OS-ELM

The OS-ELM algorithm was introduced by Liang et al. Its training process consists of two stages: an initialization phase and an online sequential learning phase. First, a small batch of data is used to compute the initial output matrix H₀ and output weights ; thereafter, these parameters are iteratively updated as new input samples arrive. The detailed procedure is as follows:

Given a dataset where each is an input vector (with n denoting its dimensionality) is an target vector (where m is the number of classes), and N is the total number of samples. We partition S into k disjoint data blocks; denote the k-th block by and let be the number of samples in that block. Select the first N₀ samples, as the initial block, where N₀ is the number of initial training samples. In the initialization phase, randomly generate the input-to-hidden weights and biases where L is the number of hidden‐layer nodes and j indexes the hidden node. Using the initial samples , construct the initial hidden‐layer output matrix

(1)

where is the activation output of the j-th hidden node given input . According to [27], the initial output weight is defined as:

(2)

where P₀ denotes the initial auxiliary matrix, in the initialization phase, it is defined as

During the online sequential learning phase, OS-ELM processes data in blocks. Suppose that when the k-th data block arrives, the hidden-layer output matrix and the corresponding target matrix are redefined as:

(3)

(4)

where is the number of samples in the i-th block. When the (k + 1)-th block arrives, these definitions become:

(5)

(6)

At this point, the auxiliary matrix and output weight vector are updated incrementally as new data chunks arrive. The recurrence relations are expressed as:

(7)

(8)

OS-ELM combines the speed and generalization advantages of ELM. As shown in Eqs. (7) and (8), its output weights are iteratively updated using the most recent result and incoming data, allowing the model to evolve continuously without retraining. This update strategy significantly reduces computational overhead and memory usage, making it ideal for online learning scenarios. Consequently, optimizing algorithms around OS-ELM has become a popular research direction in data stream classification.

Drift detection mechanism

In data streams, instances typically arrive at high speed and in unbounded volume. However, their underlying distribution or the associated labels may evolve, represented by [28], it signifies the emergence of a new concept. Here, X denotes a two-dimensional feature space, and y represents its corresponding label [29]. The Fig 1 illustrates three types of concept drift.

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Fig 1. Concept drift types.

https://doi.org/10.1371/journal.pone.0353728.g001

Research on concept drift is extensive and can be broadly divided into three categories: error rate, data distribution and hypothesis testing. DDM (Drift Detection Method) [30,31] and ECDD (Exponentially Weighted Moving Average Concept Drift Detection) [32] are typical error rate based algorithms that continuously feed all incoming training samples into the model and have demonstrated strong classification performance. Specifically, within a sliding time window, they compute the error rate between each new sample’s accurate label and its predicted label—an indicator known as the ’online average error rate’. When this error rate exceeds a predefined warning threshold, the system triggers and constructs a new classifier; only if the error rate further rises to reach the ’drift’ threshold does the new classifier replace the existing model. This mechanism ensures that the current model remains in use until an actual concept drift occurs.

Different from the above methods, conceptual drift detection based on data distribution treats the data of the past period (the reference window) and the latest arriving period of data (the detection window) as two sets of samples and determine whether their distributions have undergone a significant difference using a statistical test or a distance metric: if the two sets of data in terms of characteristic distributions (e.g., probability distributions of dimensions) or output distributions (e.g., class frequencies) the difference exceeds a predetermined threshold, concept drift is considered to have occurred. Otherwise, it is regarded as normal sampling fluctuation. For example, Guo et al [33] proposed a local drift detection for clustering, which compares the differences in data distributions between historical and current data blocks, and is used to capture new concepts and identify the corresponding drifted regions. Li et al [34] used Gaussian Mixture Models (GMMs) to obtain similarities between data distributions, which are used to assign weights to the base learner for fast adaptation to conceptual drifts.

Hypothesis testing methods employ statistical tests to determine whether observed errors (or distribution changes) are merely random fluctuations due to sampling variability or indicate genuine concept drift. For example, Sun et al [35] proposed two such detection techniques: EBTBM (Entropy-Based Thresholding Method) and EBBSM (Entropy-Based Sampling Method). Both approaches employ the uncertainty inherent in the target concept’s definition as a statistical measure of the difference between two consecutive data blocks.

Class imbalanced learning

In data streams, a common issue is class imbalance within the dataset, where one or more classes are significantly underrepresented compared to others. This imbalance becomes particularly problematic when the minority class carries more critical information since models built on such data incur higher classification costs in real-time applications [36]. Furthermore, imbalance not only complicates the construction of effective classifiers but also necessitates specialized algorithms to mitigate bias and ensure robust performance in all classes. Consequently, designing classifiers for imbalanced datasets remains a significant challenge in data mining.

