Medulloblastoma (MB) starts in the cerebellum, the part of the brain crucial for voluntary muscle movements like balance and motor control. MB is the most prevalent malignant brain tumor in children, even though it can also happen in adults [1]. The standard treatment regimen for MB, which includes surgical resection, radiation therapy, and chemotherapy, has resulted in favorable short-term outcomes. The pressing need for less toxic and more targeted therapy calls for a greater understanding of the heterogeneity within and amongst MB tumors. Sonic Hedgehog MBs (SHH-MB) molecular subgroup represents 30% of cases and is characterized by an overactive Sonic Hedgehog pathway caused by acquired or inherited mutations [2]. The SHH-MBs have been established as distinct subtypes with the advancement of molecular studies using transcriptomics and DNA methylation profiling methods. There are probably four molecular subtypes of SHH-MB: SHH_alpha, SHH_beta, SHH_delta, and SHH_gamma [3,4]. These classifications could enhance personalized plans of therapy and improve patient outcomes by improving the accuracy of predictions. Recent research into SHH-MB has significantly fostered our understanding of the molecular subtypes of the disease and their implications for tumor biology and therapy. Recent studies contributed noteworthy conclusions. A comprehensive review of current developments in SHH-MB molecular analysis has underlined significant intertumoral variability among SHH-MB tumors and led to the identification of new subtypes with distinct biochemical and clinical features [4]. The authors of [5] emphasized the combination of computational analysis with single-nucleus RNA sequencing to investigate the relationships between tumor heterogeneity and developmental trajectories in SHH MBs. The study emphasized the potential for differentiated treatment of SHH-MBs by identifying cell types associated with the various phases of granule neuron development. Investigating 96 samples of SHH-MB that were identified using RNA sequencing, targeted DNA sequencing, and genome-wide DNA methylation profiling, the authors of [2] detected molecular subtypes of this tumor and assessed their prognostic relevance. Their study concluded that adult MBs, although histologically uniform, demonstrate remarkable molecular heterogeneity. Furthermore, they deduced that the identification of two distinct molecular subtypes was integral to understanding their disparate clinical behaviors and outcomes. A cohort of 250 human SHH-MB samples was analyzed by strand-specific RNA sequencing with the incorporation of DNA methylation and whole-genome sequencing to investigate their molecular subtypes and underlying biology. The findings support the presence of four clinical subtypes based on presentation and age: SHH_alpha, SHH_beta, SHH_gamma, and SHH_delta [6].

Several factors influence the performance of any classification model, particularly in the case of multiclass classification. The effectiveness of machine learning algorithms in practical settings may be hampered by class imbalances, but this effect is dependent on class overlap as well [7]. Class overlap occurs when samples from different classes exist in common areas within the feature space, thereby creating ambiguity in the decision domains [8]. The underlying classifier’s overall performance gets worse with the increasing level of class overlap since it incorrectly classifies the samples along the boundary line [9]. Different strategies have been proposed to address the class overlap issue in the classification problem. An under-sampling-based supervised learning was created by the authors of [10] to address the class imbalance in binary datasets, where there is a class overlap issue. Their proposed under-sampling framework effectively addressed the challenge of class imbalance and class overlap in binary datasets. The visibility of the minority class was enhanced by identifying and removing overlapping majority-class samples. Their suggested method’s sensitivity was comparable to the state-of-the-art approaches. Wang et al. [11] introduced Extreme SMOTE and Synchronous Sampling Learning Methods as effective methods for improving financial distress detection in listed companies that have issues with class imbalance and class overlap. Their proposed method, while focusing on refining decision boundaries and optimizing company selection, significantly improved the classification performance of various machine learning algorithms. An under-sampling algorithm based on random forest cleaning rule (RFCL) was introduced by Zhang et al. [12]. It provides a solid solution for managing class imbalance and class overlap in classification problems. Effectively defining and applying a new decision boundary, RFCL enhances model performance, outperforming other under-sampling methods. Entropy and neighborhood-based under-sampling (ENU), a method recently developed by Kumar et al. [13] was intended to remove the majority of class samples from the overlapping regions to manage class overlap in classification tasks. ENU computes entropy scores for the majority class samples and establishes a threshold to determine which samples can be removed. The results indicate that ENU significantly outperforms several state-of-the-art methods across various metrics.

The earlier studies on class overlap in the classification task have certain drawbacks. Their primary focus is on binary classification tasks. Multiclass classification, however, adds more complexity, which, in turn, requires further research and customized methods. On the other hand, certain research avoids the majority class sample from areas where it overlaps with the samples from different classes, the selection of distance measures can make a distinction in how the overlapping regions are perceived [10,13]. Moreover, certain studies overlook the potential risk of overfitting that arises from oversampling specific types of samples, particularly when those samples are excessively similar to the training set. In our previous study, we successfully classified imbalanced MB subgroups using singular value decomposition-based penalized multinomial regression (SVDPMR) [14]. In the present study, we aimed at further sub-classifying a main molecular subgroup of MB, namely SHH-MB, by applying Tomek links-based SVDPMR (TL-SVDPMR) to analyze DNA methylation probe features from samples of SHH-MB patients. Penalized multinomial regression (PMR) model is helpful for multiclass classification tasks, mainly when working with high-dimensional data. The PMR model is a penalized generalized linear model that is fitted by minimizing the penalized loss function. Numerical techniques are employed to estimate the coefficients in the PMR model. We employed SVD and group lasso as dimensionality reduction techniques. SVD created a new low-dimensional input feature space from the training set. Then, the group lasso selected features that distinctly represent each SHH-MB subtype. Tomek Links (TL) finds the sample pairings of different subtypes that are closest to one another [15]. By eliminating the sample from the majority subtype in these pairs, TL helps to improve model performance and decision boundaries by removing samples that lead to class overlap. By reducing the number of features and selectively decreasing samples from the majority classes, these methods could enhance the clarity of class boundaries and improve the performance of the classification model. Our ultimate objective was to achieve high performance in correctly distinguishing the four subtypes within SHH-MB with less error.

