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AYATSK

AYATSK is a classification-focused TSK model from the 2025 paper by Xue, Yang, and Wang.

Reference

G. Xue, Y. Yang and J. Wang, "Adaptive Yager T-Norm-Based Takagi–Sugeno–Kang Fuzzy Systems," in IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 55, no. 12, pp. 9802-9815, Dec. 2025, doi: 10.1109/TSMC.2025.3621346.

Mathematical Formulation

Antecedent

The paper uses Composite Exponential membership functions (CEMF) for each input feature:

\[ \mu_{r,d}(x_d) = K^{-1 + \exp\left(-\frac{(x_d - m_{r,d})^2}{2\sigma_{r,d}^2}\right)} \]

The lower bound is \(1/K > 0\), which is important because AYATSK derives the adaptive Yager index from a fixed lower bound rather than from a sample-dependent minimum.

Adaptive Yager T-norm

Rule firing strengths are computed with a parameterized Yager T-norm:

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

In the paper, \(\lambda\) is computed once from the input dimensionality \(D\) and the membership lower bound \(\varepsilon\):

\[ \lambda = -\frac{\ln D}{\ln(1 - \varepsilon)} \]

highFIS follows that policy by deriving \(\lambda\) from the number of input features and the CEMF lower bound in AYATSK models.

Defuzzification

AYATSK uses a sum-based defuzzifier to normalize rule activations:

\[ \bar{f}_r = \frac{w_r}{\sum_{i=1}^{R} w_i} \]

This is implemented in highFIS by SumBasedDefuzzifier.

Consequent (first-order)

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

Final outputs are weighted averages of rule consequents:

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

Code ↔ Paper Correspondence

Concept Class / Method Notes
Adaptive Yager T-norm AYATSKClassifierModel, AYATSKRegressorModel Default AYATSK path computes lambda from dimension and lower bound
Yager exponent \(\lambda\) YagerTNorm Fixed after dataset-level adaptation; not learnable per rule
Sum-based defuzzification SumBasedDefuzzifier Normalizes \(\bar{f}_r\) across rules
Composite Exponential MF CompositeExponentialMF Provides a positive lower bound needed by the adaptive strategy
Estimator wrapper AYATSKClassifier, AYATSKRegressor Builds input MFs and handles training/hyperparameters

Implementation notes

  • AYATSKClassifierModel uses MSE loss, Adam, and zero-initialized consequents by default.
  • AYATSKClassifier defaults to n_mfs=3, mf_init="grid", rule_base="coco", epochs=300, k=2, and learning_rate="auto" (0.01 up to 1000 features, 0.001 above).
  • AYATSKClassifier and AYATSKRegressor expose k (CEMF parameter), with required constraint k > 1.
  • With the default batch_size="auto", AYATSK uses full batch for N < 500 and 0.1 * N otherwise, matching the paper's training policy. Pass batch_size=None to force full batch at any size.
  • AYATSKRegressor remains available as a framework extension, but the paper itself evaluates classification only.

Estimator wrappers

  • AYATSKClassifier wraps AYATSKClassifierModel and supports classification with paper-style adaptive Yager aggregation.
  • AYATSKRegressor wraps AYATSKRegressorModel for regression tasks, but this is outside the paper’s evaluated scope.
  • Estimators accept the usual hyperparameters: n_mfs, mf_init, sigma_scale, random_state, epochs, learning_rate, batch_size, shuffle, validation_data, and patience.
  • pfrb_max_rules exists on the shared estimator base but is unused by AYATSK.

Alignment with the paper

  • The paper evaluates AYATSK on classification datasets only.
  • highFIS implements the paper-strict classification path with CompositeExponentialMF, dataset-level adaptive YagerTNorm, MSE loss, Adam, and zero-initialized consequents.
  • The regressor remains supported as a framework extension, but it should not be presented as part of the paper’s experimental validation.

Fidelity to the source article

The defaults of AYATSKClassifier are the experimental settings of the article for article: the composite exponential membership function, the adaptive Yager T-norm, the partition with three rules, consequents at zero, and Adam on the whole training set for 300 epochs, with a learning rate of 0.01 up to 1000 features and of 0.001 above. The article studies \(K\) from 2 to 10 and finds no value that is best everywhere; the default is \(K = 2\), the one with the best mean accuracy in its Table III (1.8 points above \(K = 10\) over the 23 datasets); on Leukemia \(K = 10\) gives 74% where \(K = 2\) gives 96%. On the same folds highFIS gives the accuracies of the authors' code on SRBCT, and the published accuracies are reproduced on Wine and Wdbc; see the reproduction.

The membership degrees are computed in one batched operation, as for the Gaussian families.