GMM Clustering

The GMM is a probabilistic model that describes a dataset comprising multiple Gaussian distributions [28]. For each Gaussian component, the probability density function is expressed as: $f\left(x|\mu \mathrm{,}\text{Σ}\right)=\frac{1}{{\left(2\pi \right)}^{n/2}{\left|\text{Σ}\right|}^{1/2}}{e}^{-\frac{1}{2}}{\left(x-\mu \right)}^{\text{T}}{\text{Σ}}^{-1}\left(x-\mu \right)\mathrm{.}x\in ℝ\mathrm{,}$ (1) where n denotes the dimensions of the pulse data; μ denotes the n-dimensional mean vector; and ∑ denotes the covariance matrix of n × n. Obviously, the Gaussian distribution is determined by the mean vector μ and the covariance matrix ∑. The initial pulse has 248 features; however, such a high dimension leads to the "Curse of dimensionality". To reduce the number of pulse features, this study used only Q_tail and Q_total as the two-dimensional (2-D) features of the pulse data, that is n=2.

Neglecting pulse stacking in the n-γ discrimination, the GMM has only two components: neutrons and gamma rays. For a Gaussian mixture distribution with two mixed components, the probability density is expressed as: $f_M(x)={\displaystyle \sum _{i=1}^2{\alpha }_i}\cdot f\left(x|{\mu }_{\text{i}}\mathrm{,}{\text{Σ}}_i\right)\mathrm{.}$ (2) where αi is the "mixture coefficient", which is the probability of selecting the ith component Gaussian. Here, αi>0, and ${\displaystyle \sum _{i=1}^2{\alpha }_i}=1$.

The model parameters αi, μi, and ∑_i must be solved by the iterative optimization of the expectation-maximum (EM) algorithm [29, 30]. Each iteration of the EM algorithm comprises two steps: step E, which involves estimating the expectation of the hidden variables based on the current parameters; and step M, which involves using the computational results from step E to update the model parameters based on maximum likelihood estimation.

A large number of errors occurred when GMM clustering was performed directly on the entire training dataset. To enhance the accuracy of GMM clustering, we divided the data into three energy ranges [17]: 0–25 keV, 25–100 keV, and 100–2100 keV. The GMM soft clustering output provides the probability that a pulse belongs to either neutrons or gamma rays. If the probability of a pulse belonging to neutrons exceeded 50%, it was classified as a neutron; otherwise, it was classified as a gamma ray. The primary objective of GMM clustering was to produce a dependable training set that is subsequently utilized by the KNN algorithm. However, this classification method may encounter ambiguous pulses, which can reduce the accuracy of KNN classification.

KNN Classification and Regression Algorithm

GMM clustering can only cluster a fixed dataset; therefore, supervised learning algorithms must be used for real-time discrimination. This study used supervised learning to accurately represent the distribution of the training set samples, which followed a Gaussian mixture distribution comprising two components. The goal of supervised learning was to ensure that the test set data exhibited classification/clustering results similar to those of the training set. Among the supervised learning algorithms, such as SVM and LDA, KNN was selected for its simplicity and ease of implementation in LabVIEW programs. The KNN algorithm is known for its simplicity and accuracy, ensuring a generalization error that is not more than twice the error rate of the Bayes optimal classifier. To optimize the KNN algorithm, it is important to determine the optimal value of K, select an appropriate distance metric, and specify the decision rule.

First, the key to the KNN algorithm is the determination of the optimal K value. When the K value is excessively large, the distant points affect the prediction results, resulting in underfitting; Conversely, if K is excessively small, the model is less tolerant of noise and prone to overfitting. In this study, the method used to determine the optimal K value was 10-fold cross validation [31]. A 10-fold cross-validation means that the training dataset D was divided into ten equal parts, nine of which were used to train the model, and the remaining 1 part of the data was used to compute the test accuracy. This process was repeated for each part of the data to obtain the average accuracy. There exists a maximum value for the results of the 10-fold cross-validation, where the accuracy value first increases with increasing K and then decreases after reaching the maximum value. The K value corresponding to the highest accuracy average is the optimal K value. The dataset D must contain a sufficient number of samples, and simultaneously, the result of the GMM clustering can obtain a reliable training set. Therefore, GMM-KNN uses the training set obtained from GMM clustering as dataset D.

