Data set description

The miRNA target gene prediction dataset was extracted from TarBase v8.0. The dataset comprised data from five higher mammals: Homo sapiens, Rattus norvegicus, Mus musculus, Bos taurus, and Ovis aries. After removing the polycyclic data from the miRNA–mRNA dimer structure, 831 positive miRNA target genes and 306 negative miRNA target genes were identified. The dataset was randomly split into training (70%) and test (30%) sets, stratified by class label to preserve the original class distribution. The test set was held out entirely during model development and hyperparameter tuning and was used only for final performance evaluation.

Prediction criteria

The prediction results were divided into four types: true positive (TP), true negative (TN), false positive (FP), and false negative (FN). These four outcomes correspond to correctly-predicted positive samples, correctly-predicted negative samples, incorrectly-predicted positive samples, and incorrectly-predicted negative samples. The five commonly-used criteria for target gene prediction were:

(1)

(2)

(3)

(4)

An additional criterion, the F1 score, was introduced to balance precision and recall.

(5)

where .

Based on Equation (5), a higher F1 value indicates better recall and precision, and thus a more accurate prediction.

A receiver operating characteristic (ROC) curve is a graphical representation that visually demonstrates the performance of a model. In the current study, the area under the ROC curve (AUC) was used to evaluate the model’s performance. The horizontal and vertical axes of the ROC curve represent the true positive rate (TPR) and false positive rate (FPR), respectively, where TPR = Sensitivity, and FPR = 1 − Specificity.

Statistical analysis

To assess the statistical significance of the performance improvements achieved by our proposed model, we conducted paired bootstrap tests at the instance level with 10,000 resamples to compare the F1 scores and AUC values between our model and each baseline model. Specifically, for each comparison between the proposed model and a baseline model, we resampled the test set instances with replacement (maintaining the original sample size) and recomputed the F1 score and AUC for both models on each bootstrap sample. The p-value for each comparison was calculated as the proportion of bootstrap samples in which the performance difference (proposed model minus baseline model) was ≤ 0 (two-sided test). This paired approach ensured that both models were evaluated on identical bootstrap samples, providing a fair comparison. The 95% confidence intervals (CIs) for all evaluation metrics were computed using the percentile bootstrap method (10,000 resamples) at the instance level.

Given that our proposed model was compared against five baseline models (GMM without dependency; single Gaussian model (SGM) considering dependency; Weibull distribution (WD); the convolutional neural network model from the miRBench framework, miRBench-CNN [25]; and the hybrid architecture combining an autoencoder and a convolutional neural network, Hybrid AE-CNN) we applied the Bonferroni correction for multiple comparisons to control the family-wise error rate. The significance threshold was adjusted to  = 0.05/ 5 = 0.01. Thus, p-values < 0.01 were considered statistically significant. All reported p-values in S5 Table are raw p-values; asterisks indicate significance after Bonferroni correction (***p* < 0.001, **p* < 0.01, *p* < 0.05 before correction, with only *p* < 0.01 considered significant after correction).

Target gene feature extraction and recognition rules

In total, 90 miRNA target gene features were extracted in this study, including seed-region hydrogen bond count, elemental ratio, and continuous matching characteristics; thermodynamic features; seed-region conservation features; context sequence features; target gene accessibility penalty features; miRNA sequence 2-mer features; and binding structure features. The decision function of the proposed prediction model was:

(6)

where and are the value and weight of the i-th feature, respectively; k is the total number of features, and is the bias term. If S > 0, the sample is classified as a positive target gene; otherwise, it is classified as a negative target gene.

The normalized 90-dimensional feature vector of each sample and its label (positive/negative) were used as input data, and the optimal weight vector and bias were obtained by solving the following convex quadratic programming problem:

(7)

where represents the norm of the weight vector, denotes the margin variable, is the penalty parameter, and are the slack variables.

