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

Method	Fuzzy type	Uncertainty handling	Limitation	Proposed work advantage
Traditional Fuzzy RS	Type-1	Limited	Cannot model higher uncertainty	Uses IT2 (FOU)
FCM RS	Type-1	Partial	Sensitive, not data-driven	Histogram-based construction
Rule-Based Fuzzy RS	Type-1	Limited	Depends on expert rules	Fully data-driven approach
Neuro-Fuzzy RS	Type-1	Moderate	Complex, less interpretable	Tree-based interpretability
Existing IT2 Systems	Type-2	Good	Not integrated with trees	Combines IT2 with decision trees
Proposed Method	IT2	Strong (FOU)	—	Data-driven, interpretable, multi-criteria

[i] FCM, fuzzy C-means; FOU, footprint of uncertainty; IT2, interval type-2.

Table 2:

Comparison with non-tree fuzzy and neural recommender models

Model	Type	Uncertainty handling	Interpretability	Limitation	Proposed work advantage
Neuro-Fuzzy RS	Hybrid	Moderate	Low	Complex, less transparent	Clear rule-based decisions
Neural Recommender (Deep Learning)	Neural	Implicit	Very low	Black-box, high complexity	Interpretable fuzzy rules
Fuzzy Rule-Based RS	Fuzzy	Moderate	High	Not scalable	Scalable via clustering + trees
Proposed Method	IT2 Fuzzy Tree	Strong (FOU)	High	—	Handles uncertainty + interpretable + scalable

[i] FOU, footprint of uncertainty; IT2, interval type-2.

Block diagram of proposed methodology. FOU, footprint of uncertainty; HBIT2FMFs, histogram-based interval type-2 fuzzy membership functions; IT2, interval type-2.

Transformation of raw ratings to IT2 fuzzy sets. FOU, footprint of uncertainty; IT2, interval type-2; LMF, lower membership function; UMF, upper membership function.

Table 3:

HDBSCAN clustering quality on multi-criteria ratings

Criterion	No. of clusters	Avg. stability	Silhouette coefficient	XB index
HDBSCAN on 5D rating vector	5	0.83	0.67	0.0112

[i] XB, Xie–Beni.

Table 4:

K-means subclustering results within HDBSCAN superclusters

Supercluster ID	No. of subclusters	Avg. intra-cluster cosine similarity
1	3	0.89
2	2	0.85
3	1	0.79
4	2	0.88
5	2	0.83

Table 5:

Number of users per rating value for each criterion

Criteria	0.5	1	1.5	2	2.5	3	3.5	4	4.5	5
Overall	1,571	4,577	4,246	2,027	113	1,372	1,181	2,783	3,962	3,535
Story	3,266	522	979	3,399	3,534	839	2,155	133	482	449
Acting	2,274	1,695	906	2,008	1,253	4,098	1,153	3,442	786	1,965
Direction	1,727	4,352	625	1,371	2,995	2,745	2,157	2,991	1,750	339
Visuals	3,109	2,892	400	2,370	995	3,156	139	3,276	178	2,600
Total	11,947	14,038	7,156	11,175	8,890	12,210	6,785	12,625	7,158	8,888

Low, medium, and high HBT1FMFs for each rating criterion. HBT1FMFs, histogram-based type-1 fuzzy membership functions.

Table 6:

Bias and variance estimates across models

Model	Bias	Variance
FDT-ID3	0.078	0.021
FDT-C4.5	0.092	0.030
ID3	0.136	0.034
C4.5	0.142	0.037

Table 7:

Classification metrics comparison

Model	Precision	Recall	F1-score	F_0.5-score	F₂-score
FDT-ID3	0.95	0.96	0.95	0.96	0.95
FDT-C4.5	0.93	0.91	0.92	0.93	0.91
ID3	0.87	0.84	0.85	0.86	0.84
C4.5	0.86	0.83	0.84	0.85	0.83

Table 8:

Confusion matrix for FDT-ID3

Actual\predicted	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
Cluster 1	0.94	0.02	0.01	0.00	0.01	0.00	0.00	0.01	0.00	0.01
Cluster 2	0.01	0.92	0.03	0.01	0.00	0.01	0.00	0.01	0.00	0.01
Cluster 3	0.00	0.02	0.93	0.01	0.01	0.00	0.00	0.01	0.01	0.01
Cluster 4	0.01	0.01	0.02	0.92	0.02	0.00	0.00	0.01	0.00	0.01
Cluster 5	0.01	0.01	0.01	0.02	0.93	0.00	0.00	0.01	0.00	0.01
Cluster 6	0.01	0.00	0.01	0.01	0.00	0.93	0.02	0.01	0.00	0.01
Cluster 7	0.00	0.00	0.00	0.01	0.01	0.02	0.93	0.01	0.00	0.02
Cluster 8	0.01	0.01	0.03	0.00	0.01	0.00	0.01	0.93	0.00	0.00
Cluster 9	0.00	0.01	0.01	0.00	0.01	0.01	0.00	0.00	0.94	0.02
Cluster 10	0.01	0.01	0.01	0.00	0.00	0.02	0.03	0.00	0.00	0.92

Table 9:

MAE across models and average runtime per classification

Model	MAE	Avg. time per instance (ms)
FDT-ID3	0.5137	13.1
FDT-C4.5	0.5982	15.7
ID3	0.6720	9.4
C4.5	0.6881	10.1

[i] MAE: mean absolute error.

Table 10:

AUC for binary classification

Model	AUC
FDT-ID3	0.968
FDT-C4.5	0.936
ID3	0.906
C4.5	0.868

[i] AUC, area under curve.

The ROC curve of four models. AUC, area under curve.

Table 11:

Comparison with non-fuzzy state-of-the-art models

Model	Type	Handles sparsity	Uncertainty modeling	Interpretability	Limitation	Proposed work advantage
Matrix Factorization	Non-fuzzy	Yes	No (implicit only)	Low	Black-box, no uncertainty	Explicit uncertainty via IT2
Deep Learning RS	Non-fuzzy	Yes	Partial	Very Low	High complexity, less explainable	Interpretable decision tree
Probabilistic/Bayesian RS	Non-fuzzy	Yes	Yes (probabilistic)	Moderate	Requires distribution assumptions	No assumptions, FOU-based modeling
KNN-based CF	Non-fuzzy	Limited	No	High	Sensitive to sparsity/noise	Robust via fuzzy clustering
Proposed Method	IT2 Fuzzy	Yes	Strong (FOU)	High	—	Handles uncertainty + interpretable + robust

[i] CF, collaborative filtering; FOU, footprint of uncertainty; IT2, interval type-2; KNN, k-nearest neighbours; RS, recommender system.

Table 12:

Qualitative comparison of all models based on experimental outcomes

Criterion	FDT-ID3 (HBT1FM)	FDT-C4.5 (HBT1FM)	ID3	C4.5
Accuracy	5	4	3	2
Interpretability	5	5	5	5
Robustness to Noise	5	4	3	2
Computational Efficiency	4	3	5	5
Prediction Error (MAE)	5	3	2	1

[i] HBT1FMFs, histogram-based type-1 fuzzy membership functions; MAE, mean absolute error.

References

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Articles in this issue

DOI: https://doi.org/10.2478/ijssis-2026-0074 | Journal eISSN: 1178-5608

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Language: English

Submitted on: Feb 2, 2026

Published on: Sep 21, 2026

Published by: International Journal on Smart Sensing and Intelligent Systems

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

interval type-2 fuzzy decision tree,

HBIT2FMF,

multi-criteria recommendation,

HDBSCAN,

redundant ratings

Related subjects:

Engineering,

Introductions and overviews,

Engineering, other

© 2026 Heersh Azeez, Sadegh Aminifar, published by International Journal on Smart Sensing and Intelligent Systems
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2026): Issue 1 (January 2026)

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

Figures & Tables

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Table 2:

Figure 1:

Flowchart 1: