Basics on Random Sets and Random Fuzzy Sets

Random Fuzzy Sets

It is well known that in decision making under uncertainty, while we are guided by a general (and abstract) theory of probability and of statistical inference, each specific type of observed data requires its own analysis. Thus, while textbook techniques treat precisely observed data in multivariate analysis, there are many open research problems when data are censored (e.g., in medical or bio-...

Fuzzy Sets As Strongly Consistent Random Sets

It is known that from the purely mathematical viewpoint, fuzzy sets can be interpreted as equivalent classes of random sets. This interpretations helps to teach fuzzy techniques to statisticians and also enables us to apply results about random sets to fuzzy techniques. The problem with this interpretation is that it is too complicated: a random set is not an easy notion, and classes of random ...

Integrals of Random Fuzzy Sets

This paper tries to give a systematic investigation of integration of random fuzzy sets. Besides the widely used Aumann-integral adaptions of Pettisand Bochnerintegration for random elements in Banach spaces are introduced. The mutual relationships of these competing concepts will be explored comprehensively, completing and improving former results from literature. As a by product dominated con...

A Note on Strong Laws of Large Numbers for Dependent Random Sets and Fuzzy Random Sets

Propagation Models and Fitting Them for the Boolean Random Sets

In order to study the relationship between random Boolean sets and some explanatory variables, this paper introduces a Propagation model. This model can be applied when corresponding Poisson process of the Boolean model is related to explanatory variables and the random grains are not affected by these variables. An approximation for the likelihood is used to find pseudo-maximum likelihood esti...