Vanilla TSK

Vanilla TSK is the standard first-order Takagi-Sugeno-Kang fuzzy model with Gaussian antecedents, product aggregation, and sum-based defuzzification.

Reference

Takagi, T. & Sugeno, M. (1985). "Fuzzy identification of systems and its applications to modeling and control." IEEE Trans. Syst., Man, Cybern. SMC-15(1):116–132. DOI: 10.1109/TSMC.1985.6313399

Mathematical Formulation

Antecedent

Each rule evaluates \(D\) Gaussian membership functions and aggregates them with the product t-norm:

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

\[ w_r = \prod_{d=1}^{D} \mu_{r,d}(x_d) \tag{2} \]

Defuzzification (sum-based)

Vanilla TSK normalizes rule firing strengths by their sum:

\[ \bar{f}_r = \frac{w_r}{\sum_{i=1}^{R} w_i} \tag{3} \]

In highFIS this is implemented by highfis.defuzzifiers.SumBasedDefuzzifier.

When the antecedent is rewritten in log-space, Equation (3) is mathematically equivalent to a softmax over log firing strengths:

\[ \bar{f}_r = \frac{\exp(\log w_r)}{\sum_{i=1}^{R} \exp(\log w_i)}. \tag{4} \]

This exposes the saturation issue identified in HTSK_2021.md: as \(D\) grows, the log-domain activations become more extreme, making the normalized weights dominated by a single rule.

Consequent (first-order)

The final TSK output is a weighted sum of first-order consequents:

\[ \hat{y} = \sum_{r=1}^{R} \bar{f}_r \left( b_{r,0} + \sum_{d=1}^{D} b_{r,d}\, x_d \right) \tag{5} \]

Code ↔ Paper Correspondence

Equation	Class / Method	Description
(1)	`GaussianMF.forward()`	Gaussian membership evaluation
(2)	`RuleLayer` with `t_norm="prod"`	Product t-norm antecedent
(3)	`SumBasedDefuzzifier.forward()`	Sum-based rule normalization
(4)	`SoftmaxLogDefuzzifier`	Equivalent log-space softmax form
(5)	`ClassificationConsequentLayer.forward()` / `RegressionConsequentLayer.forward()`	Weighted first-order consequent

Implementation notes

TSKClassifier and TSKRegressor default to SumBasedDefuzzifier.
The highFIS estimator wrappers TSKClassifierEstimator and TSKRegressorEstimator construct the Gaussian antecedent MFs from input_configs, n_mfs, mf_init, and sigma_scale.
GaussianMF stores a raw parameter that is transformed with softplus to guarantee \(\sigma > 0\).
SumBasedDefuzzifier clamps rule weights to a small floor before normalization, improving numeric stability without changing the semantic sum-based normalization.
In high dimensions, vanilla TSK is prone to softmax saturation because the normalized weights in Equation (4) are equivalent to a softmax over log-domain firing strengths.