuring that each laser pulse acted on a fresh location on the target's surface. Additionally, the aluminum alloy sample's surface was positioned 1 mm in front of the focal point to minimize air plasma interference in the spectral analysis. In order to mitigate random errors in spectral detection, 50 laser pulses were accumulated for signal averaging, and 20 data sets were collected for each sample. This study focused on the LIBS classification analysis of five Al-Cu-Mg-Fe-Ni aluminum alloys (1060, 6061, 5052, 2024, 7075). To ensure the stability and reproducibility of the spectral data, the spectral data of each type of aluminum alloy samples were obtained on 1 block of samples, therefore, 5 samples in total (each containing 1 sample). Table 1 presents the precise concentrations of Cu, Mg, Fe, Ni, Mn, Si, Zn, and Ti elements in these five samples. Figure 2 illustrates the femtosecond LIBS spectra obtained for the aforementioned alloys.

 

Fig. 1. Experimental Setup

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Table 1. Elemental Composition of Samples

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Fig. 2. Schematic Diagram of Sample Spectra

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2.2 Algorithm description

The initial step in modeling the processing of LIBS data typically involves dimensionality reduction of the high-dimensional spectral intensity data. This step addresses the issue of feature redundancy and aims to identify an optimal coordinate space for selecting suitable inputs for subsequent modeling. While PCA is a classical and commonly employed technique for dimensionality reduction in LIBS data processing, it has limitations in terms of feature interpretability. Although PCA effectively reduces the dimensionality of data while retaining the maximum amount of target information through feature extraction, its features lack interpretability [30]. In the modeling process of LIBS, there is an increasing focus on the interpretability of features and the physical characteristics of spectral lines. Therefore, feature selection is adopted as the foundation for model construction in this study. Specifically, a three-layer stacking model is employed for data analysis and processing, as depicted in Fig. 3.

 

Fig. 3. Model Framework. Five-fold cross-validation is used for each base model in the second layer.

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In Level 1, illustrated in Fig. 3, the classification task encounters the challenge of a sparse number of samples, necessitating the comprehensive utilization of available data as a key objective. Therefore, the initial task of the model is to extract spectral features from the original data that effectively capture the characteristics of the samples and mitigate the risk of underfitting arising from the small sample size and data complexity inherent in the modeling process.

The core concept of Bootstrap sampling is to construct reliable estimated confidence intervals by employing resampling techniques when the complete set of samples is unknown [31]. In light of this, the first layer of our model is designed with two primary objectives: (1) to identify spectral features that possess significant characterization power for the target samples through Bootstrap sampling, and (2) to extract relevant spectral information to enhance the feature representation of the spectral data. To achieve the first objective, we employ the random forest (RF) algorithm, which combines Bootstrap sampling with decision trees. Each decision tree is constructed using a randomly selected subset of samples and features, enabling the identification of spectral features with superior characterization capability based on the principle of informativeness. Filtering samples and features using Random Forest involves the following five steps: (1) Data preparation. (2) Building the Random Forest model. (3) Feature importance evaluation. (4) Sample filtering. Based on the predictions of the Random Forest model for each sample, filter the samples according to specific criteria. (5) Feature filtering. Using the feature importance scores obtained in step 3, select and retain the features with higher importance. By employing Random Forest for sample and feature filtering, we aim to enhance the predictive performance of the model and reduce the dimensionality of the dataset. This approach allows us to focus on the most relevant features and optimize the model's performance. Regarding the second objective, it is important to note that most studies treat spectral line intensities in LIBS spectra as discrete features. However, this approach often overlooks the fact that spectral lines represent continuous variations. Consequently, the continuous spectral profile can serve as a valuable descriptor for the spectrum. One important aspect of LIBS spectra that can capture the shape of spectral peaks is the spectral line spread, typically quantified by the full-width at half maximum. By considering the FWHM as a characterization of the spectral line broadening [32], we can effectively describe the LIBS spectrum in terms of continuous variation. In order to leverage the broadening features present in LIBS spectra, we performed Voigt function fitting on the selected spectral peaks to calculate their full-width at half maximum. By incorporating the prior knowledge of FWHM for the spectral lines, we enriched the feature space of LIBS spectra by considering both spectral line intensity and line broadening as spectral features, thereby enhancing the dimensionality of the LIBS spectral data. Therefore, in the first layer, we select spectral lines with high characterization ability using random forest and combine them with the corresponding FWHM values. This ensures that the first layer fulfills the construction requirements, and the combined features are subsequently used as input for the second layer. Our expectation is that the learners in the second layer will be capable of superior learning performance by considering both spectral line intensity and spectral line spreading dimensions. The output of the first layer can be expressed as follows:

