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REVIEW 4 major objections 5 minor 88 references

Rule-Based Modeling of Low-Dimensional Data with PCA and Binary Particle Swarm Optimization (BPSO) in ANFIS

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that applying PCA to normalized firing strengths and selecting components with BPSO drastically reduces ANFIS rule counts and training time while preserving accuracy on low-dimensional data.

desk verdict A workable computational shortcut for shrinking ANFIS layer 3-to-4, but the 'fewer rules' claim conflates principal components with fuzzy rules and needs major rework before it supports interpretability. read the letter →

arxiv 2502.03895 v1 pith:T6T2V7EO submitted 2025-02-06 cs.CV

Afnan Al-Ali , Uvais Qidwai This is my paper · ORCID
classification cs.CV
keywords ANFISrulereductionBinaryParticleSwarmOptimizationPrincipalComponentAnalysisfiringstrengthsinterpretablefuzzyruleslow-dimensionaldataclassificationandregression
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a rule-reduction method for ANFIS, a five-layer neuro-fuzzy network whose grid-partitioned rule base grows exponentially with input dimensionality. The authors insert a two-stage selector between the normalization layer and the defuzzification layer: PCA first decorrelates the normalized firing strengths of all generated rules, and Binary Particle Swarm Optimization then picks a small subset of the resulting components that minimizes prediction error. Their claim is that on low-dimensional classification and regression benchmarks this keeps accuracy within a few points of the full ANFIS while shrinking rule counts from dozens or hundreds to roughly two to four rules and cutting training time by orders of magnitude. The practical payoff is that a transparent, rule-based model becomes fast enough and simple enough for settings where interpretability matters and computing time is limited.

What carries the argument

The load-bearing mechanism is the two-stage PCA-BPSO selector interposed between layer 3 (normalized firing strengths $\bar{w}_j$) and layer 4 (weighted consequents $\bar{w}_j f_j$) of ANFIS. PCA converts the matrix of normalized firing strengths, $\text{MAT}_{Nfs}$, into linearly uncorrelated principal components, and BPSO maintains a swarm of binary position vectors that act as a keys matrix, element-wise multiplying the components; the fitness function is the prediction error of the masked rule set on training data. The inertia weight and acceleration coefficients of BPSO are linearly updated each iteration so that exploration gives way to exploitation, and because the update loop touches only layers 3 and 4, the antecedent membership-function parameters stay fixed at their initialization throughout pruning.

What would settle it

On a dataset with highly skewed or poorly initialized membership functions, run ANFIS-PCA-BPSO and then re-train the premise parameters of the surviving rules; if accuracy jumps substantially, the initial firing strengths were ranking rules by importance incorrectly. Alternatively, compare the selected rule subset against a randomly chosen subset of the same size: if random subsets match its accuracy across several datasets, the firing-strength-based selection is not doing the work.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the normalized firing strengths of a fully generated ANFIS rule base contain enough information to decide which rules matter, and that this decision can be made by an embedded optimizer rather than by expert-set thresholds. The method forms a matrix of normalized firing strengths over samples and rules, applies PCA to obtain uncorrelated components (keeping those explaining about 95% of variance), and lets BPSO generate a binary keys matrix that selects components by evaluating the resulting rule set's error. Only the consequent parameters are updated during the forward pass, so the optimization loop runs entirely between layers 3 and 4 without backpropagation through the antecedent part. The reported outcome is a rule base reduced to between about 2 and 4 rules on most benchmarks, training times reduced from thousands of seconds to under a minute in most cases, and accuracy that matches or slightly exceeds the full ANFIS on several datasets, with only a small accuracy drop on a few others.

Load-bearing premise

The method assumes that the normalized firing strengths computed from the initial, un-tuned membership functions faithfully rank which rules matter, even though those membership functions are never updated after the BPSO pruning step.

