Modeling Rating Uncertainty with Interval Type-2 Fuzzy Decision Trees in Multi-Criteria Recommendation

a. Detailed procedure and evaluation metrics

a.vi. Evaluation metrics

System performance is assessed using mean absolute error (MAE), Precision, Recall, F1, F0.5, and F2-scores. For ROC-area under curve (AUC) analysis, binary splits were derived by thresholding cluster ranks at the median centroid score, ensuring balanced positive and negative partitions. Alternative binarization strategies, including percentile-based thresholds and adaptive cutoffs, were tested and yielded consistent AUC values within ±0.02. Additional analyses with confusion matrices and error distributions evaluate stability and interpretability.

To illustrate the transformation process, consider a sample user-rating vector [4, 3.5, 4.5, 3, 4]. First, each rating is mapped to its corresponding histogram bin for each criterion. Next, normalized frequency distributions are used to construct the UMF and LMF. The interval between UMF and LMF defines the FOU. These interval-valued memberships are then assigned to linguistic labels (Low, Medium, High), producing a fuzzy representation of the original ratings. This process is repeated across all users, resulting in a fuzzified dataset suitable for IT2 fuzzy decision tree learning. A flowchart 1 shows the Transformation of Raw Ratings to IT2 Fuzzy Sets.

Flowchart 1:

a. Clustering evaluation using HDBSCAN

The five-dimensional user rating vectors are clustered using HDBSCAN, which identifies natural groupings in the rating space without requiring a predefined number of clusters. These clusters serve as class labels for supervised classification [22].

The quality of clustering is evaluated qualitatively through visual inspection of t-distributed stochastic neighbour embedding (t-SNE) projections and quantitatively by cluster stability scores (provided by HDBSCAN) and Silhouette Coefficient. Table 3 summarizes the clustering quality metrics for applying HDBSCAN on the five-dimensional multi-criteria rating vectors. The algorithm identified five natural clusters, automatically determined based on data density. Clustering quality was evaluated using three metrics: Average Stability (provided by HDBSCAN), the Silhouette Coefficient, and the Xie–Beni (XB) Index. The high average stability score of 0.83 confirms consistent cluster structure across varying density thresholds. The Silhouette Coefficient of 0.67 indicates reasonable separation between clusters. Additionally, the low XB index (0.0112) reflects well-defined, compact clusters with good inter-cluster separation, supporting the robustness of using these clusters as pseudo-labels for supervised tree-based classification [23].

To further improve computational efficiency during similarity search within each HDBSCAN-derived supercluster, we performed subclustering using k-means. Since the superclusters produced by HDBSCAN exhibit homogeneous local densities, k-means is well-suited for dividing them into finer-grained, non-overlapping subclusters. The number of subclusters per supercluster was determined using the elbow method, ensuring optimal separation while maintaining internal cohesion [24]. This hierarchical clustering structure (superclusters → subclusters) significantly reduces the search space during the recommendation stage by confining similarity computations to smaller, more coherent groups. Alternative approaches, such as HDBSCAN-only clustering, may yield coarse groupings with insufficient granularity, whereas spectral clustering requires predefined parameters and incurs a higher computational cost. The combined HDBSCAN + k-means approach balances density sensitivity with computational efficiency, producing stable and fine-grained clusters suitable for supervised learning. Table 4 presents the number of subclusters generated per supercluster and their respective average intra-cluster cosine similarity scores, confirming that local grouping enhances granularity while preserving semantic coherence among users. Total subclusters are 10, which serve as the final class labels for training the classification models (ID3, C4.5, and their fuzzy counterparts). Stability of subcluster labels was evaluated across multiple runs with different random seeds and cross-validation folds. Results showed consistent cluster assignments with minor variations that did not significantly impact classification performance. The use of HDBSCAN ensures stable core clusters, while k-means refinement operates within these stable regions, minimizing label instability effects.

b. Fuzzy membership function construction

HBT1FMFs were constructed for each of the five rating criteria (10 points) based on the normalized frequency distribution of user ratings. These data-driven fuzzy membership functions were then used to fuzzify the decision nodes in the fuzzy decision trees, enabling a more flexible and uncertainty-aware decision-making process. Table 5 presents the number of users corresponding to each discrete rating level (horizontal axis) across all criteria (vertical axis), forming the basis for histogram generation [25].

This preprocessing step plays a crucial role in capturing the uncertainty inherent in user-generated discrete ratings, and it directly enhances the granularity and adaptability of split conditions within fuzzy decision tree models.