To obtain the optimal weight vector, ω, we solved the convex quadratic programming problem defined in Eq. (7) using a linear Support Vector Machine (SVM). The model was implemented using the LinearSVC class from the scikit-learn library in Python, with the liblinear solver selected. A linear kernel was used, consistent with the linear decision function shown in Eq. (6). The penalty parameter, C, was optimized using five-fold CV on the training set. We conducted a grid search over C ∈ {0.01, 0.1, 1, 10, 100} and selected the value that maximized the average G-means score across the validation folds. The optimal C value was found to be 1.

The most important k features directly associated with the biological mechanism of miRNA targeting were selected from all 90 characteristics as input features for the prediction model, whereas the remaining characteristics were treated as redundant information. This approach effectively reduced computational costs.

The feature effectiveness measure, , was used to calculate the importance of each feature:

(8)

where is the i-th element of .

The 90 features were sorted based on their values in descending order, and the top k features with the highest values were selected.

The value of k was determined using the G-means , which is the geometric mean of TP and TN; a value closer to 1 indicated better model performance. When the number of features was 18, G-means approached a peak value, indicating optimal performance. Increasing the number of features did not substantially improve the G-means but increased computational cost. Therefore, k = 18 was selected as the optimal feature set size. The values and weights of the input features selected for this study are presented in Table 1.

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Table 1. Values and weights of each input feature.

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Biological rationale for the selected 18 features

The 18 selected features can be categorized into four biologically-relevant groups based on their functional roles in miRNA–mRNA targeting (Table 1).

First, seed region features (Sm_6mer, Sm_7mer_m8, Sm_7mer_m1, Sm_7mer_A1, Rgs_match, Rgs_mismatch) capture the critical base-pairing between miRNA positions 2–8 and the mRNA target. The seed region is well-established as the primary determinant of miRNA target recognition [18], and mismatches in this region severely impair binding efficiency.

Second, thermodynamic features (Rgs_energy, Acc_energy, Rgt_energy) reflect the binding stability and accessibility of the target site. Rgs_energy quantifies seed region duplex stability, Acc_energy measures the energy required to unfold local mRNA secondary structures, and Rgt_energy represents total binding affinity. These features collectively determine whether a target site is accessible and thermodynamically-favorable for miRNA binding [6].

Third, conservation features (Consv_seed, Consv_3cntxt, Consv_5cntxt) indicate evolutionary pressure on the target site. PhyloP scores measure nucleotide-level conservation across species; higher conservation suggests functional importance and has been shown to correlate with genuine miRNA targeting [7].

Fourth, contextual features (2mer1, 2mer6, 2mer7, 2mer12, Nt1, Rgt_match) capture flanking sequence effects and non-seed region contributions. The 2-mer features represent dinucleotide pairing patterns outside the seed region that modulate binding specificity [12]. Nt1 records the nucleotide type at the first position of the target gene, and Rgt_match quantifies non-seed region complementarity. These contextual factors are increasingly recognized as important modulators of miRNA targeting efficiency [25].

Together, these 18 features comprehensively represent the multi-faceted mechanisms of miRNA–mRNA interaction, including sequence complementarity, thermodynamic stability, evolutionary conservation, and contextual modulation.

Feature number selection using G-means

To determine the optimal number of features (k), we sorted all 90 features in descending order of and evaluated the G-means (geometric mean of sensitivity and specificity) on the validation set as k increased from 1 to 90. The G-means increased rapidly from k = 1 to k = 18 (from 0.45 to 0.820), reaching a clear plateau at k = 18 (Fig 1). Beyond k = 18, the G-means improved only marginally (ΔG-means < 0.005), increasing from 0.820 at k = 18 to 0.825 at k = 90. Therefore, k = 18 was selected as the optimal feature set size, balancing predictive performance and computational cost.

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Fig 1. Feature number selection based on G-means.