In addition, the KNN algorithm must calculate the distance between an unknown pulse and each pulse in the training set (in this study, the distance is the Euclidean distance). For example, for points X and Y with n features, the distance is calculated as $D(X\mathrm{,}Y)=\sqrt{{\left(x_1-y_1\right)}^2+{\left(x_2-y_2\right)}^2+\dots +{\left(x_n-y_n\right)}^2}\mathrm{.}$ (3) Because we use Q_tail and Q_total as the 2 features of pulses, the above equation has n=2. Calculating the distance between the two pulses simply involves replacing x and y with Q_tail and Q_total, respectively.

Finally, the KNN algorithm outputs the classification or regression results for the pulses in the test set. For the KNN classification task, the result is the category that occurs most frequently in the nearest neighboring K instances [32]. For the KNN regression task, the result is the average value of each feature over the nearest neighboring K instances [33].

GMM-KNN Classification and Regression

Both GMM and KNN are effective methods for n-γ discrimination. GMM clustering performs well in PSD analysis. However, the results of GMM clustering are limited to the current dataset, and prior knowledge obtained from the training set cannot be directly applied to the test set. In contrast, KNN accurately captures the sample distribution and can be easily implemented on hardware, providing strong real-time performance and the potential for real-time n-γ discrimination. However, the drawback of KNN is that it requires prestoring the sample set.

In this study, the goal of GMM clustering was to construct a reliable training set, and KNN utilized this training set to discriminate unknown pulses. To integrate the GMM and KNN, we proposed improvements to both methods. When GMM clusters the data across the entire energy range, a significant number of misclassifications are obtained. Therefore, in GMM-KNN, the data were divided into three energy partitions to enhance the clustering performance. In addition, the GMM clustering results often include confusing pulses (low-probability events). Hence, we selected only the pulses with classification probabilities greater than 99% as the training set. The test set of GMM-KNN comprised a large number of unknown pulses, and each pulse must calculate its distance from all samples in the training set. Using the complete training set yielded the most accurate classification results but significantly increased the computational cost of the algorithm.

In the context of GMM-KNN classification, reducing the size of the training dataset has a minimal impact on the classification results, but significantly decreases the algorithm complexity. This facilitates a flexible selection of the sample quantity within the training set for the GMM-KNN classification. However, for GMM-KNN regression, a complete training set must be used to ensure the accuracy of the regression predictions. To assess the real-time performance of our method, a unified programming language or first-in, first-out (FIFO) data transfer with other devices must be employed. The LabVIEW program can call or directly read the acquired data, which facilitates the real-time implementation of the GMM-KNN for pulse discrimination. Using parallel computing, the LabVIEW program can concurrently calculate two decision values, resulting in simpler decision logic and faster computation. In addition, the real-time performance of the algorithm must consider the runtime and memory footprint of the trained model. Thus, this study proposed the GMM-KNN algorithm that employed only two features, Q_tail and Q_total, for both GMM clustering and KNN classification and regression. These features reduced data dimensionality in GMM clustering and significantly decreased the computational cost and memory footprint of the trained model.

A block diagram of the procedure of–GMM-KNN algorithm is shown in Fig. 1. First, the method divides the preprocessed training and test data into three parts separately, These three energy ranges are 0–25 keV, 25–100 keV, and 100–2100 keV respectively. Second, GMM takes Q_tail and Q_total as pulse features, and performs small batch clustering in three energy ranges. Consequently, this method selects a portion of the data of clustering results (probability > 99%) as the training set of KNN. Finally, GMM-KNN implements the classification and regression algorithms with LabVIEW programming, and subsequently outputs the category and regression prediction values.

