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AdaTSK

AdaTSK extends TSK with an adaptive softmin antecedent that stabilizes high-dimensional fuzzy inference while preserving first-order TSK consequents.

Reference

G. Xue, Q. Chang, J. Wang, K. Zhang and N. R. Pal, "An Adaptive Neuro-Fuzzy System With Integrated Feature Selection and Rule Extraction for High-Dimensional Classification Problems," in IEEE Transactions on Fuzzy Systems, vol. 31, no. 7, pp. 2167-2181, July 2023, doi: 10.1109/TFUZZ.2022.3220950

Mathematical Formulation

AdaTSK extends TSK fuzzy inference by using an adaptive softmin antecedent (Ada-softmin) together with first-order linear consequents.

Antecedent

Each rule-term membership is typically 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.

In highFIS, the default estimator wrappers build standard Gaussian MFs. The paper's proposed positive lower-bound variant can be instantiated with highfis.memberships.CompositeGaussianMF when desired.

Adaptive Ada-softmin aggregation

AdaTSK computes rule firing strengths with an adaptive softmin based on the minimum antecedent membership for each rule:

\[ \hat{q}_r = \left\lceil \frac{690}{\ln (\min_{d}\mu_{r,d}(x_d))} \right\rceil, \quad \hat{q}_r \in [-1000, -1] \]
\[ \phi_r = \left( \frac{1}{D} \sum_{d=1}^{D}\mu_{r,d}(x_d)^{\hat{q}_r} \right)^{1/\hat{q}_r} \]

The exponent \(\hat{q}_r\) is recomputed on every forward pass and is clamped for numerical stability, which avoids the fixed-parameter softmin problems of underflow and fake minimum.

Normalization

Rule firing strengths are normalized by simple sum normalization:

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

Consequent

AdaTSK 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
Adaptive softmin highfis.layers.AdaSoftminRuleLayer Computes per-rule softmin exponents from the minimum membership value
Normalization highfis.defuzzifiers.SumBasedDefuzzifier Standard sum-based rule strength normalization
Consequent ClassificationConsequentLayer / RegressionConsequentLayer First-order linear consequents
Membership functions highfis.memberships.GaussianMF Default Gaussian antecedent MFs
Optional membership highfis.memberships.CompositeGaussianMF Optional positive lower-bound MF matching the paper variant

Implementation notes

In highFIS, AdaTSKClassifier and AdaTSKRegressor implement the core AdaTSK model by replacing the standard product antecedent with the adaptive softmin operator.

Model classes

  • AdaTSKClassifier and AdaTSKRegressor use highfis.layers.AdaSoftminRuleLayer to compute rule strengths.
  • The TSK consequent remains first-order linear and is normalized with highfis.defuzzifiers.SumBasedDefuzzifier.
  • AdaTSKClassifier and AdaTSKRegressor do not expose the feature- selection / rule-extraction gates of FSRE-AdaTSK.

Estimator wrappers

  • AdaTSKClassifierEstimator and AdaTSKRegressorEstimator are sklearn-compatible wrappers around the low-level AdaTSK model classes.
  • They build Gaussian membership functions from input_configs, n_mfs, mf_init, and sigma_scale.
  • The default sigma_scale=1.0 is appropriate because the adaptive softmin operator handles high-dimensional stability.

Membership functions

  • The primary antecedent MFs are standard highfis.memberships.GaussianMF objects.
  • An optional nonzero lower-bound membership function is available via highfis.memberships.CompositeGaussianMF for paper-style stability.

Training in the paper vs. highFIS

  • The paper trains AdaTSK end-to-end by optimizing the task loss through the adaptive softmin operator.
  • highFIS follows the same gradient-based training paradigm in BaseTSK.fit().

  • eps is used to clamp membership values and stabilize log-space computations in AdaSoftminRuleLayer.

  • FSRE-AdaTSK is documented separately in docs/models/fsre-adatsk.md.

Alignment with the paper

  • The paper's key AdaTSK contribution is the adaptive softmin antecedent operator to avoid numeric underflow and fake minimum effects.
  • highFIS implements this via AdaSoftminRuleLayer with a per-rule exponent derived from the rule's minimum antecedent membership.
  • The TSK consequent remains first-order, matching the paper's model.
  • The default estimator wrappers use GaussianMF, while the paper's positive lower-bound MF can be supplied via CompositeGaussianMF.