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Dependency analysis

Clayton, Gaussian, and canonical vine copulas were used for comparison to demonstrate the superiority of the R-V-copula function. The Akaike information criterion (AIC) was used to evaluate the dependency-fitting ability of the different copula models. The smaller the AIC value, the stronger the model’s fitting capability. The AIC comparison results for the different copula functions are shown in Fig 2.

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Fig 2. Akaike information criterion (AIC) comparison results of different models.

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The Gaussian and Clayton copulas exhibited the highest AIC values, indicating the poorest ability to describe dependencies. This is primarily because both copulas could not adequately capture dependencies among high-dimensional multivariate variables, and their performance decreased substantially as dimensionality increased. In the current study, the input consisted of 18 different feature types, forming high-dimensional multivariate data; therefore, the results obtained using these two copula functions were poor.

The canonical vine copula, as a vine-structure copula, was more suitable for modeling dependencies among high-dimensional variables than the Gaussian and Clayton copulas and showed relatively good fitting performance. However, the C-vine copula had structural limitations and could not fully capture the complexity of genetic data. In contrast, the R-V-copula, as one of the most general vine structures, provided strong fitting capability for high-dimensional multivariate variables and did not suffer structural limitations, thereby yielding the best results.

Performance test of the proposed prediction model

Three benchmark models (GMM without dependency, single Gaussian model (SGM) considering dependency, and the Weibull distribution (WD)) and two state-of-the-art models (the convolutional neural network model from the miRBench framework (miRBench-CNN) [25] and the hybrid architecture combining an autoencoder and a convolutional neural network (Hybrid AE-CNN) [26]) were used to evaluate the effectiveness of the proposed prediction model (GMM with dependency).

To ensure fair and reproducible comparisons with state-of-the-art models, all baseline models were evaluated under identical experimental conditions. Specifically:

1. (1). Data partitioning: All models were trained and tested on the same stratified 70% –30% train–test split of the dataset, with the test set kept untouched until final evaluation. The same random seed (42) was used to ensure reproducibility.
2. (2). Input features: All baseline models used the identical set of 18 selected features listed in Table 1. For deep learning models (miRBench-CNN and Hybrid AE-CNN) that originally accepted different feature formats, we adapted their input layers to accept the 18-dimensional feature vector while preserving their core architectural designs as described in the original publications [25,26].
3. (3). Hyperparameter selection: For miRBench-CNN and Hybrid AE-CNN, hyperparameters were optimized using five-fold CV on the training set only, following the same procedure used for our model. The optimal configurations were: for miRBench-CNN, we used the default architecture from miRBench but reduced the input dimension to 18; for Hybrid AE-CNN, we retained the autoencoder latent dimension of 32 and CNN kernel sizes of 3 and 5, as originally reported. All other hyperparameters (learning rate: 0.001, batch size: 32, epochs: 100 with early stopping) were kept identical across deep learning models.

The comparison results and ROC curves of the different models are presented in Table 2 and Fig 3. The results demonstrated that:

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Table 2. Comparison of results under different criteria.

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Fig 3. Receiver operating characteristic (ROC) curves of different models.

TPR, true positive rate; FPR, false positive rate.

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1. (1). Of the probabilistic models, GMM outperformed both SGM and WD across all evaluation metrics, confirming the advantage of mixture modeling over single-distribution assumptions for capturing complex feature distributions in miRNA target data.
2. (2). Incorporating feature dependencies via R-vine copula improved performance, as evidenced by the comparison between the full model and GMM without dependency, highlighting the importance of modeling inter-feature correlations.
3. (3). When compared with state-of-the-art deep learning methods under identical experimental conditions (same data split, same 18 features, and consistent CV protocol), the proposed model achieved competitive performance. The model attained an F1 score of 0.8139 and an AUC of 0.85, which are comparable to those of miRBench-CNN (F1 = 0.8127, AUC = 0.83) and Hybrid AE-CNN (F1 = 0.8137, AUC = 0.81) (Table 2). All values reported in Table 2 were obtained by retraining each baseline model on our training set and evaluating on our test set, rather than being directly cited from prior publications. This ensures a fair comparison across all methods.

