In practical applications, data quality is often difficult to guarantee due to missing feature values, making it exceptionally challenging to screen high-quality features from feature streams. Taking a medical monitoring system as an example, if a sensor fails and causes data loss, traditional OSFS models may transmit erroneous signals to control devices, which could ultimately endanger patients' lives. Therefore, preprocessing the data before feature selection to impute missing entries is of crucial importance. The LFA model holds significant value for missing-data imputation, as it completes missing values by mapping the sparse matrix onto two latent factor matrices (Li J. et al., 2025; Chen J. et al., 2024; Yuan et al., 2025). Traditional methods—such as mean imputation and matrix factorization—typically fill missing values based on observed data and rely on assumptions such as linearity or local similarity, which limits their ability to capture complex non-linear relationships in high-dimensional or sparse streaming data (Chen et al., 2023; Xu et al., 2025b). In contrast, the LFA model can capture the underlying structure of the data through latent space modeling, thereby handling complex dependencies and non-linear patterns more effectively (Liao et al., 2025; Lin M. et al., 2025).

The complete latent features extracted from incomplete data can be used for missing-value imputation, classification, clustering, and other tasks (Wu et al., 2025b; Lyu et al., 2026). The extraction methods are mainly divided into linear and non-linear feature extraction. Linear feature extraction mostly employs LFA-based models that rely on matrix factorization. When dealing with sparse data, such models aim to construct a low-rank approximation of the high-dimensional incomplete matrix (Wu et al., 2024; Wu H. et al., 2025). They map the known entries of the target high-dimensional incomplete matrix to its row and column nodes, formulate a learning objective that measures the discrepancy between the actual data and the estimated data, and thereby generate a complete low-rank approximation matrix of the target high-dimensional incomplete matrix. An optimizer is then used to minimize linear error, achieving efficient representation (Wu et al., 2025a; Qin et al., 2024; Xu et al., 2025a; Chen M. et al., 2024).

The initialization procedure subjects both matrices U and V to small random values. These values, generated by scaling a random permutation to a vicinity close to zero, serve as the non-zero starting point for the iterative algorithm, for which initial conditions are crucial. The following illustration details the update method, taking matrix U as a representative case.

The LFA model constructs a low-rank approximation for R (Tang et al., 2024; Luo et al., 2021a). Typically, matrices U and V are derived from R by minimizing a loss function defined by the Euclidean distance between R and R^ (Xu et al., 2023). Building upon Equations 2, 3, the complete streaming features are predicted using the known values according to:

Subsequently, the loss function for the m-th element f_{m, j} is calculated as:

To solve this loss function, stochastic gradient descent (SGD) is employed (Ahmadian et al., 2025; Lin X. et al., 2025; Lei et al., 2024). The method computes the gradient of the loss function with respect to the combined parameters and updates them in a descending direction:

From Equations 7 and 8, the partial derivative of the loss is derived:

Here, η denotes the learning rate. U and V are optimized to minimize errors on the known values, yielding R = UV^T. The error between the estimated and actual data is errm,j=fm,j′-∑k=1Lum,kvj,k, i.e.,

The principal advantage of GA-OS²FS lies in its independence from missing value completion via an LFA model. The method sustains a feature subset whose fitness is evaluated through real classification accuracy. Consequently, the search process gains the capacity to tolerate and sidestep local misleading associations stemming from potential completion errors. As a prominent and widely implemented evolutionary optimization method, the GA offers several compelling advantages. GA maintains a diverse population of candidate solutions, enabling simultaneous exploration of multiple regions in the solution space. Through mechanisms such as selection, crossover, and mutation, it effectively combines and propagates beneficial gene patterns while continually introducing new variations. This population-based strategy significantly mitigates the risk of premature convergence to local optima, making GA particularly robust in navigating complex, multimodal search landscapes commonly encountered in feature selection. GA operates solely on the evaluation of candidate fitness, requiring no derivative information of the objective function. This characteristic renders it highly suitable for optimizing non-differentiable, discontinuous, or noisy objective functions—frequently the case in feature selection where the fitness is often a classification error rate or another performance metric derived from a learning model. The evolutionary process inherently promotes solutions that achieve an optimal balance between multiple, often competing, objectives. In feature selection, fitter individuals naturally tend to be those that maximize classification performance while minimizing the number of selected features, without needing an explicitly tuned regularization parameter. This emergent trade-off helps in discovering compact, discriminative feature subsets. The evaluation of fitness for each individual in a population is independent of others, making this computational step “embarrassingly parallel.” This allows for efficient distribution across multiple processors or cores, drastically reducing wall-clock time and enhancing the scalability of GA for large-scale or high-dimensional problems. Collectively, these advantages establish Genetic Algorithms as a powerful, flexible, and efficient metaheuristic framework for tackling the inherently combinatorial and complex problem of feature selection.