Fig. 1Full-screen View

Block diagram of n/γ discrimination based on GMM-KNN

Evaluation Metrics

The output of the GMM-KNN classification was binary, with zero representing gamma rays and one representing neutrons. Both pulse types exhibited an elliptical distribution in the feature space. Comparing the difference between the output and ground truth facilitated the qualitative assessment of the effectiveness of the n-γ discrimination. The outputs of the GMM-KNN regression comprised the average values of the nearest K pulses for Q_tail and Q_total, as well as their ratios. We used this ratio to calculate the FOM, where a higher FOM indicated better n-γ discrimination performance.

There is a difference in the ratio of slow charge to total charge between the two types of pulses, neutrons and gamma rays; therefore, the CCM selects the ratio of the tail integral (Q_tail) to the total integral (Q_total) as the PSD factor [34] and then calculates the FOM as the discrimination metric. In this study, Q_tail and Q_total were adjusted to obtain the best FOM. Q_tail was considered as 68 ns and Q_total as 124 ns (Fig. 2). For the GMM-KNN regression, the pulse features are the predicted regression values, thus, PSD = (Regressed Q_tail) / (Regressed Q_total). After calculating the PSD values for all pulses, the GMM-KNN regression used the updated FOM as the evaluation metric, which can use the FOM to evaluate the n-γ discrimination effectiveness further.

Fig. 2Full-screen View

Diagram of the tail integral Q_tail and the total integral Q_total

Two Gaussian peaks were observed in the PSD histogram after Gaussian fitting (Fig. 3), and the FOM is calculated as follows: $\text{FOM}=\frac{\text{Γ}}{FWH{M}_{\gamma }+FWH{M}_n}\mathrm{,}$ (4) where $\text{Γ}$ is the distance between the two Gaussian peaks and $FWH{M}_{\gamma }$ and $FWH{M}_{\text{n}}$ are the full width at half maximum of the gamma and neutron peaks, respectively. A larger FOM value implies a higher separation between the two Gaussian peaks. Thus, the algorithm n/γ discrimination was more effective.

Fig. 3Full-screen View

Definition of FOM

Energy division

The pulses in the test set were randomly collected in the real experiments. The pulses were divided into three parts based on three energy ranges: 0–25 keV, 25–100 keV, and 100–2100 keV. Figure 4 shows the LabVIEW program diagram for pulses division based on energy. The conditional diagram uses energy as the determinant, and the program counts the pulses in each energy range. Thereafter, by sorting the energy in ascending order and setting the index of "array subset vi" based on the count value, we can obtain three sub-arrays.

Fig. 4Full-screen View

Block diagram of LabVIEW program for energy division

GMM-KNN classification algorithm

A program block diagram of the GMM-KNN classification is shown in Fig. 5, GMM-KNN classification to determine the category of a pulse included three steps:

