BLS is a flat network structure based on RVFLNN. Unlike traditional deep learning networks, the incremental learning algorithm in BLS enables it to have efficient and fast reconstruction capabilities. This avoids the time-consuming phenomenon caused by a large number of hyperparameters in the deep learning framework. BLS utilizes pseudo-inverse and ridge regression methods to efficiently calculate weights that connect feature nodes and enhancement nodes to the output parameters.

Suppose the input samples are X, Y, where X ∈ ℝ^N×M, Y ∈ ℝ^N×C and N is the number of samples, M is the dimension of X, C is the dimension of Y. Using the feature mapping function ξ_i(XW_zi + α_zi) to map the sample data and result mapping feature Z_i. Where W_zi and α_zi are the parameters of Z_i. Different feature mapping functions can be different, and the resulting mapping feature set is Zn≡[Z1,…,Zn]∈ℝN×nk, nk is the number of feature mappings. Pass all feature nodes through enhancement functions ηj(ZnWhj+γhj) to get the enhancement nodes H_j and its parameters are W_hj and γ_hj. The collection of all enhanced nodes is Hm≡[H1,…,Hm]∈ℝN×m where m is the number of enhancement nodes. Combine the feature nodes set and the enhanced nodes set to obtain a direct connection to the output node A = [Zⁿ|H^m].

The broad learning model is expressed by the following Equations (1) and (2):

Thus through the ridge regression of [Zⁿ|H^m]⁺ we can obtain the output weight

and λ is a regularization parameter. Figure 1 shows the structure diagram of the standard broad learning system, in which the spare nodes of the model can be quickly established by only calculating the pseudo-inverse of the added nodes.

In BLS, by minimizing the Equation (4)

we obtain the expression for the output (3). Although the BLS model can be solved quickly, it does not take into account the geometric structure of the underlying data. Based on the basic BLS structure, Jin et al. (2018) proposed graph regularized BLS (GBLS). Experiments have proven that the effect of IPGBLS is superior to GBLS in most cases (Jin et al., 2018), therefore IPGBLS is utilized in this article. Add the graph regularization E_G that represents the difference between the local structure of the data into the objective function of GBLS. The objective function is as follow:

and λ₁, λ₂ are parameters in the objective function (5) to decide which part is more important. Here, λ₁ and λ₂ are weight parameters used to adjust the significance of the EG term and the ||W||² term, respectively. GBLS exhibits a high degree of tolerance to the choice of λ₁ and λ₂, with optimal values obtained when λ₁ is relatively large. This suggests that the graph regularization term significantly enhances the BLS model.

The manifold regularization method is incorporated into the optimization process, enhancing the classification ability of GBLS compared to BLS. In one of the GBLS models, specifically IPGBLS, the intrinsic graph G^w(X, V^w) and penalty graph G^p(X, V^p) are added simultaneously. The vertex matrix of the intrinsic graph G^w is constructed in Equation (6)

where Nk1 is the k₁ neighbor set of x_i and k₁ represents the neighbors' number. The intrinsic graph's Laplacian matrix is Lw = Dw − Vw, where Dw = sum_jV_ijw. Consequently, the intrinsic graph's geometric structure can be shown as follows:

The penalty graph G^p is the same as G^w, the vertex matrix of G^p is shown in Equation (8):

and Nk2 is the k₂ neighbor set of x_i and k₂ is the neighbors' number. The penalty graph's Laplacian matrix is Lp = Dp − Vp, where Dp = sum_jV_ijp. As a result, the penalty graph's geometric structure may be shown as follows:

In order to make the model has the best classification effect, minimizing the data points of similar geometric structures, that is, minimizing Equation (7). At the same time, maximizing the distance between samples of different structures Equation (9). Thus, the objective is shown in Equation (10)

To ensure the L^p reversible, add a small enough disturbance ζ on the diagonal of L^p that does not affect the model solution. Defined the LIPGBLS≜((Lp)-1/2)TLw((Lp)-1/2). The final objective function of IPGBLS is shown in Equation (11)

Then the output weight of the model is:

The idea of differential evolution (DE) comes from the earliest genetic algorithm, which explores the optimal solution in space through mutation and crossover processes (Price, 2013). Differential evolution algorithm has a good effect on function optimization (Storn, 1996) and noise reduction (Price et al., 2006). At the same time, the differential evolution algorithm has also been fully applied in industries (Rocca et al., 2011) and other fields. Assuming that the objective function is f(θ), in order to minimize the objective function, differential evolution establishes a population of NP individuals, where the individual vector is θi,G=[θi,G1,θi,G2,…,θi,GD],i=1,2,…,NP.