In the second layer, depicted in level 2 of Fig. 3, three heterogeneous learners are employed as the models, with each learner referred to as a base learner. The selection of heterogeneous learners allows for the transformation of the original data into different feature spaces, thereby harnessing the diverse characterization capabilities of each feature space to enhance prediction accuracy. Additionally, the heterogeneous learners employ distinct feature capture criteria for the same input, enabling the acquisition of data diversity, which effectively addresses the issue of underfitting caused by limited sample data in small sample learning scenarios. Given the emphasis on maximizing data utilization in this layer, cross-validation is employed within each base learner. This approach aims to mitigate the potential unreliability introduced by adding features to the small sample data and to minimize the risk of overfitting. By leveraging cross-validation, the model's performance can be further refined, compensating for the inherent limitations of small sample sizes and enhancing the reliability of the prediction results. Thus, the second layer not only leverages the distinct feature spaces offered by heterogeneous learners but also employs cross-validation techniques within each base learner, promoting the exploration of data diversity and addressing the challenges associated with small sample learning scenarios. Within the base learners, three distinct algorithms are employed: KNN, SVM, and XGBoost. Each algorithm possesses unique characteristics and contributes to the overall model's predictive capabilities.

KNN [33] is a non-parametric algorithm that classifies data points based on the distances between them. It selects the nearest neighbor nodes in the feature space for classification, allowing for local patterns to influence the predictions. By considering the proximity of data points, KNN can effectively capture the underlying relationships within the data.

SVM [34], on the other hand, is a powerful classifier that aims to find the optimal hyperplane to separate different classes based on a kernel function. It leverages the mapped feature space created by the kernel function to transform the data into a higher-dimensional space where the classes can be effectively separated. SVM's ability to handle complex decision boundaries makes it a robust classifier for diverse datasets.

In contrast, XGBoost [35] is a tree-based learner that operates within a gradient boosting framework. It builds an ensemble of decision trees to make predictions for each category, with each subsequent tree trained on the residuals of the previous tree. This iterative process allows XGBoost to continuously reduce prediction errors and enhance the overall model's performance.

By incorporating KNN, SVM, and XGBoost as base learners in the second layer, the stacked model benefits from the unique strengths of each algorithm, such as KNN's local pattern recognition, SVM's effective separation of classes, and XGBoost's iterative refinement of predictions. These algorithms collectively contribute to the enhanced learning of the spectral data and the improved performance of the model.

Cross-validation is a crucial technique in model evaluation, ensuring that the number of samples in each fold is greater than the number of categories to achieve reliable results. To assess the model's predictive ability for small sample sizes, the number of folds in cross-validation is adjusted based on the available training spectra. In this study, we have employed five-fold cross-validation for the base learners in the second layer, as indicated by “Five-fold cross-validation” in Fig. 3. Five-fold cross-validation is chosen because it strikes a balance between accurately evaluating the model's performance and efficiently utilizing the available data. The five-fold cross-validation procedure divides the training set into five subsets, with each subset used to train different base models. These trained base models are subsequently applied to the validation set to obtain prediction results, denoted as P_ij, where i (1, 2, 3) represents the base model, and j (1, 2, 3, 4, 5), represents the fold number. The prediction results of the validation set are essential for determining the output of the second layer, as defined by a specific equation. This rigorous approach ensures robust evaluation of the second layer's predictive performance in comparison with other models in terms of classification ability.

The predictions of the validation set generated by all the base learners are combined to form a new dataset, which serves as input alongside the original dataset for the second layer. This combined input, consisting of the original dataset and the new set of predictions, is then utilized as the input for the third layer. By incorporating the predictions from the base learners along with the original dataset, the third layer of the model can make use of the enriched information to further refine the classification process. This hierarchical approach ensures that the model benefits from the collective knowledge and diversity captured by the base learners in the second layer, leading to improved classification accuracy and robustness in the final predictions.

Layer 3: Following the learning process by the three heterogeneous classifiers in the second layer, the diverse and complex nature of the data is effectively captured within each heterogeneous feature space. In the input of the third layer, we incorporate the collected spectral line information, the descriptors of selected spectral peak widths, and the predictions from different machine learning models. These three types of information collectively encompass the discrete wavelength features, continuous spectral line features, and machine learning features. Therefore, the primary focus of layer 3 lies in reorganizing the features acquired from the first two layers to enhance the accuracy of the prediction outcomes. In this layer, the central concern is mitigating the risk of overfitting while leveraging the rich feature set obtained from the input of the preceding layers. This is achieved through two key approaches: feature reduction and regularization. Consequently, L2 regularized logistic regression is selected as the final classifier. By incorporating L2 regularization, logistic regression exhibits robust generalization performance, particularly in scenarios involving high-dimensional data. Moreover, logistic regression aligns with the task requirements, as it enables decision-making based on probabilities when making predictions regarding sample categories.

3. Results and discussion

In the present learning task, the available sample data is often obtained through multiple experimental iterations, aiming to augment the dataset. When the data is sparse, the collected samples may exhibit different characteristics within the corresponding feature space. These characteristics can include a more uniform distribution, a concentration with distinct classification boundaries, or a complex distribution with multiple overlapping boundaries. The uncertainty surrounding the sample distribution poses a significant challenge in small sample learning, leading to unreliable prediction results. Thus, the objective of this study is to identify optimal spectral features that effectively characterize the spectra derived from the collected data, thereby improving the accuracy of prediction outcomes. To gain insights into the approximate data distribution, PCA is initially employed to visualize the LIBS spectra, as depicted in Fig. 4. In the figure, the gray area represents the spatial visualization result, while the blue, pink, and green areas denote the projections of the spatial visualization result onto three coordinate planes. The visualization highlights the complexity of the spatial distribution for the five types of aluminum alloy samples. Notably, the 1060 (purple) and 5052 (green) aluminum alloys exhibit more distinct classification boundaries compared to the other three types. Conversely, the 2024 (blue), 6061 (orange), and 7075 (red) aluminum alloys are clustered together in space, making it challenging to achieve accurate classification solely based on boundary observation. Therefore, there is a need to explore higher-dimensional spaces to effectively differentiate these three samples.

 

Fig. 4. (a) illustrates the loadings of the first three principal components, with markings indicating the locations of several selected spectral peaks used as features. (b) displays the PCA score plots of LIBS data from five different aluminum alloy samples, showing the first three principal components. Spatial Projections on PC 1 and PC 2 planes (projection1), PC 1 and PC 3 planes (projection2), and PC 2 and PC 3 planes (projection3).

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3.1 Comparative classification results

The dataset was divided into a training set comprising 70% of the samples and a test set comprising the remaining 30%. The comparative analysis utilized three machine learning models, namely KNN, SVM, and RF, which have demonstrated good performance in LIBS classification. The reason for selecting three models (KNN, SVM, RF) is as follows: KNN is a non-parametric algorithm that classifies data points based on their distances, effectively capturing underlying relationships within the data. SVM with the RBF kernel can effectively handle non-linear relationships in spectral data and achieve good classification performance with limited training samples, known for its strong generalization ability. RF can capture valuable spectral information through feature selection, reducing the risk of underfitting caused by limited data and dataset complexity. The learning was conducted separately on the original spectral data and the reconstructed spectra after dimensionality reduction. PCA, a commonly used technique in LIBS spectral data processing, was chosen as the dimensionality reduction method. Specifically, the first ten principal components were selected as inputs for the reduced-dimensional model. The effectiveness of the models was evaluated based on metrics such as accuracy, precision, recall, and F-1 score. Grid search was conducted on all selected classification models to explore different values of their tunable hyperparameters. The models were evaluated using five-fold cross-validation, and the highest classification accuracy was recorded as the optimal hyperparameter for each model. Table S1 displays all the classification models, the range of tunable hyperparameters, and the best hyperparameter values determined for each model. Figure 5 presents the classification results. Among the models based on the entire spectrum, the SVM achieved the best performance. Compared to RF and KNN, SVM exhibits unique advantages in handling high-dimensional sparse data. Additionally, SVM employs an unbiased estimate for generalization error, indicating strong generalization capability. Unlike KNN, which is sensitive to noise and outliers, SVM can handle high-dimensional data without the need for feature selection. Compared to SVM and KNN, RF demonstrated improved classification accuracy after spectral reconstruction using PCA. The effectiveness of PCA-RF surpassed that of RF, resulting in a notable enhancement in classification performance.

 

Fig. 5. Comparison of three type of models (KNN, SVM, and RF) and their achieving dimensionality reduction using PCA (PCA-KNN, PCA-SVM, and PCA-RF).

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The category prediction results of each type are presented in Fig. 6. As depicted in the graph, the 6061 aluminum alloy demonstrates superior discriminative performance among various classifiers, as indicated in Table 1. Notably, the 6061 alloy exhibits a more distinct gradient in the content of different metallic elements compared to the other four alloys. Therefore, among the compared classifiers, the 6061 alloy consistently achieved excellent discriminative capability. Only RF (Fig. 6(e)) and PCA-reduced RF (Fig. 6(f)) exhibited errors in predicting the 6061 alloy samples, where they misclassified 16.7% (5 out of 30) of the test spectra as 7075 alloy and 16.7% of the spectra as 2024 alloy. On the other hand, the distributions of the 6061, 7075, and 2024 types, as depicted in Fig. 4, presented challenges in identifying clear decision boundaries. Consequently, these three categories frequently experienced misclassification among various types. The models that obtained the highest error rates in the classification were 2024 and 7075. Considering Fig. 4, the spatial distribution and distances between 2024, 7075, and 6061 were consistently small. Only after applying PCA for dimensionality reduction, RF and SVM achieved the best performance in predicting the 2024 alloy. This can be attributed to the fact that PCA retains the essential information of the spectral data.

 

Fig. 6. Confusion matrix for each model. a, b for KNN and PCA-KNN, c, d for SVM and PCA-SVM, e, f for RF and PCA-RF. The confusion matrix illustrates the performance of the model in correctly classifying samples into their respective categories. The diagonal elements represent the proportion of correctly classified samples in each class, while the off-diagonal elements represent the proportion of misclassified samples.

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Furthermore, the resampling mechanism of RF and the information extraction principles based on Gini impurity in decision trees allowed RF to effectively avoid overfitting during training on the training set. Given the matrix effect and self-absorption effect present in LIBS spectra, the collected spectra often contain noise and overlapping data from different samples within the same spectral band. In contrast to SVM's excellent handling of high-dimensional data and random forest's feature selection principles based on information gain, KNN are more sensitive to noise and data overlap when confronted with high-dimensional data. Therefore, through the PCA-based dimensionality reduction and subsequent spectral reconstruction, the reconstructed spectra may suffer from issues like multicollinearity or noise interference, which can lead to a decrease in the classification performance of KNN. Although PCA can effectively extract features, the dimensionality-reduced data actually loses the original structure (as the principal components are linear combinations of the original variables). This may result in the loss of some useful information for classification, causing differences between the SVM's decision planes in the principal component space and the original data space, thereby reducing classification accuracy. In summary, the comparative results indicate that different classifiers exhibit unique sensitivities to the same data. Each classifier has its advantages in distinguishing various aluminum alloy models. However, achieving accurate predictions remains challenging, especially in the case of the 7075 alloy, which poses a difficult problem in model construction.

3.2 Impact of spectral broadening features

The previous discussion has demonstrated the effectiveness of PCA-based dimensionality reduction in processing LIBS data and enhancing the classification accuracy of classifiers. However, one limitation of PCA is that the resulting features obtained after dimensionality reduction lack a clear physical interpretation. Furthermore, the computation of high-dimensional information from LIBS spectra imposes a significant computational burden and is susceptible to noise interference. Consequently, exploring alternative approaches for learning from the full spectrum becomes challenging, necessitating the use of dimensionality reduction techniques. To address this issue, some researchers have proposed integrating machine vision technology into LIBS classification. For instance, Jiujiang Yan et al. [36] have explored the utilization of image processing techniques by mapping LIBS data into visual images. By applying image processing methods, such as gradient histograms, these researchers indirectly extract classification features from the multidimensional spectral intensities in the form of pictures. This approach offers a novel way to obtain dimensional features using machine vision technology in LIBS classification. By adopting machine vision techniques, researchers aim to overcome the limitations associated with traditional feature extraction methods, such as PCA, and leverage the inherent strengths of image processing to capture informative patterns and structures within LIBS data. This not only enhances the interpretability of the extracted features but also addresses computational challenges and noise interference inherent in high-dimensional LIBS spectra. Incorporating machine vision-based dimensional features into LIBS classification opens new avenues for improving the accuracy and understanding of the classification process.

