The paper explores the application of the Determination-Based Correlation Coeficient (DBCC) for analyzing dependencies between continuous and categorical features in classification tasks. While DBCC has previously been validated in regression contexts, its suitability for classification problems remains underexplored. This study evaluates DBCC across various dependency structures, including step, multi-class, and sigmoidal relationships, demonstrating its robustness and efectivity. The findings suggest that DBCC provides a computationally eficient and flexible alternative to traditional methods such as ANOVA, point-biserial correlation, and mutual information. Additionally, classification performance metrics such as Accuracy, F1-score, and ROC-AUC are examined to assess the practical implications of using Grid-Mean Algorithm in feature selection and model evaluation. Future research directions include refining the methodology and extending its application to high-dimensional datasets.
Understanding the relationship between continuous and categorical features is a fundamental aspect
of many classification tasks in machine learning. While numerous techniques exist for dependency
analysis, their efectiveness varies based on data structure, underlying assumptions, and computational
eficiency. In our previous work, we introduced and analyzed the Determination-Based Correlation
Coeficient (DBCC) in the context of regression tasks, demonstrating its ability to capture complex
dependencies and maintain robust performance across diverse datasets [
In this study, we explore the use of DBCC for assessing feature dependence in classification tasks,
aiming to provide a robust and interpretable alternative to existing methods. We evaluate DBCC on
synthetic datasets with stepwise, multi-class, and sigmoidal dependencies, comparing its performance
against established dependency measures. Additionally, we examine classification metrics such as
accuracy, F1-score, log loss, and ROC-AUC [
Accurate dependency estimation is crucial for classification models. Recent studies have highlighted
the importance of efective dependency estimation in classification. Techniques such as ANOVA and
point-biserial correlation rely on distributional assumptions that limit their applicability in real-world
scenarios [
In our previous studies, we conducted an in-depth investigation of the Determination-Based Correlation Coeficient (DBCC) and explored its application for predicting regression metrics based on data characteristics. Our findings demonstrated that DBCC is highly efective at capturing nonlinear dependencies, making it a powerful tool for assessing complex relationships within datasets. Given that classification tasks frequently involve interactions between continuous and categorical features, which often follow stepwise or sigmoidal patterns, we hypothesize that DBCC can also provide valuable insights in this context. DBCC is calculated using the following formula: 2 = 1 − ∑︀=0 [ − ()]2 ∑︀=0 [ − ()]2 Then for each ∈ [, ], the grid is: where () – is the mean value within the grid cell that contains . This grid-based mean is defined as follows: Let the interval [, ] be divided into equally sized subintervals of width ℎ = ( − )/ . ℎ = ︂{ ∈[,+1](), = + ℎ, = 0, . . . , , ℎ = − }︂ To rigorously assess the efectiveness of DBCC in classification tasks, we designed and executed a series of controlled experiments using synthetic datasets. These experiments aimed to analyze how well DBCC captures diferent types of dependencies between continuous and categorical features. Our methodology consisted of the following key steps: • Generating datasets with diferent types of dependencies between continuous and categorical features. • Computing the DBCC correlation coeficient and interpreting its results.
• Analyzing the outcomes and assessing the adequacy of DBCC in various scenarios. We categorized dependencies into five primary types to ensure comprehensive testing: • Step function. • Twice-changing step function. • No dependency. • Multi-class categorical feature.
• Sigmoidal dependency with varying levels of noise.
Each synthetic dataset contained 10,000 data points, ensuring stable and statistically meaningful estimates. To manage the discretization of continuous variables, we set the grid parameter to n = 100. This parameter controls the level of granularity in the approximation process: lower values smooth out dependencies, whereas higher values increase noise and inconsistencies. Through extensive testing, we determined that n = 100 provides an optimal balance between approximation accuracy and computational eficiency.
Step Function. In the first experiment, the continuous variable was sampled from a normal distribution with a mean of 50 and a standard deviation of 15. The categorical variable was assigned a value of 1 if the condition > 50 was met and 0 otherwise (see Figure 1). This dependency structure is frequently observed in classification tasks where a clear threshold separates the classes.
The computed DBCC value was 0.975, indicating an almost perfect correspondence between the continuous feature and the categorical label. The approximation accurately captured the underlying 1.0 1.0 0.8 0.6 c n u f 0.4 dependency, though minor uncertainty was observed around the threshold due to the efects of gridbased discretization.
Double Step Function. For the second experiment, we considered a more complex twice-changing step function. In this scenario, the categorical feature changed its value at two distinct thresholds, simulating real-world classification problems where decision boundaries are non-monotonic (see Figure 2).