Editorial extensions

If this is right

  • Rule counts drop from 9-256 in the full ANFIS baseline to roughly 2-4 rules across the tested benchmarks, so the pruned model is far cheaper to store, inspect, and execute.
  • Training time drops from thousands of seconds to under a minute on most classification and regression benchmarks, making ANFIS practical for rapid retraining or real-time deployment.
  • Accuracy stays within a few points of the full ANFIS on most datasets and improves on Iris, Banana, Haberman, and Monk2, so the main cost of pruning is small and occasional.
  • On the ischemic-stroke application, the pruned model keeps statistically significant predictions (p-value 0.00007) with a small rule base, suggesting that the reduction does not destroy predictive signal in a real clinical task.
  • Compared with published rule-reduction baselines, the method achieves the fewest rules on 7 of 8 classification datasets and the best or near-best RMSE on 3 of 4 regression datasets, locating it on the efficient end of the accuracy-complexity frontier.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This reader's inference: because the PCA-BPSO step operates on normalized firing strengths rather than raw features, the same selector could be dropped into other rule-based systems that normalize rule activations, such as TSK fuzzy classifiers outside the ANFIS architecture.
  • This reader's inference: the fixed-premise assumption means the method's success depends on initial membership functions being reasonable; a natural follow-up is to re-tune the surviving rules' premise parameters after pruning and measure the accuracy gain.
  • This reader's inference: replacing the accuracy-only BPSO fitness with a multi-objective score that penalizes rule count could yield even smaller rule bases, since the current results already achieve near-baseline accuracy with very few rules.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes ANFIS-PCA-BPSO, a rule-reduction technique that inserts PCA and binary particle swarm optimization between layers 3 and 4 of ANFIS. The normalized firing strengths are first transformed by PCA; BPSO then selects a subset of components using a fitness function based on training error. The authors report experiments on eight classification and four regression benchmarks plus an ischemic stroke dataset, comparing accuracy/RMSE, number of rules, and training time against standard ANFIS and several published rule-reduction methods. Their main claims are fewer rules, shorter training, and preserved accuracy.

Significance. If correct, this would be a practically useful pruning strategy for low-dimensional ANFIS, and the paper has strengths: it evaluates across multiple benchmarks, reports standard deviations, includes ablation studies, and applies the method to a real medical dataset. However, the central claim is undermined by the fact that PCA components are dense linear combinations of firing strengths rather than fuzzy rules, so the reported 'rule counts' do not count interpretable fuzzy rules. The dimensional inconsistency in the formal definition and the feature-selection leakage in the real-data experiment further weaken the empirical support.