The phenomenon of spectral line broadening [32] is pervasive in various spectra, and it closely resembles the shape of spectral peaks. When the physical parameters are held constant, the peak width in LIBS spectra is primarily influenced by the characteristics of the sample. In the context of aluminum alloy LIBS spectral data, variations in elemental content lead to different peak widths in the corresponding spectra. This is because different elemental contents can affect factors such as the breakdown threshold of the sample, plasma temperature and density, and self-absorption effects. These factors, in turn, cause variations in the width and intensity of the spectral peaks. By maintaining the consistency of physical parameters, we can better focus on the impact of the sample's inherent characteristics on the peak width. By comparing the spectral data of different samples, we can study the differences in peak width among them and further understand the chemical composition, structure, or other features of the samples. And a rich feature vector is available, and the objective of increasing the feature dimension is not to burden the model with redundant features. FWHM is a commonly used parameter in spectroscopy to characterize peak width, representing significant aspects such as peak resolution and accuracy. LIBS spectral peaks can generally be categorized into two types, primarily influenced by Stark spreading and Doppler spreading. Doppler broadening arises from the thermal motion of emitting atoms or ions, resulting in Gaussian lineshapes [37]. The full-width at halfmax of a Gaussian line is proportional to the square root of the temperature and the central frequency. Thus, higher temperatures or frequencies lead to larger full-width at halfmax due to Doppler broadening. On the other hand, Stark broadening stems from the broadening of spectral lines in atomic energy levels under the influence of strong electric fields. As a form of pressure broadening, Stark broadening manifests as a Lorentzian line pattern [37]. The full-width at halfmax of Stark broadening depends on the electron density and the strength of the electric field. Consequently, higher electron densities or electric field strengths result in larger full-width at halfmax due to Stark broadening. To comprehensively describe the line shape influenced by both Stark and Doppler broadening, the Voigt function is utilized. It represents the convolution of Gaussian and Lorentzian functions, effectively capturing the spectral line shape resulting from the combined effects of Stark and Doppler broadening. By employing the Voigt function, spectral lines affected by the self-absorption effect can be more accurately approximated in terms of intensity compared to the theoretical line shape. By leveraging the information provided by spectral line broadening, the physical characteristics of aluminum alloy LIBS spectra can be better understood and harnessed. This offers potential insights into the nature of spectral peaks and enables the utilization of spectral line broadening as a valuable feature in classification and analysis tasks.

Therefore, after passing the peak-seeking algorithm, several spectral peaks that are more independent and less disturbed are selected for fitting and their full-width at halfmax are calculated, and the fitted spectral peaks are shown in Table 2 and the Voigt fit is shown in Fig. 7.

Table 2. Selected Spectral Peaks

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Fig. 7. Schematic Illustration of Voigt Fitting. Peak Fitting of the MgI285.21 nm Spectral Line.

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In order to investigate the impact of the spreading mechanism on the classification model, the RF algorithm, which exhibited superior performance in the comparative analysis, was selected. The objective was to assess whether the inclusion of spreading features would influence the learning process using the full spectrum and the reconstructed spectrum after dimensionality reduction. For the RF algorithm, which demonstrated better performance among the compared classification models, the introduction of FWHM features through PCA-RF was evaluated. The corresponding results are presented in the Table 3.

Table 3. Effect of broading mechanism on classification accuracy of random forest

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From the table, it can be observed that the inclusion of the spreading mechanism in the random forest algorithm leads to an improvement in the classification accuracy. For instance, in Fig. 6(e), the classification accuracy for the 2024 types increased from 83.3% to 100% when incorporating the spreading features. Similarly, after applying PCA dimensionality reduction, the introduction of the spreading mechanism in the random forest model resulted in an improved prediction accuracy for the 7075 type, increasing it from 83.3% to 100%. These findings suggest that integrating spreading features enhances the predictive capability of the random forest algorithm, leading to improved accuracy in classifying aluminum alloy samples.

