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: 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:

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

Selectable building blocks

The formulation above is the default of TSKClassifier and TSKRegressor. They are also the generic estimators of highFIS: the membership function, the T-norm and the defuzzifier can be chosen with the mf, t_norm and defuzzifier arguments.

from highfis import TSKClassifier

clf = TSKClassifier(n_mfs=3, mf="bell", t_norm="min", defuzzifier="softmax_log")

The available values and what each one does are described in Building Blocks of the Generic TSK. The other families keep their published combination and do not take these arguments.

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 TSKClassifier and TSKRegressor 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.

Strict paper mode

With the default batch_size="auto" the estimator applies the HTSK batch protocol: 512, falling back to min(N_t, 60) when it exceeds the training set.
Validation split is not auto-generated by the model/estimator; pass x_val and y_val explicitly when you want early stopping based on validation metrics.