Search

We consider the problem of measuring the degree of inclusion and similarity between interval-valued fuzzy sets. We propose a new idea for constructing indicators of inclusion and similarity measures based on the precedence relation, aggregation and uncertainty assessment. Furthermore, we examine selected properties of the suggested measures and their interactions. Finally, we discuss several similarity measures that appear in the literature and compare them with our novel approach.

In 2011 Grzegorzewski introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on copulas and make a bridge between probability theory and fuzzy logic. Another family of fuzzy conditional implication operators was proposed by Dolati et al. in 2013. In this paper we consider some properties of these three classes of fuzzy implications like the law of contrapositions and the law of importation. Moreover, we examine intersections of these families of implications with R-implications, ( S, N)-implications, QL-operations and Yager’s f- and g-generated implications.

Interval-valued observations arise in many real-life situations as either the precise representation of the objective entity or the representation of incomplete knowledge. Thus given p features observed over a sample of objects belonging to one of two possible classes, each observation can be perceived as a non-empty closed and bounded hyperrectangle on R p . The aim of the paper is to suggest a p-dimensional classification method for random intervals when two or more classes are considered, by the generalization of Fisher's procedure for linear discriminant analysis. The idea consists of finding a directional vector which maximizes the ratio of the dispersion between the classes and within the classes of the observed hyperrectangles. A classification rule for new observations is also provided and some simulations are carried out to compare the behavior of the proposed classification procedure with respect to other methods known from the literature. Finally, the suggested methodology are applied on a real-life situation example.

Recently, Grzegorzewski (2011) introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In his previous articles author gave a motivation to his idea and indicates some interesting connections between new families of multivalued implications and the dependence structure of the underlying environment. In this paper the laws of contraposition, the law of importation and T-conditionality are studied for these families of fuzzy implications. Furthermore, we discuss the intersections of both new families of implications with R-implications, (S,N)-implications and QL-operations.

One of the key problems in statistics is to get information about the form of the population from which a sample is drawn. To check compatibility of a set of observed values with a presumed distribution one can apply various, so called, goodness-of-fit tests. It seems that the goodness-of-fit testing problem becomes much more complicated in the presence of imprecise observations. Actually, although many statistical procedure dedicated for specified types of distributions were generalized to fuzzy environment, still there are not too many tools that help under fuzzy data from the unknown distribution. Therefore, in the paper we suggest how to generalize the well-known one-sample goodness-of-fit tests based on the empirical distribution function, like the Kolmogorov test, the Cramer-von Mises test or the Anderson-Darling test, for fuzzy data.

Such fundamental concepts of the classical theory of probability, like independence of events, the conditional probability, the law of total probability and the Bayes theorem, are generalized for IF-events.

Interval and trapezoidal approximation methods for fuzzy numbers have been intensively developed and examined recently. In this paper another approach to simplify fuzzy numbers, called shadowed set approximation, is suggested. Some basic properties of the proposed method is also discussed. It seems that the proposed approximation might be useful for granular computing as a tool that removes excessive precision in describing imprecise phenomena.

The probabilistic version of the inclusion-exclusion principle is generalized for IF-events. Two versions of the generalized formula, corresponding to different t-conorms applied for defining the union of IF-events are shown.

The problem of measuring the degree of inclusion for two intuitionistic fuzzy sets (IF-sets) is considered. Since the nature of IF-sets includes uncertainty expressed both by the membership and nonmembership functions, this very feature should be also regarded in the construction of the subsethood measures. Therefore, we suggest not a single coefficient measuring the grade of subsethood but two indicators that leave some space to the mentioned uncertainty and correspond rather to necessary and possible inclusion.