major comments (4)
  1. [Section 4.3.1, Eqs. (18)-(19)] The formal definition of the proposed integration is inconsistent. MAT_Nfs is defined as N x M and keys is defined as M x D, so the Hadamard product in Eq. (19) is undefined unless N = M = D. The displayed result also claims a size of M x D, which contradicts the row dimension N of MAT_Nfs. Even under the natural correction of a matrix product, MAT_Nfs times keys would be N x D, not M x D. Because this product is the mathematical core of the proposed pipeline, the method is not reproducible as written.
  2. [Abstract, Section 4.3, Tables 2, 6, and 7] The central claim that the method reduces the number of fuzzy rules is not supported by the described procedure. PCA on the normalized firing strengths produces principal components that are dense linear combinations of all M firing strengths. Selecting K of these components gives an output of the form sum over selected components of (sum over all original rules of loadings times firing strengths) times a consequent function; every original rule still contributes through the loadings, and the selected objects are not fuzzy rules with conjunctive antecedents. Reporting '2 rules' for Iris in Table 2, or the rule counts in Tables 6 and 7, therefore counts selected principal components, not fuzzy rules. Without evidence that the PCA loadings are one-hot or sparse—which PCA generically does not provide—the rule-reduction and interpretability claims in the abstract and contributions section are a category error.
  3. [Algorithm 1, Section 4.3] The optimization loop operates exclusively between layers 3 and 4, and only the consequent parameters are updated after the PCA-BPSO selection. The firing strengths used for PCA are therefore those of the initial grid-partitioned FIS, whose membership functions are never re-estimated. If the initial membership functions are poorly centered or scaled, the variance ranking from PCA may not reflect the true contribution of each rule to the final prediction. The paper does not provide an ablation that varies the initialization or re-tunes premise parameters, so the robustness of the selection step is unexamined.
  4. [Section 9, Table 8] For the real-world ischemic stroke experiment, the input features are selected using p-values and Pearson correlations computed on the entire dataset (Table 8) before the 5-fold cross-validation is run. This is a selection-before-validation procedure and can optimistically bias the reported results. A nested cross-validation, or feature selection performed inside each training fold, is needed to support the real-data claim.
minor comments (5)
  1. [Section 2, Eq. (5)] The statement that the consequent parameter vector 'equals (n_inputs+1) x j' is dimensionally unclear; it should say that each rule j has a consequent parameter vector of length n_inputs+1.
  2. [Section 6.1, Table 2] The training time of the baseline ANFIS has very large standard deviations (for example, 3.53e+03 +/- 6.65e+03 seconds on Iris); the source of this variance and the convergence criterion used for the baseline should be stated.
  3. [Section 5.1, Table 1] The Airfoil dataset is listed with 4 features, but the standard UCI Airfoil self-noise dataset has 5 input features; please verify the entry.
  4. [References] Reference [62] and reference [65] are the same paper and appear twice in the bibliography; the duplicate should be removed.
  5. [Section 10] The conclusion states that the model's efficacy diminishes with datasets having a large number of features; this limitation restricts the scope to low-dimensional data and should be stated explicitly in the abstract as well.

Circularity Check

2 steps flagged · score 6.0 of 10

Reported 'fewer rules' counts PCA components rather than fuzzy rules, and the real-dataset feature selection leaks test labels; benchmark accuracy/time results are otherwise independent.

  1. self definitional [Section 4.3.1 (Eqs. 18-19), Algorithm 1; Tables 2, 6, 7]
    "In this approach, the selected components of the normalized firing strengths are treated as input features to the BPSO algorithm, generating a matrix comprising candidate rules."

    PCA on the N×M normalized firing-strength matrix produces principal-component scores, each a dense linear combination of all M original firing strengths. BPSO selects a subset S of these scores, and the paper reports the size of S as the reduced rule count (e.g., '#rules 2(±1.22)' for Iris in Table 2). The final Layer-5 output can be rewritten as a sum over all original firing strengths with combined consequent functions, so no original fuzzy rule is actually removed. The 'rule reduction' is therefore a definitional consequence of counting selected PCA components as rules, not a reduction of the fuzzy rule base.

  2. fitted input called prediction [Section 9 (Real-World Application); Tables 8-10]
    "we calculated this dataset's P-value and correlation coefficient between each feature and our target (the IGRII). This calculation helped us choose the most significant features that impact the prediction of the IGR II. The final set of features selected for our model is (DYSLIP, UNCAL, TEMPORAL, INVOL1, and Growthrate_1)."

    The p-values and Pearson correlations in Table 8 are computed on the full 204-record stroke dataset before the 80/20 training/test split. These statistics select the five input features used in the model. The test folds are therefore not unseen: their target values (IGR II) have already influenced which features enter the model. The test-set metrics in Tables 9-10 are thus partly in-sample, and the reported predictive performance is statistically forced by feature selection that has seen the test labels.