To facilitate interpretation, the histograms were grouped into three overlapping fuzzy linguistic ranges: Low (0.5–2.5), Medium (1.5–4.0), and High (3.0–5.0). These overlapping boundaries reflect the fuzzy nature of user preferences and enable smooth transitions between rating categories. Figure 2 illustrates the extracted HBT1FMFs corresponding to these ranges for each criterion. Unlike traditional symmetric fuzzy sets, the derived functions show non-symmetric, data-aligned shapes, effectively preserving the distributional nuances of user evaluations [26].

Figure 2:

c. Classification performance

Since the class labels in our study are derived from the 10 subclusters generated by k-means, the classification task involves multi-class categorical prediction. We split the dataset into 80% training and 20% testing subsets, and employed 5-fold cross-validation to assess model robustness and generalization. During cross-validation, we measured variance and bias in prediction accuracy to understand the consistency and reliability of each model Table 6.

Bias was computed as the average error between predicted and true labels across folds.

In the multi-class fuzzy classification setting, bias is computed as the average misclassification rate across all classes, while variance is estimated as the variability of predictions across cross-validation folds. Fuzzy membership aggregation contributes to reduced variance by smoothing decision boundaries across overlapping classes.

Variance was derived from the standard deviation of prediction errors across folds.

Values reflect outcomes of 5-fold cross-validation over an 80/20 train-test split.

FDT-ID3 achieved the best bias–variance tradeoff, indicating strong generalization Table 7.

Fuzzy trees clearly outperform their crisp counter-parts, especially FDT-ID3, which achieves the highest F_0.5 and F₂ scores, indicating balanced performance across precision- and recall-focused scenarios.

d. Confusion matrix analysis

The confusion matrix for FDT-ID3—shown below—illustrates a strong diagonal dominance, highlighting the model's effectiveness in predicting the correct subcluster labels Table 8.

This matrix confirms the model's ability to differentiate between user segments, even with overlapping or adjacent rating patterns.

e. Prediction accuracy and computational efficiency

Prediction performance was evaluated by computing the MAE between the predicted and actual overall satisfaction scores, which were inferred indirectly from the categorical class labels (Cluster 1–10) assigned to each user. Each prediction was derived using a membership-weighted class voting mechanism, where a new instance activates multiple fuzzy paths. Each path ends in a categorical class (leaf node), and the final prediction is determined by aggregating membership degrees across activated paths:

For instance, if a sample activates three paths with membership degrees [0.6, 0.4, 0.7] leading to clusters C1, C2, and C1 respectively, aggregated scores become C1 = 1.3 and C2 = 0.4. The final prediction selects C1 as the dominant class based on maximum aggregated membership.

After determining the winning class, its corresponding overall satisfaction score (e.g., class centroid) was used for MAE calculation Table 9.

Fuzzy models, especially FDT-ID3, significantly reduce prediction error. This highlights the advantage of integrating fuzzy membership-based aggregation in uncertain multi-criteria settings.

The average classification time per test instance was used to assess computational overhead. Although fuzzy trees require additional computation due to fuzzy membership evaluation, the added cost remains practical.

While slightly slower, the fuzzy models offer improved robustness and interpretability without incurring significant runtime penalties.

f. ROC curve and AUC analysis

To assess the models' ability to distinguish between satisfaction levels (e.g., low vs high ratings), receiver operating characteristic (ROC) curves were plotted using binary splits based on thresholds in the cluster ranking Table 10.

The high AUC for FDT-ID3 shows that fuzzy trees excel at handling noisy or ambiguous user feedback Figure 3.

Figure 3:

The experimental results clearly show that incorporating HBT1FMFs and fuzzy decision trees significantly enhances both the classification and prediction capabilities of recommender systems. Among all tested models, FDT-ID3 consistently delivered the best balance of accuracy, interpretability, and robustness, validating the effectiveness of the proposed fuzzy modeling framework.

Table 11 presents a comparison between the proposed model and state-of-the-art non-fuzzy recommender systems under uncertainty and sparsity conditions. The comparison highlights limitations of existing models in handling explicit uncertainty and interpretability. The proposed approach demonstrates superior robustness by integrating IT2 fuzzy modeling with interpretable decision tree learning.

a. Impact of HBT1FMFs

A key contribution of this study is the use of histogram-derived fuzzy membership functions to model uncertainty in user rating behavior. Rather than relying on Fuzzy C-Means (FCM) for feature-level clustering, we constructed HBT1FMFs for each rating criterion (e.g., story, acting, direction), based on normalized histograms of rating distributions.

This approach avoids the need to define cluster centroids or optimize fuzzy partitions and instead directly captures rating uncertainty through overlapping membership functions. These functions offer interpretability and scalability, while still supporting fuzzy decision logic in downstream classifiers.