Given a dataset D=(xi,yi)i=1m with m samples and n features, where xi∈ℝn represents the feature vector and yi∈Y denotes the class label, the feature selection problem aims to identify an optimal subset of features S⊆1,2,…,n that maximizes classification performance while minimizing dimensionality. This binary optimization problem can be formulated as:

where b=[b1,b2,…,bn]⊤ is a binary vector with b_j = 1 if feature j is selected, and b_j = 0 otherwise, |b|0=∑j=1nbj denotes the ℓ₀-norm counting selected features, X ∈ ℝ^{m × n} is the feature matrix with Xij = xi, j, E(·) represents the classification error function, α and β are weighting coefficients balancing classification accuracy and feature sparsity, ⊙ denotes element-wise multiplication, 1_m is an m-dimensional vector of ones.

Each candidate solution (chromosome) is encoded as a binary vector b ∈ 0, 1ⁿ. The initial population P(0)=b1,b2,…,bN of size N is generated randomly:

where p₀ = 0.5 ensures unbiased initial exploration of the feature space.

The fitness evaluation metric is primarily assessed by measuring the classification error achieved using the selected features. As a wrapper-based method, this approach directly employs the performance of a target classifier—such as support vector machine (SVM)—to determine the quality of a candidate feature subset. The classification error serves as a direct and interpretable indicator of how well the selected features support the learning algorithm in discriminating between classes. Typically, to ensure robustness and prevent overfitting, the error is estimated via cross-validation or hold-out validation. This design aligns the feature selection process closely with the end classification task, thereby enhancing the relevance and discriminative power of the final feature subset.

To form the mating pool M(t) for generation t, individuals are selected probabilistically based on their fitness. The selection probability for chromosome b_i is:

Here, ι > 0 is a small constant preventing division by zero. The cumulative distribution function is:

For each selection, a random number r ~ U(0, 1) is generated, and chromosome b_k is selected where k=mini:Pisel≥r.

With probability p_c, pairs of parent chromosomes undergo single-point crossover. For parents b_p and b_q, a crossover point c is randomly selected:

Two offspring bp′ and bq′ are generated as:

If crossover is not applied (with probability 1 − p_c), the offspring are exact copies of the parents.

Each gene in the offspring undergoes mutation with probability p_m. For gene b_ij:

This operator maintains population diversity and enables exploration of new regions in the search space.

To guarantee monotonic improvement across generations, the algorithm employs an elitism strategy. The best chromosome bbest(t) from generation t is preserved by replacing the worst chromosome in the offspring population P′(t):

This ensures that:

The genetic algorithm for OS²FS begins with inputs including the feature matrix X ∈ ℝ^{m × n}, label vector y ∈ ℝ^m, population size N, maximum iterations T_max, crossover probability p_c (default: 0.8), mutation probability p_m (default: 0.05), mutation strength μ (default: 0.01), and hold-out folds k (default: 0). Initially, it sets t = 0 and generates the population P(0) via Equation 11, then evaluates each individual by computing fi=F(Xbi,y,k) for i = 1, …, N, and identifies the best individual as bbest(0)=argminb∈P(0)f with fitness fbest(0)=minf. While t < T_max, the algorithm performs selection to form the mating pool M(t) using roulette wheel selection (Equations 12, 13), generates offspring P′(t) via single-point crossover (Equations 14, 15) with probability p_c, applies bit-flip mutation (Equation 16) to P′(t) with probability p_m, and evaluates the offspring by computing fi′=F(Xbi′,y,k). It updates the best individual if minfi′<fbest(t), then applies elitism by replacing the worst individual in P′(t) with bbest^(t) (Equation 17), followed by updating P(t+1)=P′(t), f^{(t + 1)} = f′, and the convergence curve ct+1=fbest(t), and incrementing t = t + 1. After the loop, it extracts the results: Sf=j:bbest,j(Tmax)=1,j=1,…,n, nf=|Sf|, Xselected=X:,Sf, and returns (Xselected,Sf,nf,c) as output. Redundancy analysis is first performed on individual features using Markov, and then on the selected features using Equation 1.

The algorithm's convergence is guaranteed by the elitism strategy, which ensures the best fitness value is non-increasing:

Theorem 1 (Monotonic Convergence). For the GA-OS²FS algorithm with elitism, the sequence of best fitness values fbest(t)t=0Tmax is monotonically non-increasing.

Proof. By construction, the elitism strategy preserves the best solution from generation t in generation t + 1. Therefore, fbest(t+1)≤fbest(t) for all t.

The expected time complexity per iteration is O(N·(n+CF)), where CF is the cost of evaluating the fitness function for one chromosome. The overall complexity for T_max iterations is O(Tmax·N·(n+CF)).

The proposed GA-OS²FS algorithm provides an effective approach for streaming feature selection, combining the global search capability of genetic algorithms with direct performance evaluation using the target classifier.

This section presents the experimental evaluation conducted on six real-world datasets obtained from two key sources: DNA microarray repositories and the benchmark collection from the Neural Information Processing Systems (NIPS) 2003 conference. These datasets are widely recognized in the machine learning and bioinformatics communities for assessing feature selection and classification algorithms under high-dimensional, small-sample conditions. The inclusion of microarray data ensures the examination of genetic expression patterns, while the NIPS 2003 datasets provide a diverse range of problem domains and complexity levels, thereby enabling a comprehensive analysis of the proposed method's robustness and generalizability. A detailed summary of the datasets—including the number of features, samples, and classes—is provided in Table 1 for reference.