A robust machine for predicting overlapped molecular subtypes in SHH-MB using DNA methylation data is created by using SVD for dimensionality reduction, TL for undersampling the majority subtypes that are located in the overlapping region with the minority subtypes, PMR as a classifier, and Coordinate Descent for optimization. The mean accuracy of about 95% achieved shows the feasibility of this approach, indicating the utility of this strategy for precision medicine and tailored therapies.

Employing DNA methylation data, we compared our proposed method (TL-SVDPMR) with a random forest (RF) algorithm [28]and support vector machine (SVM) [29]. We used SVD to reduce the dimensionality of the features before using RF and SVM techniques. To train effective RF and SVM models, we selected the number of first features derived by SVD based on variance explained and model performance. TL is used to make the decision boundaries for separating the SHH subtypes less blurry by reducing the number of samples in the majority classes. In addition, we chose a numerical optimization method, which can take advantage of minimizing the loss function, expressed as the sum of the probabilities that each sample is assigned to the wrong class. Based on fivefold cross-validation, the results shown in Table 4 summarize the performance of the three methods in classifying SHH-MB subtypes using the DNA methylation dataset. The assessment is provided by AUC values, specificity, and sensitivity metrics in a multiclass classification context.

All of the SHH-MB subtypes, including SHH_delta, have a high average AUC value (0.997), indicating our proposed method’s exceptional classification ability. Next in line is SHH_beta at 0.996. For RF and SVM, the competitive AUC values were between 0.969 and 0.990 and 0.983 and 0.993, respectively. However, our method consistently outperformed both.

The recall metrics show the high sensitivity of our method, particularly for SHH_delta (0.988) and SHH_beta (0.948). These figures suggest that positive cases within these subtypes can be identified with a high degree of accuracy. Both RF and SVM showed lower recall values, with RF reaching a maximum of 0.975 for SHH_delta and SVM reaching 0.948 for SHH_delta. With high specificity, our method demonstrated good negative case classification for all subtypes, particularly SHH_beta (0.989). In conclusion, our proposed method performs significantly better in the classification of SHH-MB subtypes than RF and SVM. The statistical significance of these results indicates the robustness of the method, making it a valuable tool for doctors to correctly diagnose and treat SHH-MB subtypes. Applying TL-SVDPMR has several benefits. First, utilizing SVD to reduce feature dimensions enhances computational efficiency and effectively handles high-dimensional data. Second, TL enhances the clarity of decision boundaries, making it a valuable technique for tackling class overlap and class imbalance in classification tasks. Furthermore, the method obtains the probabilities of a new sample belonging to each of the four SHH subtypes, adding a layer of interpretability to the classification process.

While our study provides significant advancements in the classification of SHH-MB subtypes, several limitations should be acknowledged. Because of the small sample size of some SHH-MB subtypes, performance metrics can differ significantly between folds. This variability could lead to inconsistent results and interpretations. Although we compared our approach to RF and SVM, many more machine-learning algorithms and approaches could be explored. A broader comparison could provide a more comprehensive understanding of the model’s performance compared to the state of the art. Finally, External datasets have not yet been used to test the model’s robustness. Confirming the model’s efficacy and robustness in clinical practice requires the validation of independent datasets.

Our study demonstrates the utility of TL-SVDPMR as a method for developing precision medicine approaches toward accurate categorization of the molecular subtypes within SHH-MB based on DNA methylation data. Effectively mitigating class overlap, class imbalance, and high feature dimensionality issues, this framework improves the discriminative nature of decision boundaries which in turn allows for more accurate classifying. The combination of SVD for dimensionality reduction, TL for undersampling the majority classes, group lasso for feature selection, and MR for the classification task has resulted in an effective classification framework for SHH-MB subtypes. Using the fivefold cross-validation technique, our proposed method produced an average overall accuracy of about 95%. Converting the data into a lower-dimensional space while keeping important features, SVD reduced computational complexity. This also leads to a better relevance of information in DNA methylation data. TL improves the intelligibility of decision boundaries by resulting in a more balanced dataset, which is essential to efficiently address class overlap and class imbalance in the classification problem. We automatically selected informative features, and then used them as regressors in our classification model, by applying a group lasso that forces sparsity at the group level (a subtype of SHH-MB). Coordinate descent is used to solve our loss function associated with the group lasso, which promotes sparsity.

Through using our proposed approach, the classification model showed excellent performance on different overlapped SHH-MB subtypes in terms of the various performance metrics.