Due to inconsistencies in the number of recording channels across subjects in the CHB-MIT and Patient datasets, only subjects with a consistent channel count were selected to ensure uniform feature dimensionality. The EEG signals were segmented into 2-s epochs with an overlap of 0.5 s. This choice of epoch length and overlap ratio was motivated by several considerations. A 2-s window provides a suitable time resolution to capture transient yet meaningful patterns in brain activity without obscuring rapid fluctuations. The 0.5-s overlap was introduced to augment the data and mitigate the potential loss of critical information at segment boundaries, thereby improving the model's ability to capture patterns spanning adjacent epochs. This segmentation approach, with a 2-s epoch length and moderate overlap, is a common practice in EEG signal processing.

Time domain features are crucial for EEG signal classification as they capture the temporal variations in EEG signals, provide a comprehensive description of the data, are easy to compute and interpret, and are sensitive to changes caused by various factors. Therefore, thirteen features in the time domain are extracted from each channel of both the normal and seizure groups. These features include mean, standard deviation (std), peak-to-peak (p2p), variance (var), minimum (min), maximum (max), argminim, argmaxim, mean-square, root mean square (rms), sum of absolute difference, skewness (skew), and kurtosis (kurt).

The GBLS model has many non-optimal parameters. In order to mitigate the negative impact of a high percentage of non-optimal nodes on the model's effectiveness, this paper proposes a combination of the adaptive differential evolution algorithm with GBLS.

Our proposed approach is used to detect epileptic seizures from EEG. To evaluate the effectiveness of our approach, we employ three publicly accessible datasets: CHB-MIT (Shoeb, 2009), Kaggle (2014), and Bonn (Andrzejak et al., 2001), along with a private dataset named Patient. As for the Bonn dataset, we used the F subset for interictal (non-seizure) periods and the S subset for seizure periods. Some sample signals of CHB-MIT and the patient are shown in Figures 3, 4. The descriptions of these datasets are briefly provided in Table 1.

To validate the proposed model, the K-fold cross-validation technique is employed. The entire dataset is partitioned into five equal-sized folds, where 1-fold is designated as the test set, and the remaining 4-folds constitute the training sets. Across all experiments, 80% of the samples were allocated for training, while the remaining 20% were designated for testing. The assessment of classification performance involved the use of metrics such as accuracy, precision, sensitivity, and F1-score. Calculations for all metrics were conducted using a default threshold value of 0.5. It is important to note that the threshold value can significantly impact the classification performance, particularly in scenarios with class imbalance or varying costs associated with false positives and false negatives. A higher threshold would prioritize precision, reducing false alarms at the potential cost of missing some true seizure events, while a lower threshold would favor recall, increasing seizure detection sensitivity but potentially increasing false positive rates. The choice of retaining the default threshold of 0.5 was primarily motivated by the exploratory nature of this work and the desire to establish a baseline performance without introducing additional complexity through threshold tuning. The evaluation metrics, including accuracy, precision, recall (sensitivity), and F1-score, as shown in Equations (18–21):

Here, the TP represents the number of correctly predicted seizure segments, while the FN indicates the number of seizure segments that were mistakenly classified as non-seizure segments. Likewise, the TN represents the number of correctly classified no-seizure segments, while the FP represents the number of no-seizure segments that were mistakenly identified as seizure segments. The F1 − score provides a comprehensive evaluation of the model's capability to accurately identify instances in unbalanced datasets by considering both accuracy and recall. Accuracy provides an overall measure of the model's performance in correctly classifying both seizure and non-seizure segments. However, in the clinical domain of epilepsy monitoring, the consequences of misclassification can be severe, necessitating a more nuanced assessment beyond mere accuracy. Precision quantifies the model's ability to avoid false positives, which correspond to falsely identifying non-seizure segments as seizures. High precision is crucial in this context, as false alarms can lead to unnecessary interventions, patient anxiety, and potential overmedication. Recall, on the other hand, reflects the model's capability to detect true positive cases, i.e., correctly identifying seizure segments. High recall is essential to ensure that no actual seizure events are missed, as missed detections can have grave implications for patient safety and timely treatment. The F1-score, being the harmonic mean of precision and recall, provides a balanced assessment by capturing both the model's ability to minimize false positives and its ability to identify true positives effectively. This composite metric is particularly valuable in the epilepsy detection context, where both precision and recall are equally critical. By evaluating the model's performance using these metrics, a comprehensive understanding of its strengths and weaknesses can be obtained.

