A complex-valued neuro-fuzzy inference system and its learning mechanism

In this paper, we present a complex-valued neuro-fuzzy inference system (CNFIS) and develop its meta-cognitive learning algorithm. CNFIS has four layers – an input layer with m rules, a Gaussian layer with K rules, a normalization layer with K rules and an output layer with n rules. The rules in the Gaussian layer map the m-dimensional complex-valued input features to a K-dimensional real-valued space. Hence, we use the Wirtinger calculus to obtain the complex-valued gradients of the real-valued function in deriving the learning algorithm of CNFIS. Next, we also develop the meta-cognitive learning algorithm for CNFIS referred to as “meta-cognitive complex-valued neuro-fuzzy inference system (MCNFIS)”. CNFIS is the cognitive component of MCNFIS and a self-regulatory learning mechanism that decides what-to-learn, how-to-learn, and when-to-learn in a meta-cognitive framework is its meta-cognitive component. Thus, for every epoch of the learning process, the meta-cognitive component decides if each sample in the training set must be deleted or used to update the parameters of CNFIS or to be reserved for future use.The performances of CNFIS and MCNFIS are studied on a set of approximation and real-valued classification problems, in comparison to existing complex-valued learning algorithms in the literature. First, we evaluate the approximation performances of CNFIS and MCNFIS on a synthetic complex-valued function approximation problem, an adaptive beam-forming problem and a wind prediction problem. Finally, we study the decision making performance of CNFIS and MCNFIS on a set of benchmark real-valued classification problems from the UCI machine learning repository. Performance study results on approximation and real-valued classification problems show that CNFIS and MCNFIS outperform existing algorithms in the literature.