Realizing Deep High-Order TSK Fuzzy Classifier by Ensembling Interpretable Zero-Order TSK Fuzzy Subclassifiers

Although high-order Takagi–Sugeno–Kang (TSK) fuzzy systems have demonstrated their computational advantages and simultaneously circumvent the weakness that the number of rules with the number of input variables and membership functions grows exponentially in both zero-order and first-order TSK fuzzy systems for complex modeling tasks, they still face two serious issues: incapability for a changing environment and no interpretability of the coefficients in high-order polynomial used in the consequent part of each fuzzy rule. In order to circumvent these two challenges, a novel stacked architecture of an interpretable deep higher order TSK fuzzy classifier called DHO-TSK and its deep learning method are proposed by proving the equivalence between a high-order TSK fuzzy classifier and a deep ensemble of interpretable zero-order TSK fuzzy classifiers in this article. DHO-TSK can be built by assembling interpretable zero-order TSK fuzzy classifiers in a special stacked way. Each zero-order TSK fuzzy classifier can be learnt by randomly selecting input features, randomly assigning an antecedent fuzzy subset from a fixed fuzzy partition to each of the selected input features, and then multiplying the output of each TSK fuzzy classifier by a randomly selected feature. Except for the abovementioned solid theoretical equivalence, DHO-TSK is featured in the following aspects: first, the consequent part of each fuzzy rule in DHO-TSK becomes interpretable and the output expression of each layer in DHO-TSK becomes comprehensible due to the adopted stacked ensemble; second, its enhanced classification performance can be achieved in a stacked deep learning way; third, DHO-TSK has its adoptability for changing environments owing to random selection of both features and fuzzy membership functions. Our experimental results on the benchmarking UCI and KEEL datasets and a real dataset indicate the effectiveness of DHO-TSK and its learning method in the sense of both classification performance and interpretability.