The theoretical fundamentals of learning theory based on fuzzy complex random samples

Abstract

Statistical learning theory based on real-valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning. The key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory. In this paper, we discuss a statistical learning theory based on fuzzy complex random samples. Firstly, the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated. Secondly, we carry out further research focused on a special type of fuzzy complex number, namely rectangular fuzzy complex number and establish some properties and develop important theorems. We also prove the strong law of large numbers based on fuzzy complex random variables. Thirdly, the definitions of the fuzzy complex expected risk functional, the fuzzy complex empirical risk functional, the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed. Finally, the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed.