Wolfgang Trutschnig

A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of all compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of Bertoluzza et al. [C. Bertoluzza, N. Corral, A. Salas, On a new class of distances between fuzzy numbers, Mathware Soft Comput. 2 (1995) 71-84] in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore, useful from the practical point of view.

Combining Soft Computing and Statistical Methods in Data Analysis

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Simulation of fuzzy random variables

This work deals with the simulation of fuzzy random variables, which can be used to model various realistic situations, where uncertainty is not only present in form of randomness but also in form of imprecision, described by means of fuzzy sets. Utilizing the common arithmetics in the space of all fuzzy sets only induces a conical structure. As a consequence, it is difficult to directly apply the usual simulation techniques for functional data. In order to overcome this difficulty two different approaches based on the concept of support functions are presented. The first one makes use of techniques for simulating Hilbert space-valued random elements and afterwards projects on the cone of all fuzzy sets. It is shown by empirical results that the practicability of this approach is limited. The second approach imitates the representation of every element of a separable Hilbert space in terms of an orthonormal basis directly on the space of fuzzy sets. In this way, a new approximation of fuzzy sets useful to approximate and simulate fuzzy random variables is developed. This second approach is adequate to model various realistic situations.

Nonparametric criteria for supervised classification of fuzzy data

The supervised classification of fuzzy data obtained from a random experiment is discussed. The data generation process is modelled through random fuzzy sets which, from a formal point of view, can be identified with certain function-valued random elements. First, one of the most versatile discriminant approaches in the context of functional data analysis is adapted to the specific case of interest. In this way, discriminant analysis based on nonparametric kernel density estimation is discussed. In general, this criterion is shown not to be optimal and to require large sample sizes. To avoid such inconveniences, a simpler approach which eludes the density estimation by considering conditional probabilities on certain balls is introduced. The approaches are applied to two experiments; one concerning fuzzy perceptions and linguistic labels and another one concerning flood analysis. The methods are tested against linear discriminant analysis and random K-fold cross validation.

A strong consistency result for fuzzy relative frequencies interpreted as estimator for the fuzzy-valued probability

The unavoidable imprecision of measurements of continuous physical quantities can be modelled by using the concept of fuzzy numbers and fuzzy vectors. Concerning a quantitative usage of such data the classical concept of relative frequencies for real data has to be extended to so-called fuzzy relative frequencies for fuzzy data, whereby the fuzzy relative frequency of a set is a fuzzy number. Analogous to A. Dempsters interval-valued probabilities induced by multi-valued mappings fuzzy-valued probabilities induced by fuzzy random vectors are considered and analyzed. It is proved that fuzzy relative frequencies can be interpreted as strongly consistent estimator for the corresponding fuzzy-valued probability.

SAFD — An R Package for Statistical Analysis of Fuzzy Data

The R package SAFD (Statistical Analysis of Fuzzy Data) provides basic tools for elementary statistics with one dimensional Fuzzy Data in the form of polygonal fuzzy numbers. In particular, the package contains functions for the standard operations on the class of fuzzy numbers (sum, scalar product, mean, Hukuhara difference, quantiles) as well as for calculating (Bertoluzza) distance, sample variance, sample covariance, sample correlation, and the Dempster-Shafer (levelwise) histogram. Moreover SAFD facilitates functions for the simulation of fuzzy random variables, for bootstrap tests for the equality of means as well as a function for linear regression given trapezoidal fuzzy data. The aim of this paper is to explain the functionality of the package and to illustrate its usage by various examples.

ANOVA for Fuzzy Random Variables Using the R-package SAFD

Due to the important role as central summary measure of a fuzzy random variable (FRV), statistical inference procedures about the mean of FRVs have been developed during the last years. The R package SAFD (Statistical Analysis of Fuzzy Data) provides basic tools for elementary statistics with one dimensional Fuzzy Data (in the form of polygonal fuzzy numbers). In particular, the package contains functions for doing a bootstrap test for the equality of means of two or more FRVs. The corresponding algorithm will be described and applied to both real-life and simulated data.

Fuzzy Probability Distributions

Consider as starting point a soccer match of two different teams M 1and M 2 (compare [1]). In this situation three different outcomes are possible: Team M 1 wins (event {a}), team M 2 wins (event {b}) or the match ends in a draw (event x{c}). For none of the three outcomes it is possible to know the exact probabilities, therefore the probabilities are estimated (by using old results), or they are provided by experts. Because of the unavoidable uncertainties in the assessment of the probabilities, it seems to be more realistic to model this estimations by using fuzzy numbers, or, in the simplest case, using intervals.

Some Smoothing Properties of the Star Product of Copulas

Three indications for the fact that the star product of copulas is smoothing are given. Firstly, it is shown that for every absolutely continuous copula A and every copula B both A*B and B*A are absolutely continuous. Secondly, an example of a singular copula A such that the absolutely continuous component of A*A has support [0,1]2 and mass at least 1/4 is given. Finally, it is shown that for every copula B of the form B = (1 − α)A + αS, whereby A is an absolutely continuous copula, S is a singular copula and α ∈ [0,1), there exists an absolutely continuous idempotent copula \(\widehat{B}\) such that \(\widehat{B}\) is the Cesaro limit of the sequence (B *n ) n ∈ ℕ of iterates of the star product of B with respect to the metric D 1 introduced in [15].

Fuzzy Probability Distributions Induced by Fuzzy Random Vectors

As a matter of fact in many real situations uncertainty is not only present in form of randomness (stochastic uncertainty) but also in form of fuzziness (imprecision), for instance due to the inexactness of measurements of continuous quantities. From the probabilistic point of view the unavoidable fuzziness of measurements has (amongst others) the following far-reaching consequence: According to the classical Strong Law of Large Numbers (SLLN), the probability of an event B can be regarded as the limit of the relative frequencies of B induced by a sequence of identically distributed, independent, integrable random variables (Xn)n∈N (with probability one). Incorporating into considerations the fact that a realistic sample of a d-dimensional continuous quantity consists of d-dimensional fuzzy vectors, it is first of all necessary to generalize relative frequencies to the case of fuzzy samples, which yields so-called fuzzy relative frequencies and furthermore mandatory to consider and analyze fuzzy-valued ’probabilities’ as generalization of classical probabilities. In the sequel the definition of fuzzy relative frequencies and the most important properties of fuzzy relative frequencies regarded as fuzzy-valued set functions are stated. After that it is shown that similar to fuzzy relative frequencies every fuzzy random vector X naturally induces a fuzzy-valued ’probability’, which will be called fuzzy probability distribution induced by X . Finally a SLLN for these fuzzy probability distributions will be stated.