To comprehensively evaluate the efficacy of the proposed model, a rigorous comparative analysis is conducted against four state-of-the-art online streaming feature selection (OS²FS) methods, which are recognized as established benchmarks in the field. The selected competitors include Fast-OSFS, SAOLA, and LOS-SA. This diverse set of algorithms encompasses various strategic approaches to handling feature streams—such as leveraging pairwise feature relations, redundancy analysis, and sparsity-aware selection—thereby ensuring a robust and multifaceted comparison. Furthermore, to objectively assess the quality of the feature subsets selected by each method, the evaluation employs three fundamental yet powerful classifiers: Support Vector Machine (SVM), k-Nearest Neighbors (KNN), and Random Forest (RF). These classifiers were chosen for their distinct learning mechanisms: SVM seeks optimal separating hyperplanes, KNN relies on local similarity, and RF utilizes ensemble decision-making. Their combined use helps verify whether the selected features generalize well across different inductive biases and are not tailored to a single classification model.

Detailed parameter configurations for all compared OS²FS algorithms and the three classifiers are systematically summarized in Tables 2, 3, respectively, to ensure full reproducibility of the experiments. All algorithms are implemented in MATLAB to maintain a consistent computational environment. And all experiments utilize five-fold cross-validation, meaning each dataset is randomly divided into an 80% training portion and a complementary 20% test portion. To account for randomness in data partitioning and algorithm initialization, each dataset is executed 10 times; the final reported result is the average predictive accuracy across these runs, along with its standard deviation where applicable.

All trials were conducted on a standard personal computer equipped with an Intel Core i7 processor running at 2.40 GHz and 16 GB of RAM, ensuring that the computational demands of the online feature selection and classification processes were feasibly met within a common research setup. This controlled hardware environment also aids in the fair comparison of runtime and efficiency where relevant.

The efficacy of the GA-OS²FS model is rigorously assessed by benchmarking it against the aforementioned suite of advanced algorithms, specifically within the challenging context of sparse streaming features. This scenario is deliberately chosen to simulate real-world conditions where data incompleteness and sequential feature arrival are prevalent, thereby testing the models' robustness and adaptability. To ensure a fair and statistically grounded comparison of performance across all algorithms, a non-parametric Friedman test is conducted at a stringent 95% confidence level. This test is employed under the null hypothesis that all algorithms perform equivalently, providing a holistic view of performance rankings across multiple datasets.

Furthermore, to drill down into pairwise performance differences, a paired Wilcoxon signed-rank test is applied at a 0.1 significance level. This test is specifically designed to examine whether the observed performance differences between the GA-OS²FS model and each individual baseline algorithm are statistically significant, rather than attributable to random chance. The resulting p-values from this comprehensive statistical analysis are consistently below the significance threshold. This robust statistical evidence leads to the conclusive finding that the GA-OS²FS model significantly and consistently outperforms all competing algorithms in the evaluation, demonstrating its superior capability in selecting informative features from sparse, evolving data streams.

To investigate the impact of missing data on feature selection performance, a missing-at-random scenario with a 10% data loss rate is established as a representative and practically relevant case for detailed analysis. This specific rate is chosen to simulate a common yet challenging level of data incompleteness encountered in real-world streaming applications. As illustrated in Figure 1, the average number of features selected by each compared method under this sparse condition is presented. The bar chart reveals distinct strategies among the algorithms: some methods maintain a conservative, highly selective profile, while others retain a larger fraction of the feature stream, reflecting different trade-offs between redundancy elimination and information preservation.

Correspondingly, Table 4 documents the concrete predictive performance outcomes, quantified by classification accuracy, when applying three fundamentally different classifiers—K-Nearest Neighbors (KNN), Support Vector Machine (SVM), and Random Forest (RF)—to the feature subsets identified by each method. This multi-classifier evaluation is crucial, as it demonstrates whether the selected features provide robust discriminative power independent of a specific learning algorithm's bias. The results in the table allow for a direct, quantitative comparison of how the parsimony or comprehensiveness of a selected feature subset, as shown in Figure 1, ultimately translates into generalization accuracy across diverse classifiers. This integrated analysis of subset size (Figure 1) and classification efficacy (Table 4) provides a comprehensive view of each algorithm's effectiveness in balancing feature reduction with predictive performance under the specified missing data condition.

This study evaluates the effectiveness of the 3WDO model by comparing it against six prominent OS²FS models—Fast-OSFS, SAOLA, SFS-FI, and LOSSA—across six datasets under missing data rates ranging from 0.5 to 0.9. While LOSSA is designed to handle missing values, the other three baseline algorithms are oriented toward complete feature streams. To adapt them for sparse data, zero-imputation is applied to fill missing entries for Fast-OSFS, SAOLA, and SFS-FI. Results are highlighted where any algorithm demonstrates superior performance compared to the others. Table 6 provides a pairwise comparison between GA-OS²FS and each baseline using the Wilcoxon signed-ranks test. The average accuracy trends of all models on datasets D1-D4 are visualized in Figure 2.