The resulting DBCC value was 0.996, demonstrating DBCC’s ability to efectively capture nonmonotonic relationships. This result is particularly noteworthy, as many traditional correlation measures struggle to detect such patterns. 1.0 0.8 1.0 0.8 40 x 60 0.2 noise 0.3 0 20 80 100 0.0 0.1 0.4 0.5
No dependency. To examine DBCC’s behavior when no dependency exists between the variables, we generated independent random features (see Figure 3). As expected, DBCC returned a near-zero correlation value of 0.009, correctly indicating the absence of a relationship. This outcome confirms that DBCC does not falsely identify correlations where none exist, reinforcing its reliability as a statistical measure.
Multiclass Categorical Feature. A distinguishing characteristic of DBCC is its capacity to handle multi-class categorical variables. To test this capability, we constructed a dataset where the categorical feature could assume three distinct values (see Figure 4). The analysis yielded a DBCC value of 0.984, illustrating DBCC’s robustness in capturing relationships in multi-class classification settings. 2.00 true data estimated data 1.75 1.50 1.25 c fun1.00 0.75 0.50 0.25 0.00 true data estimated data −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5
10.0 x 0 20 40 60 80 100 0.0 0.1 0.3 0.4 0.5 x
Sigmoidal Dependence.To simulate more intricate nonlinear dependencies, we introduced a sigmoidal function: =
1 where controls the steepness of the transition between class labels. As increases, the function becomes more gradual, introducing greater uncertainty and blurring class boundaries (see Figure 5). Conversely, as approaches zero, the function approximates a power-law relationship.
Our results showed that as increased, the DBCC value decreased accordingly, reflecting the growing ambiguity in classification. This behavior suggests that DBCC is sensitive to uncertainty in decision boundaries and can serve as an efective diagnostic tool for evaluating the separability of classes in noisy data.
DBCC results showed that as class boundaries became more blurred (higher ), the correlation coeficient decreased, reflecting the growing uncertainty in classification.
The findings indicate that DBCC is an efective tool for assessing dependencies in classification tasks. It demonstrates high accuracy even in cases of complex and nonlinear dependencies. In future research, we plan to evaluate the efectiveness of this approach on real-world datasets.
Comparison with Other Methods. To provide a broader perspective, we compared DBCC with widely used methods for analyzing dependencies between continuous and categorical variables. The key characteristics and performance metrics of these methods are summarized in Table 1.
Our preliminary results suggest that DBCC ofers a unique balance between sensitivity to nonlinear patterns and robustness against noise, making it a valuable addition to the toolkit of data scientists working with classification problems.
Extending Grid-Mean Algorithm for Classification Metrics. In our other study, we explored the application of a grid-based algorithm for the approximate computation of regression metrics. The fundamental idea behind this approach was that by discretizing the continuous feature space into grid intervals and computing mean values within each interval, we could eficiently approximate regression-based evaluation metrics. Following the same logic, we propose extending this methodology to classification metrics, leveraging the Grid-Mean Algorithm as a machine learning model capable of estimating classification performance indicators.
Beyond correlation analysis, the grid-based structure of DBCC allows for the approximation of classification metrics by utilizing the probability distribution of categorical labels within each grid interval. Since each segment of the grid contains averaged feature values, it can serve as a foundation for estimating the probability of categorical outcomes, thereby supporting class prediction. This capability enables us to derive classification performance metrics directly from the grid without requiring an explicit predictive model.
We considered four primary classification metrics that are widely used for evaluating model performance: • Accuracy – The proportion of correctly classified values relative to the total number of samples.
This is a fundamental metric that provides an overall measure of predictive performance.
Accuracy =
True Positive + True Negative
True Positive + True Negative + False Positive + False Negative • F1-Score – The harmonic mean of precision and recall, balancing the trade-of between false positives and false negatives. It is particularly useful in cases with imbalanced class distributions. where – predicted probability of the i-point, and – label of this point.
To evaluate the efectiveness of this approach, we conducted experiments using a sigmoidal function with varying values: =
1 where controls the level of noise in the data. Higher values of result in smoother transitions and greater uncertainty near the decision boundary, while lower values produce sharper class separations. This setup allows us to assess how well the Grid-Mean Algorithm can approximate classification metrics under diferent levels of ambiguity in class labels.
For each experimental setting, we compared the actual and approximated values of classification metrics, ensuring a comprehensive evaluation of this method’s accuracy. The results (see Figure 6), demonstrate that the approximation error never exceeded 0.05 across all considered scenarios. This level of accuracy indicates that the proposed approach provides a reliable and computationally eficient means of estimating classification performance without the need for full-fledged model training.