To the best of our knowledge, popular approaches to addressing class imbalance can be categorized into three types: data-level, algorithm-level, and hybrid strategies [37]. At the data level, random undersampling and oversampling are two commonly used techniques. They balance class proportions by either reducing the number of majority‐class samples or increasing the number of minority‐class samples. Payel Sadhukhan [38] introduced a novel undersampling strategy applied to synthetic minority sets. Sun et al. [39] proposed an undersampling method that incorporates minority‐class density information: they use kernel density estimation to learn the probability density of minority samples, filter out majority samples located in high‐density regions of the minority class, and then define a ’sampling fitness’ measure to evaluate each majority sample’s desirability—thereby selecting the most informative examples. Others [40] have proposed an adaptive weighting and nearest‐neighbor–based region‐control method to address insufficient noise filtering in oversampling, difficulty in interpreting sample importance, and exacerbated class overlap.

Unlike the data-level approaches, algorithm-level methods start by modifying the classification algorithm itself. Zhu et al. [41] addressed the imbalanced data problem by considering class distribution through a novel hybrid resampling method, MSHR, and an improved cost-sensitive SVM model. Similarly, Liang et al. [42] introduced a pre-grouping strategy into the SVM framework to optimize data distribution and enhance the quality of minority-class samples. Moreover, Dai et al. [43] proposed a genetic algorithm that selects the optimal combination of heterogeneous clustering techniques—guided by fitness functions—to resolve class-overlap problems.

Hybrid strategies combine the advantages of both data-level and algorithm-level methods. Li et al. [44] incorporated the density information of training samples into the class‐imbalance ratio, thereby assigning different weights to samples even within the same class. By adopting the Learn++ algorithm [45], Ditzler et al. [46] further adjusted each base classifier’s voting weight according to the imbalance ratios observed in both current and historical data.

More and more research focuses on imbalanced learning [47], especially in real-world applications. Mingkuan Shi et al. [48] addressed imbalanced industrial data streams by leveraging prior distribution information to strengthen the classification decision boundary between majority and minority classes, thereby proposing a novel imbalance-aware learning system. Similarly, Wu et al [49] employed a feature‐based resampling approach to correct classifier bias induced by class imbalance.

Algorithm 1 The flowchart of EOS‐PIELM

  Input:

   S: data stream

   : k‐th data block

   E: ensemble classifier

   M: number of base classifiers

  Output:

   : updated E

1: perform

2:   by

3:  while do

4:   perform

5:    by

6:   if perform drift detection → P₀ compared P₁ then

7:    perform drift adjust → update

8:    perform adjust CF

9:   end if

10:   return and CF → update

11:  end while

OS-PIELM

To improve the nonlinear representation capability of ELM, various feature mapping methods have been proposed, among which Kernel ELM [50] and random feature-based ELM [51] are representative approaches. By mapping the original samples into a high-dimensional feature space, these methods can effectively enhance class separability and improve the approximation ability of ELM for complex nonlinear problems. Kernel ELM enhances nonlinear representation by constructing a kernel matrix that characterizes pairwise relationships among training samples. Consequently, the feature representation depends on the entire training dataset, and the nonlinear mapping process is achieved through kernel matrix operations. Although this strategy can improve classification performance, it requires kernel matrix construction and storage, making it difficult to efficiently accommodate continuously arriving data in online sequential learning scenarios. Random feature-based ELM methods improve nonlinear representation by projecting original samples into a higher-dimensional space through randomly generated mapping parameters. While these methods avoid explicit kernel matrix construction, their mapping performance is strongly influenced by randomly initialized parameters and generally follows a fixed transformation mechanism after model initialization.