We conducted paired bootstrap tests (10,000 resamples) to assess whether the performance differences between our proposed model and each baseline model were statistically significant. Our model substantially outperformed all baseline models (GMM without dependency, SGM considering dependency, WD, miRBench-CNN, and Hybrid AE-CNN) in terms of both F1 score and AUC (all P values < 0.05) (S5 Table). These results confirm that the observed improvements were not due to random chance and demonstrate the robustness of our approach.

Performance evaluation under different data sizes

To systematically evaluate the performance of the proposed model under limited data conditions and to assess the effectiveness of HD-MTD, we conducted experiments using 30%, 50%, and 100% of the training set (which originally comprised 70% of the entire dataset). The test set remained unchanged throughout. For each data size condition, we applied three data expansion techniques (HD-MTD, DD-MTD, and traditional MTD) to generate virtual negative samples, balancing the positive-to-negative ratio to 1:1. The experimental protocol, detailed in the Methods section, involved 10 repeated runs with different random seeds to ensure statistical robustness. The mean F1 scores and 95% confidence intervals are presented in Table 3. For the experiments evaluating data expansion techniques, we adopted the following rigorous protocol to prevent data leakage (Table 3):

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Table 3. Performance evaluation using different data sizes.

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1. (1). Data partitioning: For each experimental condition (using 100%, 50%, or 30% of the total data), the original dataset was first randomly split into training (70%) and test (30%) sets using stratified sampling to preserve the original class distribution.
2. (2). Data expansion: Different types of MTD were applied exclusively to the training set to generate virtual negative samples. The target ratio after expansion was set to 1:1 (positive:negative), meaning that negative samples were generated until the number of negative samples in the training set equaled the number of positive samples.
3. (3). Model training and evaluation: The model was trained on the expanded training set (containing both original and synthetic samples) and evaluated on the pristine test set (containing only original samples, with no synthetic data). This ensured that performance metrics reflected the model’s generalization ability on real-world data.
4. (4). Robustness check: The entire procedure (splitting–expansion–training–evaluation) was repeated 10 times with different random seeds, and the average F1 scores are reported in Table 3.

In the presence of sufficient sample data (i.e., when 100% of the total data were used), all data expansion techniques were able to generate virtual negative samples to balance the proportion of positive and negative samples, thereby enhancing prediction accuracy. Compared with traditional MTD and DD-MTD techniques, HD-MTD performed dataset expansion more effectively. When the number of samples was insufficient (i.e., when 30% or 50% of the total data were used), the prediction accuracy of the models decreased as the amount of data decreased. The HD-MTD technique effectively alleviated this decline in prediction accuracy because it used HDs to describe the data.

When trained on only 30% of the training data, the model with HD-MTD achieved an F1 score of 0.7931 (95% CI: [0.77, 0.81]), which was comparable to the performance of the model trained on 100% of the data without expansion (F1 = 0.8139). Statistical analysis confirmed that the difference between these two conditions was not significant (p > 0.05, paired bootstrap test), supporting the claim that the proposed approach performed effectively even with substantially-reduced training data. This finding highlights the practical utility of HD-MTD in scenarios where labeled data are scarce.

To assess whether HD-MTD generated virtual samples that preserved biologically-meaningful feature distributions, we conducted comprehensive quantitative comparisons between the original negative samples (n = 306) and the virtual negative samples generated using HD-MTD.

