Przemysław Grzegorzewski

Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric

New methods for measuring distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets, based on the Hausdorff metric, are suggested. The proposed new distances are straightforward generalizations of the well known Hamming distance, the Euclidean distance and their normalized counterparts.

Nearest interval approximation of a fuzzy number

The problem of the interval approximation of fuzzy numbers is discussed. A new interval approximation operator, which is the best one with respect to a certain measure of distance between fuzzy numbers, is suggested.

Metrics and orders in space of fuzzy numbers

Abstract Two classes of metrics δ p , q and ϱ p ,where 1 ⩽ p ⩽ ∞, 0 ⩽ q ⩽ 1, in space of fuzzy numbers are introduced. Then a method of ranking fuzzy numbers based on these metrics is proposed and investigated. Not only fuzzy numbers but also the left-sided and right-sided fuzzy numbers are considered. Applications in making decisions and statistics are suggested.

Testing statistical hypotheses with vague data

A definition of fuzzy test for testing statistical hypotheses with vague data is proposed. Then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one-sided or two-sided alternative hypotheses is shown. This fuzzy test, contrary to the classical approach, leads not to the binary decision: to reject or to accept given null hypothesis, but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypothesis, respectively. However, it is a natural generalization of the traditional test, i.e. if the data are precise, not vague, we get a classical statistical test with the binary decision. A measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed.

Trapezoidal approximations of fuzzy numbers---revisited

Fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed. The previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number.

Trapezoidal approximations of fuzzy numbers preserving the expected interval--- Algorithms and properties

Fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed. Algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed. It is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number, like its ambiguity, width, its value and weighted expected value.

Soft methods in probability, statistics and data analysis

Part I: Introductory Papers.- Part II: Soft Methods in Probability - Fundamentals.- Part III: Soft Methods in Statistics - Fuzzy Stochastic Models.- Part IV: Soft Methods in Data Analysis - Fuzzy, Rough and Other Approaches.

Discussion: Some notes on (Atanassov's) intuitionistic fuzzy sets

The paper contains few comments and positions the paper on (Atanasovs) intuitionistic fuzzy sets presenting points related to terminology, connections with other mathematical structures and possible interpretations.

Combining Soft Computing and Statistical Methods in Data Analysis

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Soft Methods for Handling Variability and Imprecision

Probability theory has been the only well-founded theory of uncertainty for a long time. It was viewed either as a powerful tool for modelling random phenomena, or as a rational approach to the notion of degree of belief. During the last thirty years, in areas centered around decision theory, artificial intelligence and information processing, numerous approaches extending or orthogonal to the existing theory of probability and mathematical statistics have come to the front. The common feature of those attempts is to allow for softer or wider frameworks for taking into account the incompleteness or imprecision of information. Many of these approaches come down to blending interval or fuzzy interval analysis with probabilistic methods. This book gathers contributions to the 4th International Conference on Soft methods in Probability and Statistics. Its aim is to present recent results illustrating such new trends that enlarge the statistical and uncertainty modeling traditions, towards the handling of incomplete or subjective information. It covers a broad scope ranging from philosophical and mathematical underpinnings of new uncertainty theories, with a stress on their impact in the area of statistics and data analysis, to numerical methods and applications to environmental risk analysis and mechanical engineering. A unique feature of this collection is to establish a dialogue between fuzzy random variables and imprecise probability theories.