In addition, several online feature-mapping methods, such as Online Sequential Kernel ELM (OS-KELM) [52] and online random feature-based learners, extend nonlinear feature mapping to streaming environments. However, these methods still rely on either kernel-based sample relationship modeling or randomly generated feature transformations. Different from the above approaches, the proposed pre-interference layer neither constructs a kernel matrix nor employs random feature mapping. Instead, it directly performs nonlinear feature reconstruction on each incoming sample before hidden-layer learning. Specifically, a set of interference nodes is introduced to transform the original input sample into a new feature representation, thereby enhancing feature separability in the transformed space. Since the transformation is performed on individual samples rather than on pairwise sample relationships, the proposed method does not depend on the entire training dataset and can naturally preserve the sequential updating mechanism of OS-ELM. Motivated by this idea, a novel pre-interference layer is inserted between the input layer and hidden layer of the original OS-ELM network, resulting in a four-layer feedforward neural network architecture. The layer consists of a set of sigmoid kernel functions, where all connection weights between the input layer and the enhancement layer are set to 1. when an input sample x passes through this interference layer, it is transformed into into a new representation (see Fig 3). In the figure, the upper part contains a pre-interference layer with two nodes, whereas the lower part contains a pre-interference layer with three nodes. The orange rectangles and green circles represent two different classes of sample data points. It can be observed that, through nonlinear kernel mapping of the original samples, the distribution complexity of the samples in the kernel space is improved. From the perspective of function approximation, a single hidden layer network theoretically possesses universal approximation capability. However, under the condition of a limited number of hidden nodes, the approximation efficiency depends on the complexity of the target function in the input space. If the mapping can nonlinearly reconstruct the original samples so that they exhibit a clearer class structure in the new space, then the complexity of the target function in the mapped space is reduced. This helps alleviate the optimization burden of the output weights in the subsequent OS-ELM network.

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Fig 3. Kernel mapping process.

https://doi.org/10.1371/journal.pone.0353728.g003

From the perspective of feature separability, let the inter-class distance between two classes in the original space be , and the intra-class scatter be . The discriminative ability can be expressed using the Fisher discriminant ratio as:

(9)

The subspace mapping is a kind of feature space transformation. If the mapping increases the inter-class distance and reduces the intra-class variance, then the discriminant ratio J in the transformed space increases, thereby improving the separability of the samples. In this case, OS-ELM performs linear combination in this space, which enables more stable classification and regression.

Algorithm 2 OS‐PIELM

  Input:

   : initial data block

   : k + 1-th data block

   : pre-interference layer activation function

   : hidden layer activation function

  Output:

  Initialization:

1:  perform pre-interference

2:  by eq. (9):

3:  by eq. (2):

  Online learning:

1:  while do

2:   perform pre-interference

3:   by eq. (10):

4:   if then

5:    by eqs. (11) and (12):

6:   else

7:    by eqs. (7) and (8):

8:   end if

9:   return

10:  end while

By using as the input vector to the OS-PIELM hidden layer, m samples are chosen during the initialization phase to recompute the initial output matrix, which is expressed as:

(10)

When the hidden‐layer output matrix H₀ is substituted into equation (2), the initial output weight can be obtained.

In the context of online sequential learning—where data arrive in batches and exhibit time‐varying characteristics with weak inter‐sample correlations—each time a new (k + 1)-th data batch becomes available, we construct the hidden‐layer output matrix for that batch as follows:

(11)

At this point, we update the inverse‐correlation matrix and the output weight using the recursive formulas Eqs. (7) and (8). In practice, the size of the (k + 1)-th batch may differ from one. When following references [53,54], equations Eqs. (7) and (8) can be rewritten in scalar‐update form as

(12)

(13)

where the After all incoming data batches have been processed, the final output weight is obtained, yielding the complete OS-PIELM model. Detailed procedural steps are provided in Algorithm 2.

Although the pre-interference layer in OS-PIELM can be used to perform nonlinear kernel mapping on the online sequential samples to enhance the discrimination of different features, the class labels of the data may change rapidly after the data stream undergoes conceptual drift, and the original model cannot respond and update in a timely manner; At the same time, if the original data suffers from severe class imbalance, the majority-class samples mapped into the high-dimensional space will still dominate, which biases the classification boundary toward the majority class and weakens the model’s ability to recognize the minority class, the majority class samples mapped into the high-dimensional space by the pre-interference layer will still dominate, which further biases the classification boundary towards the majority class and weakens the model’s ability to recognize the minority class. In order to ensure that the model can quickly respond to the changes of distribution and labeling in the data stream while taking into account the detection performance of the minority class samples, it is necessary to introduce a concept drift detection and class imbalance learning strategy.

Adaptive forgetting factor and concept drift detection

When data drift occurs, sample labels tend to change. Overstoring historical information only reduces the model’s ability to predict new data. To address this, we introduce a forgetting factor in the hidden-layer-output weight update of OS-PIELM and, on that basis, derive a new update formula. This ensures that the algorithm can adapt quickly and maintain classification performance when facing concept drift.

By incorporating the forgetting factor into Eq. (10), so that new and old data batches carry different weights, Eq. (10) becomes:

(14)

When , the model degenerates to the original OS-PIELM.

