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DombiTSK

DombiTSK replaces product aggregation with an adaptive Dombi t-norm in the antecedent, providing more flexible high-dimensional rule activation behavior.

Reference

G. Xue, L. Hu, J. Wang and S. Ablameyko, "ADMTSK: A High-Dimensional Takagi–Sugeno–Kang Fuzzy System Based on Adaptive Dombi T-Norm," in IEEE Transactions on Fuzzy Systems, vol. 33, no. 6, pp. 1767-1780, June 2025, doi: 10.1109/TFUZZ.2025.3535640.

Mathematical Formulation

Antecedent

Each rule-term membership is computed with a Gaussian function:

\[ \mu_{r,d}(x_d)=\exp\left(-\frac{(x_d-c_{r,d})^2}{2\sigma_{r,d}^2}\right) \]

where \(c_{r,d}\) is the center and \(\sigma_{r,d}>0\) is the spread.

Dombi aggregation

DombiTSK uses the Dombi t-norm to combine the antecedent membership values:

\[ \phi_r = \frac{1}{1 + \left(\sum_{d=1}^{D} \left(\frac{1 - \mu_{r,d}}{\mu_{r,d}}\right)^{\lambda}\right)^{1/\lambda}} \]

The hyperparameter \(\lambda > 0\) controls the aggregation shape:

  • small \(\lambda\) makes the aggregation softer and closer to product-like behavior,
  • large \(\lambda\) makes the aggregation sharper and closer to minimum-like behavior.

Normalization

Rule firing strengths are normalized by sum normalization:

\[ \bar{\phi}_r = \frac{\phi_r}{\sum_{k=1}^{R} \phi_k} \]

Consequent

DombiTSK uses a first-order TSK consequent for both classification and regression.

For classification:

\[ \mathbf{y}_r = W_r \mathbf{x} + \mathbf{b}_r \]

For regression:

\[ \hat{y}_r = \mathbf{w}_r^\top \mathbf{x} + b_r \]

Output aggregation

The final prediction is the normalized weighted sum of rule consequents:

  • Classification:
\[ \mathbf{y} = \sum_{r=1}^{R} \bar{\phi}_r \mathbf{y}_r \]
  • Regression:
\[ \hat{y} = \sum_{r=1}^{R} \bar{\phi}_r \hat{y}_r \]

Code ↔ Paper Correspondence

Equation Class / Method Description
Dombi aggregation DombiTSKClassifier / DombiTSKRegressor Fixed-λ Dombi antecedent aggregation with t_norm="dombi"
Normalization SumBasedDefuzzifier Sum-based rule strength normalization
Consequent ClassificationConsequentLayer / RegressionConsequentLayer First-order linear consequents
Membership functions GaussianMF Standard Gaussian antecedent MFs

Implementation notes

  • DombiTSKClassifier and DombiTSKRegressor use a fixed Dombi parameter lambda_ > 0 in the antecedent and default to SumBasedDefuzzifier.
  • DombiTSKClassifierEstimator and DombiTSKRegressorEstimator are sklearn-compatible wrappers that build the rule base and membership functions from input_configs, n_mfs, mf_init, and sigma_scale.
  • The estimators default to mf_init="kmeans" and sigma_scale=1.0.
  • The default rule_base for estimator-built models is "coco" with mf_init="kmeans" and "cartesian" with mf_init="grid".
  • CompositeGaussianMF is available when a positive lower bound on antecedent membership values is desired, supporting ADMTSK-style stability.
  • AdaptiveDombiRuleLayer is implemented in the codebase and provides per-rule adaptive Dombi exponents, but there is currently no dedicated ADMTSK wrapper class exposing this behavior directly.

Alignment with the paper

  • The paper defines a Dombi TSK baseline with Dombi antecedent aggregation and first-order consequent structure.
  • highFIS implements this baseline directly through DombiTSKClassifier and DombiTSKRegressor.
  • Rule strengths are normalized by sum-based defuzzification, matching the paper's TSK output aggregation.
  • The package also includes building blocks for the ADMTSK extension: CompositeGaussianMF and AdaptiveDombiRuleLayer.
  • A full ADMTSK-style model would require combining these components with an adaptive λ selection mechanism, which is not currently wrapped by a single model class.