full rationale

The benchmark experiments are mostly self-contained: BPSO uses training error as its fitness function, and the reported test accuracies are empirical, so the accuracy and training-time comparisons do not reduce to fitted values. The main circularity is in the central 'fewer rules' claim: the paper equates BPSO-selected PCA component scores with fuzzy rules, making the reduced rule count, by construction, the number of selected components; the output still depends on all original firing strengths through the PCA loadings, so no fuzzy rule is actually removed. A second, evaluation-level circularity appears in the real-world application, where feature selection uses p-values and correlations computed on the full dataset before the split, making the test evaluation partly in-sample. Citations to the authors' prior ANFIS-BPSO work [85] and [87] are used only as empirical baselines and are not load-bearing for the derivation. The training-time savings and benchmark accuracy results retain independent content, so the paper is partially circular rather than wholly so.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The model introduces no new physical or mathematical entities. It relies on existing ANFIS, PCA, and BPSO components. The free parameters are standard hyperparameters and heuristic choices, though some are tuned manually. The key domain assumption is that firing strengths from the initial membership functions are a valid basis for pruning.

free parameters (7)
  • PCA explained variance threshold = 95%
    The number of PCA components is determined by retaining 95% of variance, which is a user-chosen hyperparameter (Section 4.3.1).
  • Number of membership functions = 2 for benchmark, 3 for stroke dataset
    The number of fuzzy sets per input is a free choice; the paper uses 2 for most experiments and 3 for the real dataset comparison (Section 9).
  • Membership function type = Generalized Bell
    The gbell membership function is chosen, not derived (Section 6).
  • BPSO inertia weight bounds = w_max=0.9, w_min=0.4
    Taken from literature [80], not estimated from data (Section 4.3.2).
  • BPSO acceleration coefficient bounds = c1: 2.5 to 0.5, c2: 0.5 to 2.5
    Assigned values from literature [81], dynamically updated during training (Section 4.3.2).
  • BPSO swarm size = Equal to the number of rules
    Heuristic choice: 'the swarm size is equal to the number of generated rules' (Section 6.1).
  • Number of iterations = 100 for benchmarks, 150 for stroke dataset
    Iteration count is set by the authors (Sections 6 and 9).
assumptions (4)
  • domain assumption ANFIS architecture and hybrid learning algorithm are valid for the given data
    The paper relies on the standard ANFIS formulation (Section 2) without questioning its applicability to the selected datasets.
  • domain assumption PCA on normalized firing strengths captures the rule significance structure
    The premise that the firing-strength matrix contains enough information for rule pruning is assumed, not derived (Section 4.3.1).
  • domain assumption BPSO can effectively select the most informative PCA components
    The paper assumes that BPSO with training error as fitness will identify a good rule subset (Algorithm 1).
  • domain assumption The normalized firing strengths are meaningful without updating antecedent parameters
    The method fixes premise parameters after initial generation and only updates consequents, which presumes the initial membership functions are adequate (Algorithm 1, Section 4.3.2).

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Cite this review

Pith. "Pith review of Rule-Based Modeling of Low-Dimensional Data with PCA and Binary Particle Swarm Optimization (BPSO) in ANFIS." pith.science (2026). https://pith.science/paper/T6T2V7EO

@misc{pith2026250203895,
  author       = {Pith},
  title        = {Pith review of: Rule-Based Modeling of Low-Dimensional Data with PCA and Binary Particle Swarm Optimization (BPSO) in ANFIS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6T2V7EO}},
  note         = {Machine review of arXiv:2502.03895}
}
read the original abstract

Fuzzy rule-based systems interpret data in low-dimensional domains, providing transparency and interpretability. In contrast, deep learning excels in complex tasks like image and speech recognition but is prone to overfitting in sparse, unstructured, or low-dimensional data. This interpretability is crucial in fields like healthcare and finance. Traditional rule-based systems, especially ANFIS with grid partitioning, suffer from exponential rule growth as dimensionality increases. We propose a strategic rule-reduction model that applies Principal Component Analysis (PCA) on normalized firing strengths to obtain linearly uncorrelated components. Binary Particle Swarm Optimization (BPSO) selectively refines these components, significantly reducing the number of rules while preserving precision in decision-making. A custom parameter update mechanism fine-tunes specific ANFIS layers by dynamically adjusting BPSO parameters, avoiding local minima. We validated our approach on standard UCI respiratory, keel classification, regression datasets, and a real-world ischemic stroke dataset, demonstrating adaptability and practicality. Results indicate fewer rules, shorter training, and high accuracy, underscoring the methods effectiveness for low-dimensional interpretability and complex data scenarios. This synergy of fuzzy logic and optimization fosters robust solutions. Our method contributes a powerful framework for interpretable AI in multiple domains. It addresses dimensionality, ensuring a rule base.