Fig. 5Full-screen View

Block diagram of LabVIEW program for GMM-KNN classification algorithm

Step 1 - Training set composition: In the GMM-KNN algorithm, the training set processed by GMM clustering contained 26172 pulses (probability >99%), and the test pulses calculated the distances with each pulse of the training set. It is more computationally intensive when using the full training set directly; therefore, we sorted the data within the training set by probability value and take one sample every three values. These samples formed a 2-D array training set M_c. The columns of M_c represent the two features of a pulse, Q_tail and Q_total, and each row represents a pulse. The m rows in front of the 2-D array are the gamma rays and the remaining k rows are neutrons. The expression for M_c is as follows: $M_c=\left[\begin{array}{c}{\gamma }_1\left(\text{train}\right)\\ {\gamma }_2\left(\text{train}\right)\\ {\gamma }_m\left(\text{train}\right)\\ ⋮\\ n_1\left(\text{train}\right)\\ n_2\left(\text{train}\right)\\ n_k\left(\text{train}\right)\end{array}\right]=\left[\begin{array}{cc}{Q}_{\text{tail}}\left({\gamma }_1\right)& Q_{\text{total}}\left({\gamma }_1\right)\\ Q_{\text{tail}}\left({\gamma }_2\right)& Q_{\text{total}}\left({\gamma }_2\right)\\ ⋮& ⋮\\ Q_{\text{tail}}\left({\gamma }_m\right)& Q_{\text{total}}\left({\gamma }_m\right)\\ Q_{\text{tail}}\left(n_1\right)& Q_{\text{total}}\left(n_1\right)\\ Q_{\text{tail}}\left(n_2\right)& Q_{\text{total}}\left(n_2\right)\\ ⋮& ⋮\\ Q_{\text{tail}}\left(n_k\right)& Q_{\text{total}}\left(n_k\right)\end{array}\right]$ (5) Step 2 - Distance values computation: This involved computing the distance values between the test pulse feature vector $\mu =\left[Q_{\text{tail}}\left(\text{test}\right)+Q_{\text{total}}\left(\text{test}\right)\right]$ and all pulses in the training set. These distance values formed a one-dimensional (1-D) array D(M_c, μ). The elements of D(M_c, μ) are the Euclidean distance values between each row of the array within M_c and feature vector μ: $\begin{array}{c}D\left(M_{\text{c}}\mathrm{,}\mu \right)=\left[\begin{array}{c}D\left({\gamma }_1\left(\text{train}\right)\mathrm{,}\mu \right)\\ D\left({\gamma }_2\left(\text{train}\right)\mathrm{,}\mu \right)\\ ⋮\\ D\left({\gamma }_m\left(\text{train}\right)\mathrm{,}\mu \right)\\ D\left(n_1\left(\text{train}\right)\mathrm{,}\mu \right)\\ D\left(n_2\left(\text{train}\right)\mathrm{,}\mu \right)\\ ⋮\\ D\left(n_k\left(\text{train}\right)\mathrm{,}\mu \right)\end{array}\right]\\ =D_{\gamma }\left(M_{\text{c}}\mathrm{,}\mu \right)+D_n\left(M_{\text{c}}\mathrm{,}\mu \right)\end{array}$ (6) Step 3 - Pulse classification: The distance subsets $D_{\gamma }\left(M_{\text{c}}\mathrm{,}\mu \right)$ and $D_n\left(M_{\text{c}}\mathrm{,}\mu \right)$ corresponded to gamma rays and neutrons, respectively. The smallest K values were obtained from the distance subsets $D_{\gamma }\left(M_{\text{c}}\mathrm{,}\mu \right)$ and $D_n\left(M_{\text{c}}\mathrm{,}\mu \right)$. The result of the 10-fold cross-validation indicated a maximum accuracy of 99.317% at K = 5; thus, K was set as 5. The two distance subsets were sorted in ascending order separately, and the smallest five gamma rays and five neutrons in each of $D_{\gamma }\left(M_{\text{c}}\mathrm{,}\mu \right)$ and $D_n\left(M_{\text{c}}\mathrm{,}\mu \right)$ were named as γ₁, γ₂, γ₃, γ₄, γ₅; n₁, n₂, n₃, n₄, n₅, respectively.

When γ₃ < n₃, we have $\{\begin{array}{l}{n}_1<n_2<n_3\mathrm{,}\hfill \\ {\gamma }_1<{\gamma }_2<{\gamma }_3<n_3\mathrm{.}\hfill \end{array}$ (7) The above equation holds for γ₄ < n₃ and γ₅ < n₃. In these cases, the number of gamma rays must be greater than the number of neutrons in the five nearest neighbor values.

When n₃ < γ₃, we have $\{\begin{array}{l}{\gamma }_1<{\gamma }_2<{\gamma }_3\mathrm{,}\hfill \\ n_1<n_2<n_3<{\gamma }_3\mathrm{.}\hfill \end{array}$ (8) The above equation holds for n₄ < γ₃ and n₅ < γ₃. In these cases, the number of neutrons must be greater than the number of gamma rays among the five nearest neighbor values.

In summary, the 3rd value of each of the subsets (γ₃ and n₃) was the judgment value. When γ₃ < n₃, the pulses were determined to be gamma rays. Conversely, when n₃ < γ₃, the pulses were considered as neutrons. In classification, the traditional KNN algorithm must perform five judgments and five counts. However, the GMM-KNN classification requires only one judgment, which improves algorithm efficiency and reduces algorithm complexity.