We compared our approach against various classifiers commonly used in the field of electroencephalography signal classification, including K-Nearest Neighbors (KNN) (Amin et al., 2015; Shih et al., 2022), Decision Tree(DT) (Siddiqui et al., 2019; Shih et al., 2022), Random Forest(RF) (Sharma et al., 2018; Siddiqui et al., 2019), Adaboost, Gaussian Naive Bayes (GaussianNB) (Shih et al., 2022), EEGNet (Lawhern et al., 2018), Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), BLS and GBLS. We use the same data preprocessing and data input for all of the methods. The epochs in the deep learning methods are all set to 100. In addition, for the publicly available datasets CHB-MIT, Kaggle, and Bonn we have cited results from other papers for comparison, including Support Vector Machine(SVM) (Ein Shoka et al., 2021), Ensemble (Usman et al., 2021), and ANFIS-PSO (Shoeibi et al., 2022).KNN: Using the class labels of the K nearest labeled samples, an income-based learning method can identify the class label of an unlabeled sample by finding the K nearest labeled samples in the training set.DT: The goal of data screening and decision-making is achieved by constructing a mathematical model based on the characteristics of the data and utilizing the concept of classification.RF: Random Forest is a decision tree-based ensemble learning technique that combines multiple decision trees. Every time, a random selection of features is used as input, and the data collection is chosen randomly with replacement.Adaboost: Adaboost is a machine learning technique that combines several basic classifiers based on their error rate. The overall classification accuracy is significantly improved by linearly combining multiple weak classifiers and then voting based on the weight of each classifier.GaussianNB: This is one of the Naive Bayes algorithms.EEGNet: mIt is a deep learning architecture specifically designed for the analysis of electroencephalography (EEG) data. It can accurately classify EEG signals with high temporal resolution and robustness to noise.

LSTM: The Long Short-Term Memory (LSTM) model, which is a type of recurrent neural network (RNN), is known for its ability to effectively capture long-range dependencies and patterns in sequential data. Its unique architecture includes memory cells and gating mechanisms, enabling it to retain information for extended periods. LSTMs are widely used in various fields, especially in natural language processing and time series analysis, because of their ability to effectively model sequences and address the issue of vanishing gradients.GRU: The Gated Recurrent Unit (GRU) model is a variant of recurrent neural network (RNN) that is specifically designed for processing sequential data. GRUs are characterized by their simplified architecture compared to traditional RNNs and LSTMs. They feature gating mechanisms that regulate the flow of information through the network, enabling the capture of long-term dependencies while addressing vanishing gradient issues. GRUs have gained prominence in various applications, particularly in natural language processing and time series analysis, due to their computational efficiency and competitive performance in modeling sequential data.BLS: The BLS architecture first randomly maps the input into multiple feature spaces which comprise the feature nodes. The outputs of these feature nodes are then expanded extensively to constitute the enhancement nodes. Finally, the output weights of the network are analytically computed based on the mappings through the feature and enhancement nodes.GBLS: The GBLS takes into account the local invariance of the data and incorporates stream shape learning into the objective function of the standard BLS.SVM: SVMs are supervised learning models that find the optimal hyperplane that maximizes the margin between classes in the feature space, utilizing support vectors which are the data points closest to the decision boundary.Ensemble: An ensemble classifier that combines the output of SVM, CNN, and LSTM using model agnostic meta learning.ANFIS-PSO: A method of combining fuzzy theory and deep learning techniques and introducing particle swarm optimization algorithm for optimization.

Tables 2–5 present a comparison between our proposed approach and previous methods for the seizure detection task. The results of the 5-fold cross-validation are shown in Tables 6–9. We chose the best 1-fold results to compare with other models. In comparison to traditional machine learning methods, SaE-GBLS achieves the highest accuracy, precision, sensitivity, and F1-score across all four datasets. In comparison to deep learning, it is clear that both LSTM and SaE-GBLS exhibit excellent performance, on the CHB-MIT dataset, LSTM outperforms SaE-GBLS overall, suggesting that LSTM does have greater potential for time-series data, but SaE-GBLS also shows comparable results, as shown in Table 2.

As for the Kaggle dataset, Table 3 displays the results. In Table 3, SaE-GBLS achieves a 5.712% improvement in accuracy, 7.942% in precision, 2.691% in sensitivity, and 5.401% in F1-score compared to EEGNet.