Figures

Figures reproduced from arXiv: 2502.03895 by the authors.

Figure 1
Figure 1. Main Challenges in ANFIS. Addressing these challenges in ANFIS requires domain expertise, careful rule￾based development, thoughtful integration with other techniques, and considerations for scalability and interpretability. Additionally, advances in machine learning tech￾niques, methodologies, and domain-specific knowledge can contribute to overcoming these challenges and enhancing the effectiveness of ANFIS in var… view at source ↗
Figure 2
Figure 2. Detailed Traditional ANFIS Architecture with Five Distinct Layers, where the squared nodes refer to the adaptive nodes and the circle nodes refer to the fixed nodes. The fourth layer is the beginning of the consequent part, known as the defuzzification layer, see equation 4, representing the weighted values of the rules using a first-order polynomial equation of consequent parameters multiplied by the normalized fir… view at source ↗
Figure 3
Figure 3. The architecture of the proposed ANFIS-PCA-BPSO, showing the addition of the two stages of PCA and BPSO between layers 3 and 4. where 𝑁 is the number of samples and 𝑀 is the number of rules. 𝑀𝐴𝑇𝑁𝑓 𝑠 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑁𝑓 𝑠1 1 𝑁𝑓 𝑠2 1 ... 𝑁𝑓 𝑠𝑚−1 1 𝑁𝑓 𝑠𝑚 1 𝑁𝑓 𝑠1 2 𝑁𝑓 𝑠2 2 ... 𝑁𝑓 𝑠𝑚−1 2 𝑁𝑓 𝑠𝑚 2 ∶ ∶ ... ∶ ∶ ∶ ∶ ... ∶ ∶ ∶ ∶ ... ∶ ∶ 𝑁𝑓 𝑠1 𝑛 𝑁𝑓 𝑠2 𝑛 ... 𝑁𝑓 𝑠𝑚−1 𝑛 𝑁𝑓 𝑠𝑚 𝑛 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ BPSO will generate another matri… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Comparative Evaluation of ANFIS and ANFIS-PCA-BPSO Models on Various Classification Datasets [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: Comparative Evaluation of ANFIS and ANFIS-PCA-BPSO Models on Various Regression Datasets. documentation in their respective publications precluded a comparative analysis in this dimension, thereby introducing an element of analytical opacity [PITH_FULL_IMAGE:figures/f…
Figure 6
Figure 6. Figure 6: elucidates a comparative canvas, presenting a visual comparison be￾tween our propounded model and the state-of-the-art techniques, articulating accu￾racy and rule generation metrics, and providing a bifocal lens through which model performance can be appraised [PITH_F…
Figure 7
Figure 7. Figure 7: A Detailed Analysis of Algorithm Performance: Examining Rule Count and RMSE Against State-of-the-art Techniques for the Regression Benchmarks. 9. Real-World Application The ischemic Stroke dataset has been used in [86],[87], and [85], and it is not publicly available. …
Figure 8
Figure 8. Figure 8: Comparative Evaluation of ANFIS and ANFIS-PCA-BPSO on a Real Dataset Using Multiple Evaluation Metrics. comparison was made with two techniques: the approach proposed by [87], and this is for two primary reasons. Firstly, their research objective is aligned with ours, …

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Reviewed August 9, 2026 · model on record in the stance chip above.