Three complementary aspects of distributional fidelity were evaluated (S4 Table):

1. (1). Univariate distribution similarity: The Kullback–Leibler (KL) divergence between original and synthetic distributions averaged 0.039 (range: 0.012–0.087) across all 18 features, with values close to zero indicating high similarity. Kolmogorov–Smirnov (KS) tests yielded *p* > 0.05 for all features, indicating no statistically significant differences between the original and synthetic samples.
2. (2). Moment preservation: Key statistical moments—including mean, variance, skewness, and kurtosis—showed relative differences below 5% for all 18 features, confirming that the synthetic data faithfully reproduced the central tendency, dispersion, and shape characteristics of the original data.
3. (3). Feature correlation preservation: To verify that interdependencies among features (e.g., the positive correlation between seed region matching and thermodynamic stability) were maintained, we compared the pairwise correlation matrices of original and synthetic samples. The average absolute difference in correlation coefficients was 0.042 ± 0.031 (S4 Table), indicating that feature correlation structures were well preserved without introducing artificial relationships.

Detailed results for each of the 18 features, including KL divergence, KS test p-values, moment comparisons, and feature correlation preservation metrics, are provided in S4 Table.

Together, these quantitative analyses demonstrate that HD-MTD generated virtual negative samples that closely approximated the statistical properties, distributional characteristics, and feature correlation patterns of real biological data. The improved predictive performance achieved with augmented data (Table 3) can therefore be attributed to effective data supplementation rather than synthetic data bias.

Experimental validation: A proof-of-concept case study

To demonstrate the biological relevance of predictions generated by the model, we selected a top-ranked prediction—miR-8485 targeting JAK2—for experimental validation. It is important to note that this single-case validation is intended as a proof-of-concept to illustrate that the model can generate biologically plausible hypotheses, rather than as a statistical validation of the model’s overall predictive accuracy. The following dual-luciferase, cellular, and animal experiments were performed to evaluate the biological validity of this specific predicted interaction.

Dual-luciferase assay

We performed a luciferase assay using miR-8485 and its corresponding target gene, Janus kinase 2 (JAK2) (predicted by the model in the previous section), to validate the effectiveness of the prediction model proposed in this study. The reagents used are listed in S1 Table.

An endotoxin-free mini plasmid extraction kit was used to perform endotoxin-free extraction of the constructed recombinant reporter gene plasmid pmirGLO-JAK2-WT/pmirGLO-JAK2-Mut. After resuscitation, HL-1 cells were seeded evenly into 24-well plates. Transfection was initiated when the cells reached approximately 80% confluency. The cell transfection procedure was performed according to pre-designed experimental groups (JAK2 gene was divided into wild-type (WT) and mutant-type, with miR-8485 treated with mimics and inhibitor, respectively). After 24 h of transfection, the cells were digested and collected. Cell lysis was performed using the dual-luciferase reporter assay kit. The lysed samples were analyzed using GloMax 20/20 luminometer to measure firefly and Renilla luciferase activities separately. Firefly luciferase/Renilla luciferase ratio was calculated, and statistical analysis was performed using GraphPad Prism software. The results are shown in Fig 4.

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Fig 4. Dual-luciferase assay results.

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Treatment with miR-8485 mimics significantly reduced luciferase activity in the JAK2 WT group, demonstrating that miR-8485 specifically binds to the 3′UTR of JAK2 to suppress its expression. Conversely, treatment with the miR-8485 inhibitor increased the luciferase activity in the JAK2 WT group, confirming its negative regulatory effect on JAK2. Notably, these regulatory effects were abolished in the JAK2 mutant-type group, demonstrating that miR-8485–mediated regulation depends on specific binding to the JAK2 3′UTR sequence. These results indicate that JAK2 is a direct target of miR-8485, providing initial proof-of-concept support for this specific prediction generated by our model.

Cellular experiments

Cellular experiments were performed to verify the validity of the proposed prediction model by investigating the targeted regulation of JAK2 by miR-8485. The main instruments and reagents are shown in S2 and S3 Tables. Cell culture was performed by thawing and passaging, which served as the basis for subsequent experiments. The cells were divided into groups as shown in Table 4. After treatment, RNA extraction, reverse transcription, and qPCR were performed sequentially, and the results are shown in Fig 5.

