A Feasibility-Constrained Treble Opposite Algorithm for Missing Data Imputation in Clinical Diabetes Prediction

Diabetes is one of the most widespread chronic diseases worldwide. Its etiology is multifactorial, involving a combination of genetic predisposition, sedentary lifestyle, poor dietary habits, and other contributing factors of environmental, metabolic, or physiological origin [1]. Diabetes is a chronic condition causing uncontrolled blood glucose (hyperglycemia), which can lead to serious complications such as heart disease, stroke, kidney failure, and nerve damage [2]. There are two main types of diabetes: Type 1, is characterized by the body’s inability to produce insulin and accounts for about 10% of all diagnosed cases; and Type 2, occurs when the body either does not produce enough insulin or cannot use it effectively, representing approximately 90% of cases worldwide [3].

According to recent data from the International Diabetes Federation (IDF) in April 2025 [4], approximately 589 million individuals aged between 20 and 79 are currently affected by diabetes across the globe. This number is projected to rise to 853 million by 2050, with over 3.4 million deaths attributed to diabetes each year. Moreover, one in every eight adults faces a high risk of developing Type 2 diabetes, while an estimated 1.8 million children and adolescents under 20 suffer from Type 1 diabetes. In light of the alarming statistics and the significant impact of diabetes on both individual health and public health systems, early detection has become a critical necessity to enable timely diagnosis and effective treatment.

In this context, machine learning (ML) has emerged as a powerful tool for the early prediction and diagnosis of diabetes, thanks to its ability to learn complex patterns from multidimensional medical data. Numerous studies have explored ML techniques for diabetes prediction. For instance, early efforts compared neural networks with traditional methods such as logistic regression and linear perceptron [5], while others evaluated a variety of classifiers including SVM, KNN, Decision Trees, Naïve Bayes, and Random Forests [6, 7]. Notably, Abdulhadi and Al-Mousa [8] combined logistic regression with feature selection techniques to achieve up to 94.25% accuracy using national Health and nutrition examination survey data.

Despite these advances, the reliability of ML models in medical applications remains heavily dependent on the quality and completeness of input data [9]. In real-world clinical datasets, missing values are a pervasive issue. They often result from incomplete medical tests, human error during data entry, or loss of information during data transfer [10]. These missing values can severely affect the performance of ML models by introducing bias, reducing statistical power, and distorting learned patterns [11]. Addressing this issue is therefore a critical step in the data preprocessing pipeline.

A wide range of imputation strategies have been developed to handle missing data. Simple techniques, such as replacing missing values with the mean or median, are computationally efficient but may over simplify the data structure and ignore feature correlations [12]. More advanced methods, including those based on machine learning and optimization, attempt to estimate missing values more accurately by exploiting underlying data relationships [13]. However, these methods often suffer from high computational complexity or sensitivity to noise, limiting their effectiveness in robust clinical prediction scenarios.

While various imputation techniques exist, many are either too simplistic to capture the complexity of medical data or too sensitive to noise and outliers. Moreover, few approaches are designed with optimization under feasibility constraints—a necessary condition in clinical datasets where imputed values must remain within physiologically valid ranges. Furthermore, metaheuristic algorithms have shown promise in solving complex optimization problems, yet their potential remains under explored in the specific context of robust imputation for diabetes prediction.

To bridge this gap, this paper proposes a novel missing data imputation method based on the Treble Opposite Algorithm (TOA), a recent swarm-based metaheuristic. We develop two feasibility-constrained variants GFC-TOA and CSFC-TOA, which are designed to generate realistic imputations while preserving the data structure. The effectiveness of these methods is evaluated using five machine learning classifiers on the Pima Indians Diabetes Dataset (PIDD). The results demonstrate that our proposed CSFC-TOA approach significantly outperforms both classical and state-of-the-art imputation methods, enhancing the accuracy and robustness of diabetes prediction models.

The remainder of this paper is organized as follows: Section 2 presents the related work on missing data imputation approaches for diabetes detection. Section 3 details the proposed TOA-based imputation method along with its two variants. Experimental results are reported in Section 4. In Section 5, we compare the performance of our two proposed approaches, followed by a comparative analysis with classical and state-of-the-art imputation techniques. Classifier performance is also evaluated and discussed. Finally, Section 6 concludes the paper and outlines directions for future research.