At this stage, the selected top ten features with the highest feature importance scores in the random forest incorporating FWHM were visualized, as shown in Fig. 8(a). It can be observed that in the traditional random forest model, which is based on the full spectrum with numerous dimensions (27398 dimensions), the FWHM of Mg I 285.21 nm and Cu I 324.75 nm have entered the top ten in terms of feature importance. This indicates that the introduced spreading features effectively enhance the descriptive information in the spectral data. Moreover, it is worth noting that the spectral intensity at 324.8251 nm is the second most important feature in the traditional random forest model, which is partially attributed to the significance of the FWHM of Cu I 324.75 nm. The classification results in the confusion matrix, as shown in Fig. 8(b), demonstrate the effect of introducing the FWHM features. In Fig. 8(b), without incorporating the FWHM, 16.7% of the samples of the 2024 alloy type were misclassified as 7075. However, with the inclusion of the FWHM, the presence of a distinct gradient relationship between the copper (Cu) and magnesium (Mg) elemental contents in the two alloy types allows for better differentiation based on the FWHM of the corresponding emission peaks. This improvement is observed in the performance of the random forest classifier. In the case of PCA-RF, as shown in Fig. 8(c), the FWHM of Cu I 324.75 nm is considered the most important discriminative information. Additionally, the FWHM of Mn II 279.52 nm is identified as the second most important discriminative information. The enhanced recognition of the 7075 alloy type can be attributed to the significant variation in copper content compared to other alloy types. By incorporating the FWHM of Cu I 324.75 nm into the PCA-RF model, the spectral features of the 7075 type are effectively enhanced, leading to improved model performance. Additionally, the manganese (Mn) content in the 7075 alloy type exhibits a range of values compared to other alloy types. This can result in variations in peak intensity and peak width due to the varying content, aiding the model in better discrimination. In conclusion, the introduction of the broadening mechanism in the qualitative identification of aluminum alloys yields targeted improvements in model performance. However, it should be noted that different models exhibit varying sensitivity to the selected FWHM features. Therefore, a more in-depth analysis is necessary to select appropriate FWHM features that can enhance the classification accuracy based on the established models.

 

Fig. 8. Feature Importance and Model Confusion Matrix. (a) and (c) display the feature importance obtained from the RF model and PCA-RF, which provides insights into the relative contributions of different features in the classification task. (b) and (d), the classification results of the FWHM-RF model and PCA-FWHM-RF are presented.

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3.3 Stacking model

Based on the learning performed by different models using full-spectrum and downscaled reconstruction of spectra, it is evident that traditional models struggle to accurately predict test spectra even with a sufficient sample size. Although the enhanced model shows improved performance by incorporating spreading features to capture spectral continuity information (as discussed in Section 3.2), it still falls short of achieving precise sample prediction. Moreover, it remains challenging to ensure reliable predictions when training spectra are limited in number. Therefore, to address these limitations, we propose the construction of a three-layer Stacking model, leveraging the respective advantages of the aforementioned models and combining and filtering their outputs. Stacking models can adapt flexibly to different types of data. By incorporating diverse base models, the stacking model can accommodate different data distributions and relationships between features, thereby demonstrating excellent performance in qualitative analysis. By combining the predictive information from multiple base models, the stacking model can achieve more accurate and robust classification performance. Leveraging the complementary nature of different models, the stacking model can effectively capture complex patterns and relationships in the data. In the first layer, we employ a random forest, which consists of multiple decision trees and is capable of capturing valid spectral information through feature selection. This approach mitigates the risk of underfitting caused by limited data and the complexity of the dataset. Additionally, an automatic peak-seeking algorithm is employed to select spectral peaks with better independence. The full-width at halfmax of these peaks are calculated using the Voigt function, effectively incorporating relevant features and reducing the likelihood of underfitting. The experiments demonstrate the strong classification performance of the random forest in handling LIBS spectra. In the construction of the second layer, we aim to include the random forest as a heterogeneous learner to capture the diversity and complexity of features in different sample feature spaces. Here, XGBoost, which integrates a tree learner in a boosting-based manner, is chosen due to its ability to prevent overfitting while maintaining good generalization, similar to the random forest. Alongside XGBoost, two other models, KNN and SVM, are included to form the second layer of heterogeneous learners. Cross-validation is employed within each base learner to mitigate the risk of overfitting arising from limited data. The outputs of these three learners, along with the output of the first layer, constitute the new training data for the third layer. In the third layer, logistic regression is employed as the classifier to effectively combine the rich features obtained from the first two layers and output the probabilities for each category. L2 regularization is applied to further reduce the risk of overfitting in the final prediction. The third layer plays a crucial role in leveraging the diverse information from the previous layers and enhancing the model's predictive capabilities. By constructing a three-layer Stacking model with careful consideration of the strengths of different learners and employing appropriate filtering and regularization techniques, we aim to improve the accuracy and reliability of predictions, especially when faced with limited training spectra. The hyperparameters of the base models in the stacked model were determined based on the experiments in Section 3.1. For the third layer, the regularization parameter of the logistic regression was set to 1, the maximum number of iterations was set to 100, and the optimization algorithm used was the quasi-Newton method.

