Gaussian Function Representation of 2nd-Order Normal Cloud Model.

Natural language is a fruit of human cognition, and its basic units are linguistic terms. Linguistic terms' representation and processing have played a fundamental role in applications such as risk assessment. Cloud Model, which integrates the approaches of probability theory and fuzzy sets, is a model for expressing the uncertainties of linguistic terms. Its advantage is that it is a bidirectional cognitive model. The 2nd-order normal cloud (2Ord-NC) is a specific Cloud Model. It can represent the linguistic concept's intension using three statistics parameters (Expectation Ex, Entropy En, and Hyper-entropy He) and its extension using cloud drops (instances). A cognitive transformation between a qualitative concept and its quantitative instances can be implemented by probabilistic programming (FCT and BCT). In the current works, Upper and Lower membership functions characterize Cloud Model-s. However, scholars have yet to explain this representation method's rationality theoretically. This paper proposes the 2nd-order normal cloud model's Gaussian function representation. Firstly, it is proved that the 2nd-order normal cloud model (2Ord-NCM) can be determined by its outer envelope and expectation membership functions. The properties of the 2Ord-NCM's fuzzy membership functions are analyzed when the concept is clear and unclear, and the definitions of inner envelope membership function family, outer envelope membership function family, and representative membership function family are given. Secondly, it is proved that the 2Ord-NCM can also be determined by any two of its representative membership functions. This work enriches the Cloud Model's mathematical theoretical foundations, thereby better guiding its practical application.