Modelling distributions of the temporal membership grades for non-stationary fuzzy sets

Non-stationary fuzzy sets have been introduced to model variation in human reasoning. While they have some similarities to type-2 fuzzy sets, there are many detailed differences. One of these differences is that they may feature horizontal random perturbations of membership functions, which result in distributions across the membership grades over time, termed temporal membership grades, which are somewhat akin to secondary membership grades of type-2 fuzzy sets. In this paper we study the distribution of the temporal membership grades of non-stationary fuzzy sets with Gaussian underlying membership functions. We derive the distribution of these grades when the non-stationary fuzzy sets are obtained by variations in location using a normal distribution. We show how to estimate the parameters of the derived distribution using numerical optimisation. In addition, we investigate the changes in the distributions obtained when using variations in location using a uniform distribution. Finally, we exhibit the performance of the beta and logistic normal distribution using examples of simulated data. In doing so, we further explore the relationships between non-stationary and type-2 fuzzy sets.