In order to verify the robustness of Stacking model, the results of Stacking model were compared with other models, and the comparison results are shown in Table 4, and the category prediction results of Stacking are shown in Fig. 9.

Table 4. Performance Comparison of Different Models

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Fig. 9. Confusion Matrix for the Stacking Model.

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Table 4 demonstrates that the variation in accuracy reflects the model's overall prediction ability at a macro level, consistent with the results obtained from the preceding experiments. However, in practical applications, the concern often lies in the success rate of the model when predicting a specific alloy type, as well as the success rate when predicting across various alloy types. Hence, evaluating the model based on accuracy and recall is a reasonable criterion, as these metrics are often considered together to assess the model's robustness, which can be intuitively assessed using the F1-score. In the case where the recognition accuracy is the same, PCA-RF has better precision than SVM when the recall rate is the same. Therefore, the F1-score is slightly higher for PCA-RF compared to SVM. Consequently, the PCA-RF model displays slightly better overall robustness than SVM with the same accuracy rate. Notably, the Stacking model achieves the highest performance across all four evaluation metrics, as illustrated in Fig. 9, where accurate predictions are made for each sample category. In the qualitative identification of multiple aluminum alloy types, accuracy serves as a macroscopic evaluation criterion, reflecting the model's prediction results. Meanwhile, the comparison of the remaining three evaluation metrics enables a more precise measurement of the model's robustness. By integrating the strengths of various learner types and addressing specific challenges encountered during the modeling process, the Stacking model enhances the generalization capability of the LIBS spectral model. Consequently, it offers an effective modeling approach for LIBS analysis. Overall, evaluating the model based on accuracy, recall, and F1-score provides a comprehensive assessment of its prediction performance and robustness, enabling a more accurate and reliable characterization of the model's capabilities in qualitative identification of aluminum alloys. The Stacking model, with its integrated learning approach and targeted solutions, presents a significant advancement in LIBS spectral modeling and offers a valuable methodology for LIBS analysis.

3.4 Exploration of small sample learning

In general, data-driven models require a large amount of data to establish reliable qualitative analysis models. The significance of small sample learning lies in addressing the issue of data scarcity in practical applications and constructing robust models. During the modeling process, the selection of features largely determines the final model performance. In this work, different levels of features were considered for the spectra, including the original spectral lines selected based on Gini impurity, the width of artificially selected spectral lines, and the predicted labels from three heterogeneous classifiers. The incorporation of multiple dimensions of features provides a reliable basis for qualitative analysis models, enabling accurate classification even with limited data. In this study, the number of spectra in the training and test sets was redefined. When the training sample size was reduced to 25 spectra, the cross-validation fold was adjusted from 5-fold to 3-fold. Similarly, when the training spectra reached 10, 2-fold cross-validation was used. The minimum training sample size was set to 10. Grid search was employed to determine the hyperparameters of all models.

In quantitative analysis, increasing the number of sample spectra enhances the accuracy of element content estimation, leading to more reliable identification results. However, in qualitative analysis, the model relies on spectral learning from each alloy type to establish predictions of qualitative results. This becomes a challenging task when the number of samples is extremely limited while the classification categories are diverse. In previous discussions, the incorporation of FWHM in PCA-RF resulted in performance improvement. By comparing PCA-RF and PCA-FWHM-RF, the relationship between qualitative recognition accuracy and the number of training spectra is depicted in Fig. 10. The experimental settings and considerations mentioned above demonstrate the challenges and strategies employed in the qualitative analysis of aluminum alloys with limited data. The combination of different feature dimensions and modeling techniques allows for accurate qualitative analysis even in scenarios with limited spectral samples.

 

Fig. 10. Relationship between Training Spectral Quantity and Accuracy.