Our findings indicate that DBCC is an efective tool for assessing dependencies in classification tasks. It demonstrates high accuracy, even in cases of complex and nonlinear dependencies. Moreover, its grid-based approach extends to approximate classification metric computation, making it a versatile tool in data analysis.
This study demonstrates the efectiveness of the Determination-Based Correlation Coeficient (DBCC) in evaluating dependencies between continuous and categorical features in classification tasks. Accurately assessing such dependencies is essential for improving classification models, as it influences feature selection, model interpretability, and predictive performance. Traditional methods often struggle to capture complex relationships, making the development of more flexible and eficient dependency measures a crucial research direction.
Through extensive experimentation on synthetic datasets, our results confirm that DBCC efectively identifies various types of relationships, including stepwise, sigmoidal, and multi-class dependencies, while also correctly recognizing the absence of correlation where applicable. The ability to handle both simple and intricate dependency structures makes DBCC a valuable tool for feature evaluation. Furthermore, compared to conventional approaches such as ANOVA, point-biserial correlation, and mutual information, DBCC ofers a favorable balance between computational eficiency, interpretability, and adaptability to diverse data distributions.
Key Advantages of DBCC: • Efective Handling of Nonlinear and Complex Dependencies: Unlike traditional methods that often assume linearity or specific distributions, DBCC can capture intricate relationships without imposing restrictive assumptions. This makes it particularly useful in real-world classification tasks, where dependencies are rarely straightforward. • Robustness Across Various Dependency Structures: DBCC performs well across diferent types of relationships, including stepwise transitions, non-monotonic dependencies, and multi-class interactions. This generalizability enhances its applicability in diverse classification scenarios. • High Computational Eficiency: One of DBCC’s strengths is its ability to process large datasets eficiently, making it scalable for applications involving extensive feature analysis. Unlike computationally expensive techniques such as mutual information, DBCC provides a streamlined approach that balances precision with speed.
Future Research Directions. To further enhance DBCC’s efectiveness, several avenues for future research should be explored. One key area is refining the methodology to improve precision and robustness across more diverse data structures. Advanced techniques for parameter tuning and optimization could enhance its sensitivity in detecting subtle dependencies, ensuring more reliable results even in challenging classification settings.
Another important direction is extending DBCC’s applicability to high-dimensional datasets, where feature interactions become increasingly complex. Many real-world classification tasks involve numerous variables, and a robust dependency measure must be capable of eficiently handling such multi-feature relationships. Developing scalable implementations of DBCC that leverage parallel computing and optimized algorithms could significantly enhance its usability in large-scale machine learning workflows.
Moreover, integrating DBCC into automated feature selection frameworks could greatly benefit the ifeld by providing a systematic way to identify relevant features based on their dependency structures. By incorporating DBCC into feature engineering pipelines, practitioners can streamline the process of selecting informative variables, ultimately improving model performance while reducing computational overhead.
Finally, exploring the potential of DBCC for approximate computation of classification metrics presents an exciting opportunity. If DBCC can serve as a proxy for evaluating classification performance, it could enable rapid model assessment without requiring extensive training and validation steps. Such an approach would be particularly valuable in iterative machine learning workflows, where quick feedback is essential for model optimization.
By addressing these research directions, DBCC can be further established as a powerful and versatile tool for analyzing complex dependencies in modern data science, contributing to more efective and interpretable classification models.
During the preparation of this work, the authors used OpenAI GPT-4 in order to: check grammar and spelling, improve formal writing style, and enhance overall readability of the text. After using this tool, the authors reviewed and edited the content as needed and takes full responsibility for the publication’s content. Cryptographic Hardware and Embedded Systems - CHES 2009, volume 5747 of Lecture Notes in Computer Science, Springer, 2009, pp. 429–443. [19] Q. Liu, M. A. Charleston, S. A. Richards, B. R. Holland, Performance of akaike information criterion and bayesian information criterion in selecting partition models and mixture models, Systematic Biology 72 (2023) 92–105. doi:10.1093/sysbio/syac081. [20] J. E. Cavanaugh, A. A. Neath, The akaike information criterion: Background, derivation, properties, application, interpretation, and refinements, Wiley Interdisciplinary Reviews: Computational Statistics 11 (2019) e1460. doi:10.1002/wics.1460. [21] H. Cheng, B. Sterner, Error statistics using the akaike and bayesian information criteria, Erkenntnis (2024). doi:10.1007/s10670-024-00897-2. [22] C. Sutherland, D. Hare, P. J. Johnson, et al., Practical advice on variable selection and reporting using akaike information criterion, Proceedings of the Royal Society B: Biological Sciences 290 (2023) 20231261. doi:10.1098/rspb.2023.1261.