On Two-Dimensional Fuzzy Random Data as Vague Perceptions of Random Phenomena

Abstract

In this paper, the author investigates numerically a class of two-dimensional fuzzy random sets(abbreviated as FRSs), which is one case of those proposed by the author[1] as models of vague capricious perceptions of random phenomena. First, the basic results concerned with two-dimensional FRSs as vague capricious perceptions of random phenomena are reviewed. Expectations of two-dimensional FRSs, their estimators, and distances between expectations and their estimators are also reviewed briefly. Finally, estimates for expectations of two-dimensional FRSs and their estimation errors are numerically examined by simulation studies, when their level sets are given by disk-form ones at each levels.

On Two-Dimensional Fuzzy Random Data with Polygon-form Level Sets as Vague Perception of Crisp Random Phenomena

“…Hereafter the fuzzy set with polygon-form level sets is called a polygonal fuzzy set. Let here B be another polygonal fuzzy set in F (p) cc (R 2 ), whose vertices v (k) B are similarly determined by (5) to (9).…”

“…In ( 65), [ D U,0 ] α k is the level set of the fuzzy set D U,0 ∈ F (p) cc (R 2 ), which is the disk with the center (0, 0, α k ) and the same radius as [ D U ] α k ; and c (k) D U is the center of [ D U ] α k (see e.g. [9]).…”

“…where the stepwise approximation of set representations are used. Therfore, applying the same procedure for the case of disk-form level sets [9], the upper bound of ρ 1 ( D…”