Although in theory OS-PIELM can accommodate various degrees of concept drift by adjusting the forgetting factor, it still suffers from several shortcomings:

1. A fixed forgetting factor cannot strike a balance between discarding outdated information and retaining useful historical knowledge.
2. While it adapts well to gradual drift (slow, continuous changes in the data distribution), it cannot adjust quickly enough during abrupt drift (especially concept reversals), resulting in delayed updates and a significant drop in classification performance.
3. In most real-time datasets, the type of concept drift is unknown, so it is impossible to tune the forgetting factor optimally for each specific drift pattern.
4. Compared with classical drift detection methods such as DDM and ADWIN, the proposed Gmean-based mechanism exhibits distinct advantages in imbalanced data stream scenarios. DDM primarily relies on monitoring the overall error rate, which may be dominated by the majority class and thus fail to reflect performance degradation on minority classes. ADWIN detects changes by analyzing data distribution shifts within adaptive windows, but it does not explicitly consider class-wise performance.

Algorithm 3 Based Gmean Drift Detection Mechanism

 Input:

  : default forgetting factor

  : concept drift detection threshold

  : predicted Gmean for the k-th data block

  : historical max Gmean

  i: total number of data blocks

 Output:

  new(): updated

1:  for k = 1 to i do

2:   Compare and

3:   if then

4:    

5:    if then

6:     

7:    else

8:     

9:    end if

10:   end if

11:  end for

To address the above issues, we incorporate a concept drift detection mechanism based on the geometric mean (Gmean) into OS-PIELM. Since Gmean simultaneously captures the classification performance of both positive and negative classes and reflects the balance between majority and minority classes, it is less sensitive to performance fluctuations dominated by the majority class compared to metrics that rely solely on overall error rate. As a result, it can effectively suppress spurious drift alarms caused by transient variations in error rates, while also mitigating false alarms and missed detections commonly observed in imbalanced data scenarios. When the Gmean value falls below a predefined threshold, concept drift is detected, and the decay rate of the forgetting factor is adaptively increased. This mechanism assigns higher weights to newly arriving samples, thereby enabling the model to respond more rapidly to abrupt environmental changes. The derivation for setting this threshold is as follows:

By applying Hoeffding’s inequality to concept drift detection, the relationship between the allowable performance degradation threshold and the sample size n at significance level is derived:

(15)

where , setting the right-hand side equal to the significance level yields:

(16)

this controls the detection confidence level. Considering the historical maximum Gmean_max as the expected performance , and define the relative deviation to characterize the magnitude of performance degradation:

(17)

Using typical parameters and , the threshold is obtained as:

(18)

In the OS-PIELM algorithm, the default forgetting factor is initialized at . When the Concept Drift Index (CDI) satisfies CDI , the forgetting factor is updated according to:

(19)

where the Concept Drift Index is defined as . Consequently, under concept drift detection, the ranges within [0.9, 0.99], with the update rule defined as:

(20)

When concept drift leads to a decline in classifier performance, the CDI value decreases, thereby reducing the forgetting factor. Unlike traditional drift detection methods that mainly rely on changes in classification accuracy or error rate, the proposed approach uses Gmean as the drift evaluation metric. Since Gmean considers the recognition performance of both majority and minority classes, the drift detection process can not only reflect distribution changes caused by concept drift, but also capture performance degradation induced by class imbalance. Therefore, the proposed Gmean-based drift detector can uniformly evaluate the model state in imbalanced data stream environments and provide more effective feedback for subsequent adaptive model adjustment. The detected drift information is further used to adaptively adjust the forgetting factor, enabling the model to more appropriately balance the influence of historical data and newly arrived data. steps are provided in Algorithm 3.

Dynamic weighting strategy

In real‐world applications, class imbalance and concept drift often occur simultaneously. Concept drift causes the data distribution and class labels to change over time. Under class imbalance—where the number of majority‐class samples far exceeds that of the minority class—traditional classifiers tend to shift the decision boundary toward the majority class, reducing sensitivity to the minority class. In this scenario, misclassifying minority‐class instances is not only easier to overlook but also typically incurs a higher cost than misclassifying majority‐class instances. To address this, we build on concept drift detection by incorporating a dynamic weighting strategy: by assigning weights to both the initial training samples and incoming online samples, we mitigate the bias introduced by class imbalance.

