Granular Trapezoidal Type-2 Shallow Fuzzy Neural Net-Work

In the paper, we use trapezoidal type-2 fuzzy setsTrapezoidal type-2 fuzzy sets (TraT2FS) for granular fuzzy modeling, the model using symmetric trapezoidal structure could simplify the network’s computation burden for tensor based fuzzy neural networks. Since fuzzy sets can be used as expressions of information granulesInformation granules, applying trapezoidal type-2 granules to the antecedent of a fuzzy system is appealing. The endpoints of the trapezoidal fuzzy sets defined on foot of uncertaintiesUncertainty could form interval granules, combines with the inner fuzzy intervals, trapezoidal type-2 fuzzy set could be formed by two types of interval granules. TraT2FS could also be represented by a series of interval type-2 fuzzy sets with their \(\alpha -\) cut, which decomposed the fuzzy sets into a series of type-1 fuzzy sets. Extending interval type-2 information granulesInformation granules to trapezoidal type-2 information granules could be finished. To GraT2SFNN, the limitations of numerical models such as low interpretabilityInterpretability or single output results are alleviated. The GraT2SFNN’s consequent parameter learning uses Moore–Penrose inverse of even order tensors and a tensor based conjugate gradient-like methodConjugate gradient-like method. Finally, 21 benchmark datasets are used in the regressionRegression analysis, the results show that GraT2SFNN has strong generalization ability and robustness, and its performance on small and medium-sized datasets can reach the level of RNN-KM and RNN-BFGS networks.