Quaternion neuro-fuzzy for real-valued classification problems

In order to generate or tune fuzzy rules, neuro-fuzzy learning algorithms (RNF) with Gaussian type membership functions based on gradient-descent method are well known. For this NF, we have proposed the quaternion neuro-fuzzy learning algorithm (QNF) extended it to four-dimensional space. This paper presents the QNF for real-valued classification problems, and introduces two new activation functions. In this QNF, four real-valued inputs are used as one quaternion input, and calculated an antecedent grade by a quaternion membership function. The antecedent grade is multiplied by a quaternion weight (singleton), and weighted sum of antecedent grades are divided by a sum of antecedent grades in an output layer. The quaternion net-input is then given to an activation function. Both activation functions map quaternion values into real values. We firstly show the ability of QNF with two-class problems, such as three-input Boolean problems, and the symmetry detection with four-input. We then tested the QNF on several real world benchmark problems. The results show that the QNF can classify each dataset, despite the QNF has smaller number of parameters than the RNF.