During the initialization phase, the imbalance ratio (IR) is computed from the proportion of minority class samples to majority class samples in the initial data chunk. The minority class sample weight W₀ is set to IR, while the majority class weight is fixed at 1. This weighting scheme amplifies the influence of minority class samples on model fitting within the initial matrix H₀. Using the recursive least squares (RLS) method, we derive the recurrence relation for :

(21)

Where is the diagonal weight matrix, C denotes the regularization parameter, I represents the identity matrix, and is the target vector.The initial output weight is therefore given by:

(22)

(23)

During the online learning phase, when a new batch of examples arrives, we first calculate the class imbalance ratio for the current data block and assign the sample weight as:

(24)

Then we construct the hidden-layer output vector:

(25)

using the Woodbury matrix identity, the recursive update formula for the projection matrix is derived:

(26)

the recursive update for the output weight during online learning is:

(27)

where,

Online ensemble learning algorithm

To enhance algorithm robustness for complex real-time data streams, we propose an online ensemble EOS-PIELM algorithm. Building upon the OS-PIELM base classifier, this approach integrates an adaptive forgetting factor, concept drift detection mechanism, and dynamic weighting strategy.

At initialization phase, we construct an ensemble of M OS-PIELM base classifiers. Each base classifier has a different number of pre-interference layer neurons and hidden neurons, specifically , and each uses one of the activation functions sigmoid,softplus,tanh, in this experiment, M was set to 12. By combining different hidden‐layer sizes with different activation functions, we ensure that the base classifiers differ both in structural complexity and in their nonlinear‐mapping capabilities.

Given a data stream of N₀ samples, we train all M base classifiers on these samples to obtain their initial weights . Once the first batch of examples has been seen, new samples are classified using a weighted‐voting scheme. The prediction for each instance is given by

(28)

where is the class‐vote (i.e., the output) of the m-th base classifier on input , is the voting weight of the m-th base classifier in the current round. In the very first round of voting, we set for all classifiers. The final predicted label is the class with the highest total vote. This weighted‐voting mechanism helps mitigate the large errors that might arise from a single classifier’s local bias.

To enable the ensemble classifier to better adapt to changes in the data stream – thus striking an effective balance between fully utilizing recent information and rapidly forgetting outdated knowledge – we adjust the class weights of the ensemble during the online learning phase. First, based on the concept drift detection result, we apply a forgetting factor to the entire ensemble model and incorporate it into the confusion matrix CM. This ensures that, when the new sample arrives in iteration k + 1, the matrix is updated in an exponentially decayed fashion, thereby emphasizing the impact of recent observations:

(29)

Next, in the decayed matrix, we increment by one the cell corresponding to the true label and predicted label of the current sample:

(30)

We then update each base classifier’s voting weight according to its confusion‐matrix performance:

(31)

Because base classifiers vary in their sensitivity to the minority class, we introduce a class‐weight correction factor for the minority‐class weight :

(32)

where . Finally, the updated weight of each base classifier:

(33)

The schematic of the complete update procedure is shown in Fig 4.

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Fig 4. Flowchart of the adaptive update process.

https://doi.org/10.1371/journal.pone.0353728.g004

Time complexity analysis

This subsection analyzes the computational complexity of EOS-PIELM from the perspective of time complexity. Since the proportion of data required in the initialization phase accounts for less than 3% of the total data in the experiments, and data streams are continuously generated, we mainly focus on the time complexity of the online learning phase.

For the standard OS-ELM, the per-instance computational cost mainly includes hidden layer output computation and recursive parameter updates, resulting in a time complexity of O(dL + L²). Therefore, for a data stream with N instances, the overall time complexity is O(N(dL + L²)).

In OS-PIELM, a pre-interference layer is introduced before the hidden layer, which adds an additional mapping cost of , where m is the number of kernels. Since this term is typically dominated by O(dL), the overall per-instance complexity remains O(dL + L²). Let T₁ denote the per-instance computational cost of OS-PIELM, i.e., . Therefore, OS-PIELM maintains the same order of time complexity as OS-ELM.

For the EOS-PIELM algorithm, an ensemble learning strategy with M base classifiers is adopted. Each classifier performs prediction and incremental updating, and their outputs are aggregated through a voting mechanism. Therefore, the total computational cost consists of three parts: the prediction and update cost of base classifiers, i.e., , the voting cost O(M), and the component update cost O(M). The overall time complexity can thus be expressed as Since the voting and component update operations mainly involve simple aggregation and bookkeeping, their computational costs are much smaller than that of model training and updating, i.e., . Therefore, the overall time complexity of EOS-PIELM can be simplified as

Compared with the standard OS-ELM, EOS-PIELM increases the per-instance computational cost from O(dL + L²) to O(M(dL + L²)) due to the ensemble learning mechanism. However, since M is typically a small constant, the additional overhead grows linearly and remains computationally efficient. Therefore, EOS-PIELM achieves improved performance while maintaining a comparable computational complexity to OS-ELM.