In this study, we present a novel missing-data imputation framework built upon the recently developed TOA [25], a metaheuristic optimization method designed to balance exploration and exploitation through a triple-opposite learning strategy. By leveraging TOA’s adaptive search capabilities, our approach aims to generate accurate and robust estimates for missing values, thereby preserving data integrity and enhancing downstream prediction performance.

Figure 1 illustrates the overall pipeline of our proposed approach and the integration of the TOA-based imputation within the machine learning workflow for diabetes prediction. After loading the dataset, a structured pre-processing workflow was employed to enhance data quality and ensure compatibility with the learning algorithms. This process begins with the detection and precise localization of missing values (NaN), a crucial step that enables targeted and accurate imputation rather than applying unnecessary transformations to complete records.

The imputation procedure, which constitutes the core contribution of this study, is described in detail in Section 3.2. To ensure that all features contribute equally during model training, Min-Max scaling is applied to normalize all variables within the [0, 1] range, as show in Eq. (1).

$Normalised _{ {value }}=\frac{{ Original }_{ {value }}+ { Min }}{ { Max }+ { Min }}$                       (1)

where, Min and Max denote, respectively, the lowest and highest observed values for the given feature.

Following normalization, the dataset was partitioned using 5-fold cross-validation, in which the data is split into five equal folds. In each iteration, four folds are used for training and one fold for testing, ensuring that every instance is used for both training and evaluation. This approach provides a robust and unbiased assessment of the predictive performance of the developed models.

3.1 Original Treble Opposite Algorithm

The TOA is a new metaheuristic optimization algorithm proposed in 2024 [25]. It belongs to the family of swarm-based metaheuristics, meaning it operates with a population of candidate solutions, as represented in Eq. (2), which evolve collectively to find the global optimum.

$X=\left\{x_1, x_2, x_3, \ldots, x_n\right\}$                     (2)

TOA begins by randomly initializing a population of solutions uniformly across the search space to ensure unbiased and well-distributed exploration. Each solution is defined within lower (x_l,j) and upper (x_u,j) bounds for dimension j, and generated using a uniform random number as in Eq. (3).

$x_{i, j}=U\left(x_{l, j}, x_{u, j}\right)$                      (3)

The algorithm proceeds through three sequential search phases, each generating new candidate solutions and selecting the best among them.

Guided Search Toward Best Solution

This phase guides the search using the current global best solution x_b. Two candidate solutions are generated using Eq. (4) and Eq. (5): one moving toward x_b and the other away from it. The better of the two, x_c11 or x_c12 is selected as x_c1, as defined in Eq. (6).

$x_{c 11, j}=x_{i, j}+U(0,1) .\left(x_{b, j}-2 x_{i, j}\right)$                         (4)

$x_{c 12, j}=x_{i, j}+U(0,1).\left(x_{i, j}-2 x_{b, j}\right)$                         (5)

$x_{c 1}=\left\{\begin{array}{c}x_{c 11}, f\left(x_{c 11}\right)<f\left(x_{c 12}\right) \\ x_{c 12},{ else }\end{array}\right.$                     (6)

Guided Search Using Midpoint

Two solutions (x_s1 and x_s2) are randomly selected from the population (Eq. (7)), and their midpoint xt (Eq. (8)) serves as the reference. Two candidates are then generated: one moving toward x_t (Eq. (9)) and the other away from it (Eq. (10)). The better of the two, x_c21 or x_c22, is selected as x_c2 (Eq. (11)).

$x_{s 1}, x_{s 2}=U(X)$                   (7)

$x_{i, j}=\frac{x_{s 1, j}+x_{s 2, j}}{2}$                     (8)

$x_{c 21, j}=x_{i, j}+U(0,1).\left(x_{t, j}-2 x_{i, j}\right)$                (9)

$x_{c 22, j}=x_{i, j}+U(0,1).\left(x_{i, j}-2 x_{t, j}\right)$                  (10)

$x_{c 2}=\left\{\begin{array}{c}x_{c 21}, f\left(x_{c 21}\right)<f\left(x_{c 22}\right) \\ x_{c 22}, { else }\end{array}\right.$                     (11)

Purely Random Search

This phase introduces random, directionless perturbations to maintain diversity and avoid premature convergence. Two random candidates, x_c31 and x_c32, are generated using Eq. (12) and Eq. (13), respectively, and the better one is selected as x_c3 (Eq. (14)).

