Maximum likelihood estimation from fuzzy data using the EM algorithm

Abstract

A method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample. This method is based on maximizing the observeddata likelihood defined as the probability of the fuzzy data. It is shown that the EM algorithm may be used for that purpose, which makes it possible to solve a wide range of statistical problems involving fuzzy data. This approach, called the Fuzzy EM (FEM) method, is illustrated using three classical problems: normal mean and variance estimation from a fuzzy sample, multiple linear regression with crisp inputs and fuzzy outputs, and univariate finite normal mixture estimation from fuzzy data.

Clustering and classification of fuzzy data using the fuzzy EM algorithm

“…As mentioned previously, the convergence of the FEM algorithm to a local maximum for the observed log-likelihood has been proved in [62]. Under some conditions on the initial values of the parameters, L is bounded from above.…”

“…Our approach is based on an extention of the EM algorithm for fuzzy data proposed by Denoeux [62,63]. Given a sample of fuzzy numbers, the likelihood of a mixture of Gaussians is computed using Zadeh's definition of the probability of a fuzzy event.…”

“…It is possible to overcome this problem by using an extension of the expectation-maximization (EM) algorithm for fuzzy data, known as the fuzzy EM (FEM) algorithm [62], which we briefly recall. Then, we assume that fuzzy instances are described by Gaussian possibility distributions.…”

“…The FEM algorithm alternatively repeats the E-and M-steps until the relative increase of the log-likelihood becomes smaller than some threshold. In [62], it is shown that the FEM algorithm converges towards a local maximum of the observed log-likelihood. The proof is similar to that used in [2] for assessing the convergence of the EM algorithm.…”

“…Then, the fuzzy EM algorithm may be used to estimate the parameters maximizing this likelihood. This procedure was recently proposed by Denoeux as an extension of the EM algorithm for imprecise data represented by fuzzy numbers [62] and by belief functions [63] 2 . Thus, unlike in [52,53,55], we consider here both attribute and class uncertainty in our estimation process.…”