GMM-KNN regression algorithm

GMM-KNN regression requires determining the nearest neighbor of the K(K=5) values and calculating the average of the five values. The algorithm does not divide the training set into two groups but requires a complete training set. Similar to the GMM-KNN classification, the implementation of the GMM-KNN regression with LabVIEW involves three steps (Fig. 6):

Fig. 6Full-screen View

Block diagram of LabVIEW program for GMM-KNN regression algorithm

Step 1: The distance values between the test pulse $\mu =\left[Q_{\text{tail}}\left(\text{test}\right)+Q_{\text{total}}\left(\text{test}\right)\right]$ and the n pulses in the training set M_r were calculated. Consequently, a 1-D array of distance values $D\left(M_{\text{r}}\mathrm{,}\mu \right)$ was obtained. $M_{\text{r}}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]=\left[\begin{array}{cc}{Q}_{\text{tail}}^{\left(1\right)}& Q_{\text{total}}^{\left(1\right)}\\ Q_{\text{tail}}^{\left(2\right)}& Q_{\text{total}}^{\left(2\right)}\\ ⋮& ⋮\\ Q_{\text{tail}}^{\left(m\right)}& Q_{\text{total}}^{\left(m\right)}\end{array}\right]$ (9) $\begin{array}{lll}D\left(M_{\text{r}}\mathrm{,}\mu \right)\hfill & =\left[\begin{array}{c}D\left(v_1\mathrm{,}\mu \right)\\ D\left(v_2\mathrm{,}\mu \right)\\ ⋮\\ D\left(v_n\mathrm{,}\mu \right)\end{array}\right]\hfill & =\left[\begin{array}{c}{D}_1\\ D_2\\ ⋮\\ D_n\end{array}\right]\hfill \end{array}$ (10) Step 2: The 1-D array $D\left(M_{\text{r}}\mathrm{,}\mu \right)$ was combined with the 2-D array M_r of the training set into a new 2-D array 𝑴n. The first row of 𝑴n represented the distance value, the second row represented the Q_tail value of the pulses in the training set, and the third row represented the Q_total value of the pulses in the training set. We then sorted 𝑴n in ascending order according to the first row and intercepted the first five columns to obtain 𝑴₅. $\begin{array}{l}{M}_n=\left[\begin{array}{cccc}{D}_1& D_2& \dots & D_n\\ Q_{\text{tail}}^{\left(1\right)}& Q_{\text{tail}}^{\left(2\right)}& \dots & Q_{\text{tail}}^{\left(n\right)}\\ Q_{\text{total}}^{\left(1\right)}& Q_{\text{total}}^{\left(2\right)}& \dots & Q_{\text{total}}^{\left(n\right)}\end{array}\right]\\ M_5=\left[\begin{array}{cccc}{D}_{\text{min1}}& D_{\text{min2}}& \dots & D_{\text{min5}}\\ Q_{\text{tail}}\left(1\right)& Q_{\text{tail}}\left(2\right)& \dots & Q_{\text{tail}}\left(5\right)\\ Q_{\text{total}}\left(1\right)& Q_{\text{total}}\left(2\right)& \dots & Q_{\text{total}}\left(5\right)\end{array}\right]\end{array}$ (11) Step 3: The regression values of KNN were calculated. The first eigenvalue E₁ was the average of the first five columns in row 2, the second eigenvalue E₂ was the average of the first five columns in row 3, and the third eigenvalue PSD_GMM-KNN was the division of the first two eigenvalues. $\begin{array}{l}{E}_1=\frac{{\displaystyle \sum _{i=1}^5{Q}_{\text{tail}}}\left(\text{min}i\right)}{5},E_2=\frac{{\displaystyle \sum _{i=1}^5{Q}_{\text{total}}}\left(\text{min}i\right)}{5}\\ PS{D}_{\text{GMM}-\text{KNN}}=E_1/E_2\end{array}$ (12) The PSD values of all pulses were stored in.csv files, and the GMM-KNN regression used PSD as a discriminating index. The PSP values were used to compute the FOM, which can evaluate the n-γ discrimination ability of the GMM-KNN regression. The GMM-KNN regression first calculated the distance value of the pulses and subsequently calculated the regression value of the nearest K value. The regression values for all pulses did not need to be calculated. Moreover, the regression values of Q_tail and Q_total were calculated in parallel, which significantly reduced the computational complexity.