Datasets

The characteristics of 9 synthetic data streams and 2 real-time data streams are listed in Table 1. All synthetic data streams are generated by the Sea and Sine generators from the Scikit-Multiflow library, and their parameters can be customized.

1. (1) Sine_IR generator [55]: This dataset comprises a stream of binary classification synthetic data with 20000 samples, each sample has four features, the class label are binary, and in various experiments the imbalance ratio IR is set to 2, 4, or 9, with abrupt concept drift simulated by flipping the decision boundary at the sample indices 5000, 10000, and 15000, with the drift width set to 1, meaning that the change in the data distribution occurs instantaneously.
2. (2) Sea_IR_T generator [56]: This dataset is designed based on the SEA algorithm to simulate concept drift by dynamically adjusting decision thresholds. The resulting dataset contains 3 attributes (only 2 are relevant), and each stream consists of 20,000 instances with 3% noise added. We generate both abrupt-drift and gradual-drift streams, each containing 20,000 samples, with the imbalance ratio (IR) set to 2, 4, or 9. Concept drift occurs at sample indices 5000 and 15000. For the abrupt-drift setting (T = a), the drift width is set to 1, indicating that the change in the data distribution occurs instantaneously. For the gradual-drift setting (T = g), the drift width is set to 1000, meaning that the transition between concepts occurs progressively over a window of samples.
3. (3) Sine_4_N generator [55]: here N indicates the total number of samples; we generate datasets with to evaluate algorithm performance on large-scale data, width is set to 1000, meaning that the transition between concepts occurs progressively over a window of samples.

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Table 1. Dataset summary.

https://doi.org/10.1371/journal.pone.0353728.t001

Considering that drift and imbalance ratio are Unknown in real-time data, this paper only introduces their basic characteristics.

1. (4) Weather Dataset [57]: This dataset contains weather information collected in Bellevue, Nebraska, from 1949 to 1999 and comprises 18159 instances. It includes eight relevant attributes, and the objective is to predict whether it will rain on a given date.
2. (5) Elec Dataset [57]: This is a widely used real‐world dataset in data‐stream learning. It consists of partial records from the New South Wales electricity market in Australia, spanning 1995–1998, and contains 45312 instances. The dataset includes six relevant attributes. Since electricity prices there fluctuate according to supply and demand rather than remaining fixed, the goal is to predict daily price movements (1 = increase, 0 = decrease).

Comparison methods

In this study, we compare EOS-PIELM with six representative online learning algorithms, which are briefly described as follows:

• WOS-ELM [24]: A weighted OS-ELM approach that dynamically adjusts sample weights based on class distribution, improving minority-class recognition.
• VWOS-ELM [25]: VWOSELM is an improved OS-ELM method designed for imbalanced data stream classification. By introducing a weighting strategy, it increases the influence of minority-class samples during model updating, while the voting mechanism enhances the overall decision performance. In this way, VWOSELM alleviates the tendency of standard OS-ELM to favor majority classes. However, VWOSELM mainly focuses on class imbalance handling and does not explicitly model concept drift, which may limit its adaptability in dynamically evolving streaming environments.
• FROS-ELM [21]: The forgetting factor-based OS-ELM introduces a forgetting mechanism to gradually reduce the influence of outdated data, thereby enhancing the model’s adaptability to evolving data streams.
• OS-ELM [26]: The standard online sequential extreme learning machine that incrementally updates model parameters as new data arrives. It serves as a baseline for evaluating incremental learning performance.
• LPP [58]: Proposed by Elwell et al. (2011), LPP is an ensemble classifier designed for non-stationary environments (NSE). It utilizes a unique multi-classifier voting mechanism to incrementally learn from streaming data while adapting to concept drift.
• SRP [55]: Introduced by Gomes et al. (2019), SRP combines bagging and random subspace techniques for online ensemble learning. The default base learner is the Hoeffding tree, and the algorithm employs ADWIN for drift detection. SRP can also accommodate alternative base classifiers, providing flexibility for different streaming scenarios.

These models represent a diverse set of strategies for handling class imbalance, concept drift, and high-dimensional streaming data. By comparing EOS-PIELM against these methods, we aim to demonstrate the advantages of our proposed approach in simultaneously addressing both concept drift and class imbalance in dynamic data streams.