This step proved crucial in enhancing the semantic richness of input features, improving the granularity of gain calculations at decision nodes, and leading to more nuanced tree splits.

b. Classification performance and generalization

The proposed FDT-ID3 model achieved the highest classification performance (Precision: 0.95, Recall: 0.96, F1: 0.95), significantly outperforming its crisp counterparts (ID3 and C4.5) and the Type-1 fuzzy decision tree (FDT) baseline. This improvement stems from the use of fuzzy membership aggregation along tree paths, which better accommodates the ambiguous nature of multi-criteria user ratings.

By leveraging minimum-membership logic across serial decision nodes, the model dynamically adjusts path contributions, producing more informed and generalizable classifications. These findings rein-force the notion that collaborative filtering outcomes are strongly dependent on the fidelity of intermediate classification stages.

c. Prediction accuracy and practical relevance

The accuracy gains in classification directly translated into a lower prediction error (MAE = 0.5137) for predicted user ratings, outperforming both fuzzy and crisp baselines. This result underscores the practical relevance of our hybrid methodology in applications where accurate predictions are critical for user satisfaction.

For instance, in the case of a movie with polarized ratings—high visuals but low story—the crisp models predicted an average satisfaction score, whereas the IT2 fuzzy tree correctly emphasized the uncertainty, producing a recommendation aligned with users who valued visuals more strongly. This qualitative example illustrates how uncertainty handling leads to more contextually reasonable recommendations.

Importantly, this model supports rating prediction without relying on matrix factorization or deep models, making it a lightweight and explainable alternative for personalized recommendation tasks.

d. Interpretability and robustness

The inherent rule-based structure of decision trees is preserved in our fuzzy models, ensuring interpretability. Each decision path can be traced through linguistic rules derived from histogram-based fuzzy sets (e.g., “If direction is high and acting is medium…”), which enhances transparency for end-users and system designers.

For example, an extracted rule can be expressed as: “If visuals is High and acting is Medium and story is Low, then user belongs to Cluster 4 with confidence [0.6, 0.8].” This interval-valued confidence reflects the uncertainty captured by the IT2 fuzzy model, providing interpretable and nuanced decision boundaries.

The confusion matrix and ROC curve analysis (AUC = 0.968) confirm that the model maintains high class-wise fidelity and performs robustly even in the presence of noise or ambiguous inputs. This robustness is essential in real-world systems where user feedback is often incomplete or noisy.

e. Computational trade-offs

Although the fuzzy models incur higher classification time than crisp trees (FDT-ID3: 13.1 ms vs ID3: 9.4 ms), the trade-off is minimal and justified by the significant gains in accuracy and robustness. Compared to other fuzzy variants, our approach remains efficient due to the lightweight construction of HBT1FMFs and the use of efficient minimum-membership evaluations rather than iterative optimization procedures like FCM.

Thus, the proposed system achieves a favorable balance between performance and runtime cost, making it viable for online recommendation settings Table 12.

The computational complexity of IT2 fuzzy decision trees scales linearly with the number of users and items, and quadratically with the number of criteria due to membership interval evaluations. Empirical runtime analysis confirmed that training time increases moderately with dataset size, while prediction remains efficient, averaging 13 ms per instance. This scalability makes the model practical for large-scale recommendation tasks. As the number of users or criteria increases, clustering remains efficient due to density-based pruning, while histogram construction scales linearly with data size. Decision tree depth grows moderately with feature dimensionality, ensuring manageable computational complexity even in large-scale environments.

f. Summary of advantages

The proposed FDT-ID3 model offers the best over-all performance, especially in terms of classification accuracy, robustness, and low prediction error, while preserving the clarity and traceability of decision trees.

Our work introduces a novel, data-driven fuzzification method through histogram-derived membership functions. This enables non-parametric, scalable fuzzification, making the model suitable for high-dimensional and noisy rating data.

Additionally, the integration of unsupervised HDBSCAN clustering for label generation complements the fuzzy tree learning phase, producing a hybrid model that is both unsupervised and interpretable, which is rare in the recommender systems literature.

These combined innovations represent a meaningful step forward in building transparent, robust, and accurate MCRS.

Extending the framework to general type-2 fuzzy sets introduces challenges in computational over-head and parameter calibration, as the secondary membership functions require complex footprint modeling. However, such an extension could capture richer uncertainty structures beyond interval bounds, potentially improving recommendation robustness in highly ambiguous rating environments. Future work may explore hybrid optimization strategies to mitigate computational costs while leveraging the expressive power of general type-2 sets.