GMM Clustering

Depending on whether the probability value exceeded 50%, we classified the GMM clustering results into two categories. Figure 7(a) shows the results of direct clustering, where the red squares and blue circles represent neutrons and gamma rays, respectively. There were significant errors in the pulse discrimination, with numerous neutrons misclassified as gamma rays. Figure 7(b) shows the results of small-batch clustering in 3 energy ranges, where only a few ambiguous pulses were observed. Small-batch clustering reduced the misclassification rate and improved the effectiveness of n-γ discrimination.

Fig. 7Full-screen View

(Color online) Results of GMM clustering. (a) Results of the direct GMM clustering. (b) Results of the small batch GMM clustering after segmentation by energy

Although small-batch clustering yielded better results than direct clustering, Fig. 7 exhibits the red downward protrusion at the energy segmentation boundary (25 keV and 100 keV). To obtain a reliable training set, these ambiguous pulses must be removed. Within 0–25 keV, we progressively increased the probability of pulses in the training set (Fig. 8). Figures 8(a)-(c) correspond to complete pulses, pulses with a probability above 90%, and pulses with a probability above 99%, respectively. Figure 8 demonstrates that a higher probability results in a better separation of pulses in the training set; however, it also decreases the number of pulses in 0–25 keV.

Fig. 8Full-screen View

(Color online) n-γ distribution in the training set within the 0–25 keV at different probabilities

Table 1 presents the quantity distributions in the training set for different probabilities. At 25–2100 keV, there was minimal variation in the number of pulses. Most of the ambiguous pulses were concentrated at 0–25 keV. As the probability increased, the number of pulses in 0–25 keV decreased significantly. When the training set comprised pulses with a probability exceeding 99.9%, the number of pulses in the 0–25 keV range was even lower than that in the 25–2100 keV. In addition, the PSD histogram of the test set exhibited three distinct peaks and poor fitting performance at this stage (Fig. 9). Therefore, this study used a training set composed of pulses with a probability exceeding 99%.

Fig. 9Full-screen View

PSD histogram of the GMM-KNN regression, when the training set comprises pulses with a probability greater than 99.9%

Figure 10 shows the training set comprising pulses with a probability above 99%. In this case, the samples with PSD values below the threshold (PSD_threshold) were removed at the energy segmentation boundaries. At this stage, the neutrons and gamma rays are completely separated, and the GMM-KNN algorithm only needed to select the remaining pulses from this portion to construct the training set. Once the training set was constructed, the GMM-KNN can flexibly select a subset of data from the training set and implement n-γ discrimination using LabVIEW program.

Fig. 10Full-screen View

Training set comprising pulses with probability above 99%

GMM-KNN Classification

In the context of real-time n-γ discrimination, the algorithmic efficiency is a crucial factor that is influenced by both the size of the training set and the number of pulse features.

This study investigated the time consumption and error rates of the GMM-KNN algorithm by employing average sampling techniques to select subsets of the complete training set with proportions of 1/2, 1/3, 1/4, 1/5, and 1/6 (Table 2). By comparing the time consumption and error rates of differently sized training sets, it was observed that when only 1/4 of the complete dataset was used, the classifier’s time consumption was reduced to approximately 1/4 of the original, resulting in an average execution time per pulse of only 67 μs compared to 294.27 μs for the complete training set. The discriminative results between the reduced dataset and the complete training set exhibited a difference of only 0.13% (41 pulses, with 21 falling in the 0–25 keV). This sampling method significantly reduced the computational costs while ensuring reliable discrimination results. This facilitated flexible selection of the training set based on the experimental latency requirements.

KNN algorithms typically use the nonzero-amplitude portion of pulses as feature points, which encompass 64 points. Table 3 shows that the time consumption of the KNN algorithm was 4325 ms in the 0–25 keV. In contrast, GMM-KNN employed only two pulse features, Q_tail and Q_total, resulting in a reduced execution time of 2021 ms for the same test volume. The program was parallelized, and the selection of Q_tail and Q_total significantly improved the processing speed. Notably, even when using the 64-points KNN algorithm, the execution time remained at 4325 ms, indicating that the algorithm was not very demanding in terms of the number of pulse features and could adapt to different pulse types and experimental requirements.