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Table 4. Treatment groups for cellular experiments.

https://doi.org/10.1371/journal.pcbi.1014402.t004

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Fig 5. Comparison of (a) miR-8485 and (b) JAK2 expression in cell experiments.

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As shown in Fig 5a, compared with the control group, miR-8485 expression was significantly decreased in the LPS-induced inflammatory model (Group 2), whereas it was significantly increased in the miR-8485 agomir (Group 3) group. No significant difference in miR-8485 expression was observed between Group 4 (JAK2 complementation) and Group 3, indicating successful overexpression of miR-8485, which was unaffected by JAK2 complementation.

As shown in Fig 5b, JAK2 expression was significantly higher in Group 2 than that in the control group. However, after transfection with miR-8485 agomir (Group 3), JAK2 expression significantly decreased. JAK2 expression in Group 4 (JAK2 complementation) was significantly higher than that in Group 3, indicating that miR-8485 overexpression inhibited JAK2 expression, and this inhibition was reversed by JAK2 complementation. The results in Fig 5 collectively verify the targeted regulation of JAK2 by miR-8485, consistent with this prediction from our model.

Verification using animal experiments

Animal experiments were performed to further verify the validity of the proposed prediction model by investigating the targeted regulation of JAK2 by miR-8485. Mice aged 8–12 weeks and weighing 20–25 g (C57BL/6J, male, clean grade) were purchased from Beijing SiPeiFu Biotech Co., Ltd. (Certificate No.: 110324241105032918, License No.: SYXK(Ji)2018–001). Animals were housed in an environment with constant temperature (22 ± 2°C) and humidity (60%), and a 12-h light/dark cycle, with free access to food and water and were acclimatized for 1 week. All animal experiments in this study complied with the Animal Experiment Ethics Standards of Hebei Medical University.)

The mice were divided into groups as shown in Table 5. Results of total RNA extraction from myocardial tissue, reverse transcription reaction, and qPCR detection, are shown in Fig 6.

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Table 5. Treatment groups for the animal experiments.

https://doi.org/10.1371/journal.pcbi.1014402.t005

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Fig 6. Comparison of (a) miR-8485 and (b) JAK2 expression in animal experiments.

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As shown in Fig 6a, after establishing the sepsis mouse model (Group 2), miR-8485 expression was significantly decreased compared with that in the control group. Upon further addition of the agomir (Group 3), miR-8485 expression was significantly increased. After JAK2 rescue (Group 4), miR-8485 expression showed no significant change compared with that in Group 3. This confirmed that miR-8485 could be successfully overexpressed in mice, and this overexpression was not affected by subsequent JAK2 rescue. As shown in Fig 6b, after establishing the sepsis mouse model (Group 2), JAK2 expression was significantly increased compared with that in the control group. Upon further addition of the miR-8485 agomir (Group 3), JAK2 expression was significantly decreased. After JAK2 rescue (Group 4), JAK2 expression was significantly increased compared with that in Group 3. This demonstrated that overexpressed miR-8485 could inhibit JAK2 expression, and JAK2 supplementation could reverse this inhibition. Together, the results presented in Fig 6 provide further evidence that miR-8485 targets and regulates JAK2, supporting the specific prediction generated by our model.

Ethics

This study followed the 3R principles. The experimental protocol was approved by the Institutional Ethics Committee of the Fourth Hospital of Hebei Medical University (No. 20240032) and Ethics Committee of the Third Hospital of Hebei Medical University (No. 20221182), in compliance with GB/T 35892–2018. Surgical and procedural interventions were performed under anesthesia and analgesia with humane endpoints, and euthanasia was carried out in accordance with the AVMA Guidelines at the conclusion of the experiment. As this study involved only animal experiments (C57BL/6J mice) and commercial cell lines (HL-1 cells), and did not include any human participants, no informed consent was required.

