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AYATSK

AYATSK extends TSK by using an adaptive Yager T-norm aggregation and optional positive lower-bound membership functions to improve stability and performance in high-dimensional settings.

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

AYATSK uses Gaussian or composite exponential membership functions for each input feature:

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

The paper additionally recommends a Composite Exponential MF (CEMF) with a positive lower bound:

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

This lower bound is important because AYATSK derives the adaptive Yager index from a fixed minimum membership value.

Adaptive Yager T-norm

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

\[ w_r = \max\left(0,\; 1 - \left[\sum_{d=1}^{D} (1 - \mu_{r,d}(x_d))^{p_r}\right]^{1/p_r}\right) \]

Each rule has its own learnable exponent \(p_r > 0\). In highFIS, this is implemented by t_norm="yager" together with YagerTNorm.

The paper’s adaptive strategy sets \(p_r\) based on input dimensionality \(D\) and a lower bound \(\varepsilon\) on the membership values. This keeps the Yager aggregation numerically stable in high dimensions while preserving T-norm properties.

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 AYATSKClassifier, AYATSKRegressor Default t_norm="yager" in highFIS
Yager exponent \(p_r\) YagerTNorm Learnable per-rule exponent controlling softmin behavior
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 AYATSKClassifierEstimator, AYATSKRegressorEstimator Builds input MFs and handles training/hyperparameters

Implementation notes

  • AYATSKClassifier and AYATSKRegressor default to t_norm="yager".
  • The adaptive Yager index is compatible with standard Gaussian MFs, but the paper’s recommended CEMF ensures a positive lower bound and more stable high-dimensional behavior.
  • rule_base is typically "coco" for k-means initialization and "cartesian" for grid initialization.
  • AYATSKClassifierEstimator and AYATSKRegressorEstimator follow the standard highFIS estimator pattern and expose the same fitting parameters as other estimators.
  • The model is trained end-to-end with backpropagation; BaseTSK.fit() uses mini-batch Adam optimization with optional early stopping.

Estimator wrappers

  • AYATSKClassifierEstimator wraps AYATSKClassifier and supports classification with adaptive Yager aggregation.
  • AYATSKRegressorEstimator wraps AYATSKRegressor for regression tasks.
  • 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 defines AYATSK through an adaptive Yager T-norm that avoids product underflow in high dimensions.
  • highFIS implements the same core idea with YagerTNorm and first-order TSK consequents.
  • CompositeExponentialMF is available in highFIS to reflect the paper’s use of a positive lower bound membership function.
  • The estimator wrappers preserve the paper’s training regime of gradient-based optimization over antecedent parameters, Yager exponents, and consequent weights.