$x_{c 31, j}=x_{i, j}+0.1 \cdot U(-1,1).\left(1-\frac{t}{t_m}\right).\left(x_{u, j}-x_{l, j}\right)$                (12)

$x_{c 32, j}=x_{i, j}+U(-1,1).\left(1-\frac{t}{t_m}\right).\left(x_{u, j}-x_{l, j}\right)$                       (13)

$x_{c 3}=\left\{\begin{array}{c}x_{c 31}, f\left(x_{c 31}\right)<f\left(x_{c 32}\right) \\ x_{c 32},{ else }\end{array}\right.$                    (14)

After each phase, the current solution is updated if the new candidate performs better (Eq. (15)), and the global best x_b is replaced if an even better solution is found (Eq. (16)). Solution quality is evaluated using the objective function f(x) where lower values indicate better solutions, as TOA is designed for minimization problems.

$x_i^{\prime}=\left\{\begin{array}{c}x_c, f\left(x_c\right)<f\left(x_i\right) \\ x_i, { else }\end{array}\right.$                    (15)

$x_b^{\prime}=\left\{\begin{array}{c}x_i, f\left(x_i\right)<f\left(x_b\right) \\ x_b, { else }\end{array}\right.$                    (16)

In TOA, the objective function f(x) measures the quality of a solution, with lower values indicating better performance, as the algorithm is primarily designed for minimization problems.

3.2 TOA-based imputation

We propose an adaptation of TOA to effectively address the challenge of missing data imputation in diabetes detection.

In this formulation, the initial population X, as defined in Eq. (17), consists of n candidate imputation vectors x_i, where each x_i each vector x_i represents a complete set of replacement values for the missing entries in the dataset.

$X=\left\{x_1, x_2, x_3, \ldots, x_n\right\}$                (17)

The population is initialized using Eq. (18), where x_l,j and x_u,jdenote the lower and upper bounds of the j^th dimension of x_i, as specified in the next sections.

$x_{i, j}=U\left(x_{l, j}, x_{u, j}\right)$                    (18)

Here, the dimensionality of each candidate solution corresponds to the total number of missing values in the dataset.

The optimization objective is to impute these missing values so as to maximize the predictive performance of a classification model. Therefore, we redefine the fitness function in Eq. (19) as the classification accuracy obtained after imputing the dataset with a candidate solution x_i.

$f\left(x_i\right)={Accuracy}\left(x_i\right)$                  (19)

Unlike the original TOA, which is inherently designed for minimization problems, our adaptation is formulated for maximization. To accommodate this change, we reformulate the core update rules and selection criteria (originally given in Eqs. (6), (11), (14), (15), and (16)) into their maximization-oriented counterparts, shown in Eqs. (20)-(24). These modifications direct the search toward imputations that yield the highest classification accuracy.

$x_{c 1}=\left\{\begin{array}{c}x_{c 11}, f\left(x_{c 11}\right)>f\left(x_{c 12}\right) \\ x_{c 12}, { else }\end{array}\right.$                   (20)

$x_{c 2}=\left\{\begin{array}{c}x_{c 21}, f\left(x_{c 21}\right)>f\left(x_{c 22}\right) \\ x_{c 22}, { else }\end{array}\right.$                   (21)

$x_{c 3}=\left\{\begin{array}{c}x_{c 31}, f\left(x_{c 31}\right)>f\left(x_{c 32}\right) \\ x_{c 32}, { else }\end{array}\right.$                   (22)

$x_i^{\prime}=\left\{\begin{array}{c}x_c, f\left(x_c\right)>f\left(x_i\right) \\ x_i, { else }\end{array}\right.$                      (23)

$x_b^{\prime}=\left\{\begin{array}{c}x_i, f\left(x_i\right)>f\left(x_b\right) \\ x_b, { else }\end{array}\right.$                    (24)

Constraint Feasibility

To maintain the feasibility of candidate solutions, we apply the feasibility check defined in Eq. (25). For each element x_i,j, the algorithm ensures that it lies within the permissible range of its corresponding feature. If a value violates these bounds, it is replaced by a random value drawn from the valid interval. The pseudo-code of the TOA-based missing data imputation method is presented in Algorithm 1.