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The accuracy of PCA-RF and PCA-FWHM-RF fluctuated continuously, as depicted in the Fig. 10. When the sample size was 20, neither method achieved a recognition success rate of 75%. As the sample size gradually increased, the accuracy of both methods steadily improved, until reaching the same level when the training spectra reached 35. The change in trend for both methods occurred around 50 spectra. However, at 45 training spectra, the recognition accuracy of PCA-RF continued to increase while PCA-FWHM-RF's accuracy decreased. Subsequently, both methods exhibited similar trends at 50 training spectra. When the number of spectra increased to 70, the recognition accuracy of PCA-FWHM-RF surpassed that of PCA-RF.

Observing the trend of recognition rates for both methods, it can be noted that as the number of spectra increased, the distribution of sample points continuously changed, leading to model weight adjustments. Consequently, after reaching the optimal recognition accuracy for the first time, the accuracy slightly decreased and fluctuated. As the number of training spectra increased, the model's reliability improved, resulting in a subsequent increase in recognition accuracy. This situation is commonly encountered in practical problems. Random forest, as a powerful learner for handling high-dimensional data, still struggles to accurately perform qualitative recognition on reconstructed spectra using PCA when the training sample size is small. Despite extracting features from the spectra that can enhance qualitative recognition results, PCA-FWHM-RF only achieved a slight improvement in accuracy compared to PCA-RF when the training spectra were fewer than 35. However, both methods still require more samples to demonstrate better performance. Furthermore, at 45 and 70 spectra, the recognition accuracy of both methods fluctuated. These observations indicate that the increase in training spectra also introduces challenges to model establishment, leading to overfitting and hindering accurate predictions.

Based on the results, it can be observed that obtaining more dimensions and richer spectral descriptive features from the original spectral information can provide better support for model establishment. Effective feature information enhances the model's reliability, thus addressing the challenges of small sample recognition. However, the Stacking model achieved improved qualitative analysis for all five categories of aluminum alloys with only 10 training spectra, surpassing the performance of the enhanced random forest model using 35 training spectra. When the training spectra were 10, the accuracy of Stacking exceeded 85%, whereas both PCA-RF and PCA-FWHM-RF had recognition accuracies below 65%. The predictive capability of the model established based on 15 spectra already surpassed the performance of the other two models when using 70 training spectra, and the qualitative recognition accuracy fluctuated less as the number of training spectra increased, consistently exceeding 96.25%.

The Stacking model addresses the challenges of data complexity and limited sample size in small sample LIBS spectral learning through a layered model construction. It effectively resolves the issue of underfitting caused by data complexity and limited sample size, as well as the overfitting problem caused by redundant spectral features. In summary, compared to the improved RF method and other traditional algorithms, the Stacking model demonstrates better robustness and accuracy in qualitative analysis of aluminum alloy samples. It also enables more accurate qualitative analysis in small-scale spectral modeling compared to most models.

4. Conclusion

This paper proposes a small-sample machine learning algorithm based on Stacking, which enhances the qualitative analysis of aluminum alloys by hierarchically extracting spectral feature information and reconstructing the training spectra. The research investigates prediction models based on reconstructed spectra incorporating intensity, FWHM, and machine learning features. Compared to other model methods such as improved random forest modeling, this algorithm significantly improves the performance of the prediction model. The model established by this algorithm using only three spectra for each alloy category demonstrates comparable performance to the improved random forest algorithm using 14 training spectra for each category. In the new prediction model with 15 training spectra, the error rate for qualitative recognition is less than 4%. When the training spectra are limited to 10 (with only two spectra for each alloy category), the accuracy of aluminum alloy qualitative analysis improves by approximately 20% compared to the improved random forest. Additionally, with sufficient training spectra, the new model achieves a qualitative analysis accuracy, precision, recall rate, and F1 prediction score of 1.0, representing a 6.7%, 6.19%, 6.7%, and 6.69% improvement over the best-performing PCA-RF in the experiments.

This improvement can be attributed to two main factors. Firstly, the new algorithm extracts richer feature information from the spectra, providing different dimensional descriptions of the spectra. Secondly, the hierarchical objectives of the stacking model correspond to the underfitting caused by the lack of samples in small samples and the overfitting caused by excessive dimensional information. The proposed approach for accurate modeling of small-scale LIBS spectra in this paper enables reliable predictions even with a small number of training samples. It provides an effective qualitative analysis method for experimental objects that are difficult to obtain spectra for or have limited data collection. In the future, this algorithm can be extensively validated for performance in various datasets.