This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social and behavioral sciences and are especially relevant when observed data are classified using fuzzy categories—as for socioeconomic studies, clinical evaluations, content analysis, inter-rater reliability analysis—or when imprecise observations are classified into either precise or imprecise categories—as for the analysis of ratings data or fuzzy-coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of expectation–maximization-based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.
In this paper, we define signed zero-divisor graphs over commutative rings and investigate the interplay between the algebraic properties of the rings and the combinatorial properties of their corresponding signed zero-divisor graphs. We investigate the structure of signed zero-divisor graphs, the relation between ideals and signed zero-divisor graphs, and the adjacency matrices and the spectra of signed zero-divisor graphs.
In this work, we consider numerical approximations of the phase-field model of diblock copolymer melt confined in Hele–Shaw cell, which is a very complicated coupled nonlinear system consisting of the Darcy equations and the Cahn–Hilliard type equations with the Ohta–Kawaski potential. Through the combination of a novel explicit-Invariant Energy Quadratization approach and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme. The introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role in obtaining the full decoupling structure while maintaining energy stability. The scheme is also linear and unconditional energy stable, and the practical implementation efficiency is also very high because it only needs to solve a few elliptic equations with constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.