Evaluation metrics

To quantitatively compare and analyze algorithm performance, this section describes the evaluation metrics used in experiments:

1. Accuracy (Acc): Measures the proportion of correct predictions among all predictions, reflecting the model’s classification capability under current concepts.

(34)

2. Recall Rate (Rec): Measures the proportion of correctly predicted positive samples among all actual positive samples, indicating the model’s coverage of minority class samples.

(35)

3. Specificity (Spe): Measures the proportion of correctly identified negative samples among all actual negative samples, reflecting the model’s ability to exclude majority class samples.

(36)

4. Gmean: Represents the geometric mean of Recall and Specificity, indicating the model’s balanced performance across both classes.

(37)

5. Classification Distance D(Rec, Spe): Measures the absolute difference between Recall and Specificity, quantifying the model’s bias in class-wise recognition capability.

(38)

For classification tasks, accuracy (Acc) is the most widely used performance metric, as it measures how well an algorithm labels the overall sample set. However, when a data stream exhibits class imbalance, accuracy is no longer an ideal indicator. The Gmean [59] metric proposed by Kubat and colleagues captures a classifier’s overall performance and is considered the most important measure for imbalanced data stream classification. Therefore, in our experiments, greater emphasis is placed on Gmean performance.

Analyzing the sensitivity of parameters

The classification performance of a data stream with concept drift and class imbalance mainly depends on two parameters: the block size and the imbalance‐change threshold . We select these parameters from the following ranges: According to the report of the previous literature [60], we set the default and then conduct a series of experiments (for different values) on all datasets except Sine_4_N.

From the Fig 5, As shown in Fig 5, the parameter significantly influences the stability–adaptivity trade-off of EOS-PIELM. When , the model becomes overly sensitive to minor fluctuations in class distribution, leading to large oscillations in . Conversely, when , the model reacts too slowly to real imbalance changes, resulting in delayed adaptation. Notably, the interval yields consistently lower across both synthetic and real datasets, indicating a balanced performance between minority and majority classes. As shown in Fig 6, increases, the Gmean initially improves due to reduced variance and more reliable parameter estimation. However, beyond a certain point (approximately = 150), the Gmean begins to decline. This is because excessively large blocks reduce the update frequency, weakening the model’s responsiveness to concept drift.

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Fig 5. D(Rec,Spe) of EOS-PIELM under different for synthetic and real-time data streams.

(a) SINE, (b) SEA_Abrupt, (c) SEA_Gradual, (d) Weather-Elec.

https://doi.org/10.1371/journal.pone.0353728.g005

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Fig 6. Gmean of EOS-PIELM under different sizes of data chunks for synthetic and real-time data streams.

https://doi.org/10.1371/journal.pone.0353728.g006

The optimal combinations of parameters on 11 data streams are obtained as follows:

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https://doi.org/10.1371/journal.pone.0353728.t007

Analyzing contribution of components

To assess how the pre-interference layer in EOS-PIELM enhances both classification performance and concept drift detection, we ran experiments on all datasets except Sine_4_N. Keeping every other condition constant, we then replaced EOS-PIELM’s OS-PIELM base classifier with OS-ELM renaming the resulting model EOS-ELM and compared their results.

The experimental results listed in Table 2 show that, on both synthetic and real datasets, the pre-interference layer brings considerable improvements in classification performance. In particular, the relatively high Gmean values indicate that introducing the pre-interference layer enhances the algorithm’s adaptability to imbalanced data streams and to those undergoing concept drift. By comparing classification performance on synthetic and real datasets under identical conditions, EOS-PIELM exhibits notably superior results when handling high-dimensional and complex data. Furthermore, taking the Weather dataset from the real data as an example, we present its real-time Gmean variation plot. From Fig 7, EOS-PIELM exhibits faster recovery after drift points and smaller performance degradation than EOS-ELM. This indicates that the proposed Gmean-based drift detector and adaptive forgetting mechanism enable the model to rapidly adapt to distribution changes while preserving stable classification performance. in terms of time-wise Gmean. In particular, at drift points, EOS-PIELM exhibits faster recovery and smaller performance degradation, indicating its superior adaptability to abrupt distribution changes. Furthermore, the fluctuations in Gmean are significantly reduced, demonstrating improved stability under noisy and imbalanced conditions. These results confirm that the proposed interference layer, combined with adaptive forgetting and dynamic weighting mechanisms, effectively enhances both drift detection and class imbalance handling.