And in Table 4, we can observe that SaE-GBLS outperforms EEGNet, LSTM, and GRU by a significant margin in all four metrics and achieves a perfect score of 1. In addition to SaE-GBLS, GBLS and BLS also achieved good performance. The small size of the Bonn dataset, which consists of only one sampling channel and relatively simple data, could contribute to this phenomenon. On the other hand, EEGNet, LSTM, and GRU are deep neural networks that may lead to overfitting. Meanwhile, SaE-GBLS inherits the advantages of BLS and has a simple structure, which makes it less likely to overfit even with small data and can yield superior outcomes.

Table 5 shows the results of the actual dataset named “Patient”. From Table 5, we can see that both LSTM and SaE-GBLS excel, but the latter performs more consistently. All in all, SaE-GBLS demonstrates the most consistent performance across the four datasets. Furthermore, other methods also demonstrate satisfactory performance on this dataset. To some extent, we can suggest that the data we collected is of good quality.

We can observe that deep learning methods and the integrated learning method Adaboost outperform other machine learning methods. However, on the Bonn dataset, which contains a small amount of data, EEGNet, LSTM, and GRU tend to overfit. On the other hand, our proposed SaE-GBLS performs well on all four datasets, surpassing the other methods overall. As can be seen from Tables 2, 4, SaE-GBLS achieves the highest values in all evaluation metrics on the CHB-MIT and Bonn datasets. Additionally, it achieves the highest accuracy, precision, and F1-score on the Kaggle dataset (see Table 3). In general, our method can effectively detect the onset of epileptic seizures.

To date, extensive research has been conducted on the diagnosis of epileptic seizures using artificial intelligence techniques. Due to the presence of complex structures, deep learning models excel at extracting the abstract and underlying characteristics of data. As a result, they have gained widespread popularity and are favored by numerous researchers. In a study conducted by the authors in Hossain et al. (2019), during their preliminary studies on diagnosing epileptic seizures, Taqi et al. (2017) introduced a novel 2D-CNN model that aimed to extract spectrum and time features from EEG recordings. This model was specifically designed to capture the overall structure of seizures. When applied to the Bern-Barcelona dataset, the model successfully extracted features and achieved remarkable results. However, it is worth noting that deep learning methods typically require a significant amount of data for training and consume a considerable amount of training time.

Our research is based on BLS, and because of BLS's ability to learn incrementally, our model can be updated dynamically when new data is added, eliminating the need for retraining. It is undeniable that BLS has certain limitations in terms of feature extraction and may not perform as well as deep learning methods. Firstly, the random mapping feature layer in BLS, although endowing the network with some nonlinear mapping capability, may have relatively limited feature extraction power compared to deep neural networks. Deep models can automatically learn hierarchical feature representations through multiple nonlinear transformations, ranging from low-level edges and textures to high-level semantic abstractions. Moreover, BLS's feature extraction process heavily relies on the generation of random mapping matrices, making it sensitive to changes in data distribution and quality. If the input data distribution undergoes significant shifts, the current random mapping matrices may fail to effectively capture the features of the new distribution, thereby impacting model performance. In contrast, deep learning models, through end-to-end training, exhibit a certain degree of adaptability to input distribution changes. There is a wide range of methods used for detecting epileptic seizures, but none of them can be definitively considered superior to others. The optimal choice of structure for seizure detection should be made with careful consideration of the dataset and the specific characteristics of the problem.

Although our approach has shown commendable effectiveness across the four datasets we have used, it is still hindered by certain limitations. For example, there is significant potential for enhancement in the design of the feature extraction module, and integrating this method into real-world clinical medical scenarios remains a challenging task. One potential limitation of the current feature extraction approach lies in its reliance solely on statistical features derived from the EEG signals. While these handcrafted features capture essential statistical properties, they may fail to fully characterize the intricate spatio-temporal patterns and nonlinear dynamics inherent in the epileptic brain activity. To address this limitation, future work could explore the incorporation of more sophisticated feature extraction techniques. First, connectivity measures quantifying the functional relationships between different brain regions, such as coherence and phase-locking values, could be incorporated. These features may provide insights into the propagation patterns of epileptic activity across the brain. Furthermore, the application of deep learning techniques for automated feature extraction could be investigated. Convolutional neural networks (CNNs) and recurrent neural networks (RNNs) have demonstrated remarkable ability to learn hierarchical representations directly from raw signal data, potentially capturing intricate spatio-temporal patterns that may be overlooked by handcrafted features. It is worth noting that the adoption of deep learning approaches may require larger datasets and computational resources, but could potentially lead to more robust and generalizable models by leveraging the ability of these methods to learn discriminative features directly from the data. It is our goal to continuously improve and implement it in real medical situations in our forthcoming endeavors.