$x_{i, j}=\left\{\begin{array}{c}x_{i, j}, x_{l, j}<x_{i, j}<x_{u, j} \\ U\left(x_{l, j}, x_{u, j}\right){ else }\end{array}\right.$                (25)

We consider two types of feasibility constraints:

1. General Feasibility Constraint

The minimum and maximum values for each feature are extracted from the entire dataset, ensuring that imputations remain within biologically and statistically plausible ranges.

2. Class-Specific Feasibility Constraint

Feature bounds are computed per class (diabetic vs. non-diabetic), enabling the generation of imputations that better preserve class-specific distributions.

Accordingly, we evaluate two variants of the proposed method:

• TOA-based imputation under General Feasibility Constraints (GFC TOA imputation)
• TOA-based imputation under Class-Specific Feasibility Constraints (CSFC TOA imputation).

The pseudo-code of the proposed TOA-based missing data imputation approach is provided in Algorithm 1, and the comparative effects of GFC and CSFC on model performance are discussed in Section 4.

This section presents a comprehensive performance comparison of classifiers using our proposed imputation approaches. The evaluation includes intra-comparisons between the two TOA-based variants, as well as comparisons with classical imputation techniques and state-of-the-art methods. Finally, we analyze the relative performance of classifiers when using the CSFC-TOA imputation strategy.

5.1 Comparison of GFC-TOA vs. CSFC-TOA

The comparative analysis between the two imputation variants, GFC-TOA and CSFC-TOA, according to Table 6, demonstrates that CSFC-TOA consistently outperforms GFC-TOA across all classifiers and evaluation metrics. Quantitatively, CSFC-TOA achieves an average improvement of approximately 7.2% in accuracy, 12.1% in recall, 10.4% in precision, 11.4% in F1-score, and 8.4% in AUC over GFC-TOA. The most significant gains are observed for Decision Tree and Random Forest classifiers, where accuracy improvements reach up to 11.5% and 9.7%, respectively, accompanied by substantial increases in F1-score and AUC.

Figure 4 presents these average performance gains across all evaluation metrics, highlighting consistent improvements, particularly in recall and F1-score. These results indicate that incorporating class-specific constraints during the imputation process leads to more discriminative feature representations, reduces the noise caused by missing values, and enhances the predictive performance of machine learning models for diabetes detection. Consequently, CSFC-TOA can be considered the dominant imputation strategy in this experimental setting.

5.2 Comparison of TOA-based imputation against classical methods

To rigorously evaluate the effectiveness of the proposed imputation strategies, we evaluated the performance of five widely used machine learning classifiers: NB, SVM, DT, RF and KNN, on the PIMA dataset. The dataset was preprocessed using both conventional imputation techniques (zero-filling, mean, median, and mode) and the proposed TOA-based strategies (GFC-TOA and CSFC-TOA). Performance results, obtained using 5-fold cross-validation, are summarized in Table 7 and measured in terms of Accuracy, Recall, Precision, F1-score, and ROC-AUC.

Table 7 compares the performance of classifiers KNN, RF, SVM, DT, and NB using different imputation methods: classical approaches (Mean, Median, Mode, Without) and the proposed variants GFC-TOA and CSFC-TOA. Classical methods show limited performance, with marginal differences in accuracy, F1-score, and AUC (<2%). The GFC-TOA approach improves results, but CSFC-TOA consistently outperforms all other methods, with significant gains, particularly for DT (+11.5% in accuracy) and RF (+9.7%), along with notable increases in F1-score and AUC.

On average, CSFC-TOA improves performance metrics by 7-12% in accuracy, 6-18% in recall, 3-15% in precision, 5-15% in F1-score, and 5-13% in AUC, confirming that incorporating class-specific constraints reduces noise from missing values and produces more discriminative feature representations. These results establish CSFC-TOA as the dominant imputation strategy for diabetes prediction Our two proposed imputation variants outperform classical methods, with CSFC-TOA achieving the highest performance. These results highlight the robustness of optimization-driven, class-aware strategies in improving data quality and enhancing predictive performance across diverse machine learning algorithms when handling missing values.

Table 7. Classification performances using different missing value imputation methods

Note: KNN = K-Nearest Neighbors; RF = Random Forest; SVM = Support Vector Machine; DT = Decision Tree; NB = Naïve Bayes; GFC= General Feasibility Constraints; TOA = Treble Opposite Algorithm; CSFC= Class-Specific Feasibility Constraints.