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Table 2. Comparison of EOS-PIELM and EOS-ELM.

https://doi.org/10.1371/journal.pone.0353728.t002

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Fig 7. Time-wise Gmean Comparison between EOS-PIELM and EOS-ELM.

[a] Result of EOS-ELM. [b] Result of EOS-PIELM.

https://doi.org/10.1371/journal.pone.0353728.g007

Comparison with other methods

As shown in Table 3 and 4, our proposed EOS-PIELM method outperforms the other six online classifiers: VWOS-ELM [25], WOS-ELM [24], FROS-ELM [61], OS-ELM [26], LPP [58], and SRP [55] in most data streams. We evaluated all algorithms using six performance metrics (Accuracy, Recall, Specificity, Gmean and D(Rec, Spe)), and EOS-PIELM consistently achieved the highest Gmean across most datasets. Since Gmean simultaneously evaluates the recognition performance of majority and minority classes, the superior Gmean values indicate that the proposed dynamic weighting strategy effectively alleviates the bias toward majority classes and improves the classification performance of minority-class samples under imbalanced data streams. Furthermore, EOS-PIELM obtained the smallest D(Rec,Spe) values in most datasets. This demonstrates that the proposed method maintains a more balanced recognition capability between majority and minority classes, further confirming its effectiveness in handling class imbalance.

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Table 3. Performance Comparison of all methods.

https://doi.org/10.1371/journal.pone.0353728.t003

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Table 4. Performance Comparison of all methods.

https://doi.org/10.1371/journal.pone.0353728.t004

On nine synthetic datasets (Sine_IR, Sea_IR_a, and Sea_IR_g among them), four algorithms: FROS-ELM, OS-ELM, LPP, and SRP exhibited rapidly declining minority-class accuracy (Specificity) as the imbalance ratio (IR) increased. In those scenarios, D(Rec, Spe) widened substantially and Gmean dropped sharply, indicating that these four methods lack the ability to adapt to class imbalance. By contrast, EOS-PIELM demonstrated a significant advantage on the Sine datasets in both Gmean and D(Rec, Spe). This illustrates that our incorporation of concept-drift detection and minority-class weighting enables the classifier to react quickly to abrupt concept drifts and adjust its model in a timely manner. On the Sea datasets, EOS-PIELM maintained strong minority-class accuracy and achieved superior Gmean values compared to the other algorithms. These results confirm that EOS-PIELM can effectively handle both gradual and abrupt concept drifts in streaming data.

Based on the statistical analysis framework, we employed the Friedman test to evaluate the experimental results in terms of Gmean and D(Rec,Spe). The p-value for Gmean is , and the p-value for D(Rec,Spe) is . The statistical results confirm that there are significant differences among all the compared algorithms. To further evaluate the statistical significance of the performance improvements, the Wilcoxon signed-rank test was conducted across multiple datasets, as shown in Table 5. The results indicate that the proposed EOS-PIELM achieves statistically significant improvements over most baseline methods in terms of both Gmean and D(Rec,Spe). Specifically, for VWOS-ELM, EOS-PIELM shows significant improvement in both metrics (p < 0.05), demonstrating its superior capability in handling imbalanced data streams. In comparison with FROS-ELM, OS-ELM, LPP, and SRP, the proposed method achieves highly significant improvements, indicating its strong and consistent performance advantages across different datasets. For WOS-ELM, EOS-PIELM exhibits statistically significant improvement in Gmean, while the difference in D(Rec,Spe) is not statistically significant. This suggests that although the two methods have comparable performance in terms of class-wise balance consistency, EOS-PIELM provides more stable and reliable classification performance overall. Overall, the statistical analysis confirms that EOS-PIELM not only improves the average performance but also achieves consistent and statistically significant superiority across multiple data stream scenarios, validating its effectiveness in addressing class imbalance and concept drift simultaneously.

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Table 5. Wilcoxon signed-rank test results of EOS-PIELM against baseline methods.

https://doi.org/10.1371/journal.pone.0353728.t005

Dealing with complex, real-time data streams is one of the biggest challenges in online classification. On the Weather and Elec real-time datasets, EOS-PIELM again performed very well: its Gmean values were at least 5–10 percentage points higher than those of the competing algorithms. This demonstrates that EOS-PIELM can maintain robust classification performance in complex, real-time streams. Moreover, as shown in Table 6, EOS-PIELM remains stable and continues to deliver excellent performance even when confronted with large-scale datasets.

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Table 6. Performance of EOS-PIELM in large-scale datasets.

https://doi.org/10.1371/journal.pone.0353728.t006