GMM

The GMM has multiple centers and can easily fit multi-peak or multi-valley distributions, allowing approximation of almost all distribution types. Therefore, although the GMM is a parametric model, it does not require strong prior assumptions, which effectively addresses the limitations of parametric models. Additionally, the GMM has a closed-form solution, providing strong interpretability of the data [31]. Therefore, in this study, probabilistic prediction was performed using the GMM.

The GMM PDF can be represented as follows [32]:

(9)

(10)

The dataset is denoted as D = , where M is the number of components, and d is the dimensionality. , , and denoted the mean, covariance matrix, and mixing coefficient of the k-th component, respectively. Accordingly, . The number of components for the GMM was selected from {1, 2, 3, 4, 5} based on the average log-likelihood in cross-validation.

GMM parameter estimation can be represented as follows:

The GMM parameters (, , and ) were estimated using the Expectation-Maximization (EM) algorithm, which can be divided into two steps:

E-step：Compute the posterior probability that each data point belongs to component k:

(11)

M-step: Update the parameters using the responsibilities:

(12)

The EM algorithm was initialized using k-means clustering and iterated until the change in log-likelihood was less than or a maximum of 100 iterations was reached. To avoid local optima, the EM algorithm was run 10 times with different random initializations, and the solution with the highest log-likelihood was selected.

R-V-copula

The copula function is the most commonly used tool to establish dependencies between marginal PDFs [33]. To construct the predictive model, we adopted a class-conditional generative approach. Specifically, we built two separate R-V-copula models: one for the positive samples (miRNA target genes) and one for the negative samples (non-target genes). For each class , we used a GMM to estimate the marginal PDFs of the 18 features, and an R-V-copula to capture dependencies among these features. This yielded two class-conditional joint PDFs, , and , where is the feature vector of a given miRNA-target gene pair.

1. (1). Construction of the basic structure of a regular vine

A nested tree was constructed, wherein each with N nodes had E edges, and d is the dimension of the input variables. Each edge is expressed as {y, z|M}, where M is the conditioning set and {y, z, y ≠ z} is the conditioned set. All elements, including y, z, and M, are nodes. In the first tree, M is an empty set.

Each edge e = {y(e), z(e) | M(e)} in the tree depends on two edges, and , where and share a common node. Therefore, the relationship between the edges is as follows:

(13)

(14)

where is the set of edges. For each edge , under the given conditioning, the corresponding bivariate copula density functions are .

Thus, , , and represent the subsets of the d-dimensional vector D corresponding to the elements M(e), y(e), and z(e), respectively.

1. (2). Modeling the R-V-copula function

To avoid information leakage, all vine structure selection and copula family determination were performed within each cross-validation fold using only the training data.

First, the edge set ({y, z|M}) of each tree was determined. Considering both the accuracy and computational complexity of the model, edges with high dependency in the first few trees were selected. Dependency was measured using Kendall’s rank correlation coefficient [34]. We used the maximum spanning tree algorithm to select edges with the largest sum of absolute empirical Kendall rank correlation coefficients.

Second, we considered the choice of the binary copula corresponding to each edge. Additionally, we calculated the AIC for all candidate binary copula families [35]. The binary copula function with the smallest AIC was considered to have the best goodness-of-fit and was therefore selected for each edge. The AIC was obtained using the following formula:

(15)

where d is the number of parameters and is the maximized likelihood of the sample set. The higher the value of d, more complex the model.

For copula parameter estimation, for each selected bivariate copula family, the copula parameters were estimated using maximum likelihood estimation (MLE). Given a set of observationsfor the two variables, the log-likelihood function is:

(16)

Where denotes the copula parameters for edge e. The MLE was obtained by numerical optimization using the sequential quadratic programming (SQP) algorithm implemented in the VineCopula package.

For samples belonging to a given class , the class-conditional joint probability density function of the 18 features can be expressed as:

(17)

(18)

Where is the marginal density of the i-th feature for class, and are the class-specific conditional distribution functions. is the bivariate copula density for the pair conditioned on the set .