5.3 Comparison of TOA based imputation against state-of-the-art approaches

In this section, we compare the performance of our proposed imputation strategies with several state-of-the-art imputation methods reported in the literature. To ensure a scientifically sound and fair comparison, we selected studies that used the same PIMA dataset without modifying its structure through sample deletion, feature selection, or class balancing techniques. This allows us to isolate and rigorously assess the impact of the missing data imputation methods alone. The approaches considered for comparison are detailed in Section 2.

To ensure a fair and scientifically rigorous comparison with state-of-the-art imputation approaches, all methods were evaluated under a unified experimental framework using 5-fold cross-validation on the Pima Indians Diabetes Dataset. No feature selection or alternative datasets were used in any method, and all classifier parameters were kept consistent across experiments. This unified setup allows for confident and unbiased assessment of the relative performance of each method. Table 8 presents the classification accuracy of the state-of-the-art methods and compares them with the accuracy achieved by our proposed approach in its two variants.

Table 8. Accuracy of several state-of-the-art imputation methods

Note: KNN = K-Nearest Neighbors; RFM= Random Forest with Mean; RFCM= Random Forest with Class' Mean; CART= Classification and Regression Tree; GMM= Gaussian Mixture Models; RFR= Random Forest Regressor; GFC= General Feasibility Constraints; TOA = Treble Opposite Algorithm; CSFC= Class-Specific Feasibility Constraints.

A Friedman non-parametric test was conducted to evaluate the effectiveness of different imputation methods using classification accuracy from five classifiers (DT, KNN, NB, RF, and SVM) under the same cross-validation protocol. Classifiers were treated as blocks and imputation methods as treatments. Table 9 shows the average ranks of the imputation methods based on the Friedman test.

Table 9. Average ranks of the imputation methods based on the Friedman test

Note: KNN = K-Nearest Neighbors; RFM= Random Forest with Mean; RFCM= Random Forest with Class' Mean; CART= Classification and Regression Tree; GMM= Gaussian Mixture Models; RFR= Random Forest Regressor; GFC= General Feasibility Constraints; TOA = Treble Opposite Algorithm; CSFC= Class-Specific Feasibility Constraints.

The Friedman test revealed statistically significant differences among the compared imputation methods (p < 0.05). CSFC-TOA obtained the best overall performance, achieving the lowest (best) average rank of 1.2, followed by GMM with an average rank of 1.8. RFR and RFCM formed a second performance tier with comparable average ranks of 4.2 and 4.25, respectively, while IGWO (4.66) and GFC-TOA (5.4) demonstrated moderate competitiveness. Classical imputation techniques, including CART (5.8), KNN (8.4), MF (8.5), and RFM (9.5), consistently ranked lower, indicating inferior performance across classifiers. Among recent swarm-based approaches, CSFC-TOA clearly outperformed IGWO and ISSA (average rank 7), confirming its superiority in terms of robustness and classification accuracy, while GFC-TOA provided stable yet less competitive improvements.

Overall, these findings confirm the effectiveness of the TOA-based imputation strategies for handling missing data. The consistent top performance of CSFC-TOA across a wide range of classifiers underscores its robustness and positions it as a promising, reliable imputation approach for predictive modeling tasks in medical and clinical applications.

The comparative study demonstrates that the proposed CSFC-TOA imputation method consistently outperforms both classical techniques (Mean, Median, Mode, Without) and other state-of-the-art and metaheuristic approaches (GMM, RFR, RFCM, IGWO, ISSA) across multiple classifiers for diabetes prediction. Quantitative improvements are substantial, with CSFC-TOA achieving average gains of 7–12% in accuracy, 6–18% in recall, 3–15% in precision, 5–15% in F1-score, and 5–13% in AUC. Statistical validation using the Friedman test confirms these differences are significant (p < 0.05), with CSFC-TOA attaining the best average rank (1.2), followed by GMM (1.8), while classical methods consistently rank lower (8.4–9.5). These results highlight the effectiveness of class-specific constraint-based imputation in reducing noise from missing values, generating more discriminative feature representations, and significantly enhancing predictive performance. CSFC-TOA thus emerges as the dominant and most robust imputation strategy in this experimental setting.