An imbalanced classification is a significant challenge for classification algorithms. In this study, we applied GMM clustering to partition a test dataset containing 10,000 gamma rays and 10,000 neutrons into 20 subsets, each comprising 1,000 pulses. We varied the gamma/neutron (γ/n) ratios from 10:1 to 10:10 and from 10:10 to 1:10, resulting in 19 different ratios. We compared the results of the GMM-KNN classifier with the prior knowledge to obtain the error rates and average execution times per pulse for each ratio, as shown in Fig. 11. All seven ratios ranging from 10:8 to 6:10 exhibited error rates below 5%. The lowest error rate of 1.01% (1.23%) and average execution time of a cycle of 301.84 μs (289.60 μs) were achieved at a ratio of 9:10 (10:9), which is consistent with the γ/n ratio of approximately 9:10 (12,542 gamma rays to 13,619 neutrons) observed in the training set. The GMM-KNN classifier demonstrated excellent real-time performance, strong adaptability, and robustness. Thus, it exhibits great potential for real-time discrimination and can be utilized for onsite analysis, serving as a reference for offline n-γ discrimination analysis.

Fig. 11Full-screen View

Error Rate and runtime for various gamma/neutron ratios

Using the complete training set, the GMM-KNN performed the classification task simultaneously in three different energy ranges (0–25 keV, 25–100 keV, and 100–2100 keV). The discrimination results in these three energy ranges were concatenated to obtain a discrimination result across the complete energy range (as shown in Fig. 12). The classification exhibited a small error at the energy boundaries, which is consistent with the GMM clustering results. This indicated that the test results accurately reflected the distribution of the data in the training set. In contrast to the CCM, which judges the pulse categories using the threshold PSD_threshold, the results of the GMM-KNN classification were more consistent with the Gaussian mixture distributions of the two components. In the low-energy range, an overlap between neutrons and gamma rays was observed, and we could not directly observe the classification results of these two pulse types in the low-energy region in Fig. 12. Only by comparing the classification effects of CCM and GMM-KNN in the feature space containing Q_tail and Q_total can we effectively evaluate their classification performances.

Fig. 12Full-screen View

(Color online) Classification results of GMM-KNN in a 2-D space comprising PSD and Energy

Using Q_tail, Q_total and energy as the X, Y, and Z axes, respectively, we obtained a three-dimensional (3-D) plot of the classification results in the feature space (Fig. 13). The red cubes and blue spheres represent neutrons and gamma rays, respectively. Figure 13(a) shows the classification results of CCM. Figure 13 shows the classification results of GMM-KNN. As evident, for the majority of pulses, the neutrons and gamma rays exhibited distinct cone-shaped distributions, which rendered them easy to distinguish. However, for certain lower-energy pulses, there was no clear boundary between the neutrons and gamma rays (indicated by the flattened red region in Fig. 13), resulting in their mixture and making differentiation difficult. To further compare the classification results of the CCM and GMM-KNN in the low-energy range, we focused on the performance of the two classification algorithms in the feature space.

Fig. 13Full-screen View

(Color online) Scatterplot in 3-D space comprising eigenvalues and energies. (a) CCM classification results. (b) GMM-KNN classification results

Figure 14 shows the projection of the 3-D visualization from Fig. 13 onto the X-Y plane. In the 2-D feature space, we can observe the classification results of the CCM (left) and GMM-KNN (right). The two types of pulses exhibited approximately elliptical distributions, with blue representing gamma rays and red representing neutrons. The CCM determined the pulse categories based on the threshold, PSD_threshold. It constructed a histogram of the pulse PSD and fit it with a Gaussian distribution, thereby facilitating the separate fitting of peaks for neutrons and gamma rays. The midpoint between the two peaks was considered as the threshold PSD_threshold, which served as the criterion for distinguishing between neutrons and gamma rays. In Fig. 14, PSD_threshold is represented by a simple straight line (the equation of the line is $y=\frac{1}{PS{D}_{\text{threshold}}}x$). In contrast, GMM worked better for elliptical clustering, such that GMM-KNN classified 1657 (5.52%) gamma rays correctly. Although GMM-KNN still cannot completely discriminate neutrons and gamma rays in the low-energy range, it performs better classification compared to CCM. In the feature space, the GMM-KNN had a larger number of discriminable pulses.