Finally, prediction was performed using Bayes’ rule. For a new instance with a feature vector , the posterior probability of being a positive target was computed using:

(19)

Where, and are the class-conditional joint densities estimated by the GMM and R-V-copula models, and and are the prior probabilities estimated from the training data. Based on the class distribution in our dataset (831 positive and 306 negative samples), the priors were set to =831/1137 ≈ 0.731, and =306/1137 ≈ 0.269.

The final deterministic prediction was obtained by thresholding the posterior probability at 0.5: if >0.5, the instance was classified as a positive target; otherwise, it was classified as negative.

Sample data expansion technique

In this study, we used HD-MTD for sample data expansion, which was conducted as follows:

Step 2: Clustering of original data.

Based on the above eight attributes, the k-means algorithm was used to cluster the original data by grouping samples with similar attributes into the same cluster. We assumed that the original dataset contained N samples , each with eight attributes. First, min–max normalization was conducted to map the values to the interval [0,1] using the following formula:

(20)

where is the minimum value of the sample data, and is the maximum value.

Second, the cluster centers were initialized. M initial cluster centers were selected: .

The Euclidean distance to each cluster center was calculated for each sample :

(21)

where denotes the j-th cluster center, denotes the t-th attribute of the i-th sample, and denotes the t-th attribute of the j-th cluster center.

The optimal number of clusters, M, was determined using the elbow method based on the within-cluster sum of squares (WCSS). Specifically, k-means clustering for M from 2–10 was conducted and the WCSS was plotted against the number of clusters. The optimal M was selected as the point where the decrease in WCSS began to diminish, forming an ‘elbow’ in the curve. The optimal number of clusters was determined to be M = 3.

Considering the eight input attributes of the training set as inputs to the k-means algorithm, the original data were all clustered into M clusters.

To prevent information leakage, the optimal number of clusters (M) was determined independently within each training fold using the elbow method, based solely on the training data of that fold. The validation fold and test set were never used in this process. Across all five CV folds and 10 independent runs with different random seeds, M was consistently equal to three, indicating that the clustering structure was stable and not sensitive to the specific composition of the training data.

Step 3: Assignment of corresponding distribution types to each cluster.

For each cluster obtained in Step 2, we aimed to select the optimal distribution that best characterized its data. To achieve this, we used log-likelihood as the goodness-of-fit measure. For a given cluster, we considered a set of candidate distribution families, including, but not limited to the normal, Weibull, gamma, and lognormal distributions. For each candidate distribution, we first estimated its parameters, , via maximum likelihood estimation based on the data points in that cluster.

The log-likelihood (LL) of the cluster data under the candidate distribution was computed as:

(22)

where is the PDF of the candidate distribution with the estimated parameters, . The LL quantifies how likely it is to observe the given cluster data under a specific distribution; a higher value indicates a better fit. For each cluster, we selected the candidate distribution that yielded the maximum log-likelihood as its optimal distribution type. This process was performed independently for each cluster. Crucially, to prevent data leakage, this entire selection process—including parameter estimation and log-likelihood calculation—was conducted within each CV fold using only the training data. The validation folds remained untouched during this step.

Computational complexity

For GMM: For a dataset with n samples and d features, the expectation maximization (EM) algorithm for GMM had a complexity of , where is the number of mixture components and I is the number of EM iterations. In our experiments (n = 1137, d = 18, and , GMM estimation for a single feature was completed within seconds. Since this process was repeated for each of the 18 features and for both positive and negative classes, the total GMM estimation time was 2–3 minutes.

For R-V-copula: The vine structure selection using the maximum spanning tree algorithm had a complexity of for each tree level. For d = 18, this step was negligible. Copula parameter estimation for each edge required operations for likelihood evaluation and numerical optimization. With approximately edges in total, the entire vine construction and estimation process took 2–3 minutes on a standard desktop computer.