Fig. 14Full-screen View

(Color online) Classification results of CCM and GMM-KNN in 2-D feature space. The left and right sides present the CCM and GMM-KNN classification results, respectively

For the portion of pulses that is difficult to distinguish for both methods (the flattened red region in the 3-D visualization), both the CCM and GMM-KNN tended to classify these ambiguous pulses as neutron pulses. This is because gamma rays exhibit a more concentrated distribution, resulting in a higher peak in the PSD histogram corresponding to gamma rays. As the PSD_threshold moved towards the gamma peak, more pulses are classified as neutrons. In the case of the GMM-KNN classification method, the training set was also generated by the GMM, resulting in classification preferences similar to those of the CCM.

A quantitative comparison of the classification results of CCM and GMM-KNN revealed that GMM-KNN improved the n-γ discrimination. For quantitative analysis, we used the GMM-KNN regression algorithm.

GMM-KNN regression

To obtain a more generalizable metric, GMM-KNN employs regression to compute a quantifiable FOM value. This is because the difference between the two types of pulses, neutrons and gamma rays, increases with higher energies. Consequently, the FOM values vary across the three energy ranges.

For the CCM, we created a histogram of the PSD (Fig. 15). Figure 15(a), (b), and (c) correspond to the test results within 0–25 keV, 25–100 keV, and 100–2100 keV, respectively. Figure 15(a) corresponds to the lowest 0–25 keV range, where the two Gaussian-fit peaks were least separated. In this energy range, neutrons and gamma rays were the most difficult to discriminate, and the number of misclassified pulses was the highest. As the energy increased, the difference between the distributions of the two types of pulses increased, and the separation of the two Gaussian-fit peaks increased, as shown in Fig.15(b) and 15(c). These two parts of the pulse can be easily distinguished. The CCM is a simple and effective discrimination method in the energy range of 25–2100 keV; therefore, the CCM can discriminate pulses with high energies well. However, at lower energy levels, there are numerous ambiguous pulses, and the level of ambiguity increases as the energy decreases.

Fig. 15Full-screen View

Histograms of PSD from the CCM for the three energy levels

For the GMM-KNN regression, we generated PSD histograms (Fig. 16). Figure 16(a), (b) and (c) correspond to the test results within 0–25 keV, 25–100 keV, and 100–2100 keV, respectively. As shown in Fig. 16(a), for GMM-KNN, the separation of the two Gaussian fitting peaks increased, and the FOM value greatly improved compared with that in case of CCM. The discrimination ability of the GMM-KNN was significantly better than that of the CCM in the energy range of 0–25 keV. However, in the higher energy ranges (25–100 keV and 100–2100 keV), the FOM values improved less and the separation of the two Gaussian-fitted peaks became slightly larger. When the effectiveness of n-γ discrimination improved, we observed that the PSD is highly concentrated in certain bins. This is attributed to the fact that the training set was not a standard Gaussian distribution, but there were several small protrusions of varying heights. After the KNN regression, pulses in certain intervals became more concentrated, leading to a more pronounced peak in the histogram.

Fig. 16Full-screen View

Histograms of PSD from the GMM-KNN for the three energy levels

The results of the CCM and GMM-KNN regressions are presented in Table 4. In the 0–25 keV, neither method could completely separate the two types of pulses. However, the FOM of the GMM-KNN method improved by 32.08%, thereby significantly enhancing the discrimination between neutrons and gamma rays. This method facilitated a further reduction in the energy threshold for discriminable pulses, implying that neutrons and gamma rays can be distinguished in a larger energy range. In addition, the improvement in FOM by GMM-KNN decreased as the energy increased. At 25–2100 keV, the CCM could achieve basic separation between neutrons and gamma rays, thereby diminishing the additional benefits of machine learning methods.