Fuzzy data, prevalent in social sciences and other fields, capture uncertainties arising from subjective evaluations and measurement imprecision. Despite significant advancements in fuzzy statistics, a unified inferential regression-based framework remains undeveloped. Hence, we propose a novel approach for analyzing bounded fuzzy variables within a regression framework. Building on the premise that fuzzy data result from a process analogous to statistical coarsening, we introduce a conditional probabilistic approach that links observed fuzzy statistics (e.g., mode, spread) to the underlying, unobserved statistical model, which depends on external covariates. The inferential problem is addressed using Approximate Bayesian methods, mainly through a Gibbs sampler incorporating a quadratic approximation of the posterior distribution. Simulation studies and applications involving external validations are employed to evaluate the effectiveness of the proposed approach for fuzzy data analysis. By reintegrating fuzzy data analysis into a more traditional statistical framework, this work provides a significant step toward enhancing the interpretability and applicability of fuzzy statistical methods in many applicative contexts.

In experimental practice, especially where the human factor plays an important role, we are faced with the need to analyze phenomena burdened with two types of uncertainty simultaneously: randomness and a lack of precision. While probability theory deals with randomness, and we cope with imprecision using the theory of fuzzy sets, combining both these descriptions is neither straightforward nor simple. Even seemingly simple tasks, such as calculating the probability of an event for a fuzzy random variable, are not obvious. In this contribution, we propose a method of calculating probabilities related to the so-called LR fuzzy random variables. Besides indicating the general method, we show how, in situations where it is difficult to obtain an analytical solution, it is possible to determine the desired probability using numerical methods, referring to the Monte Carlo simulations. The considered approach is illustrated with examples, also based on the real-life dataset.

Various two-sample tests are used in statistics to verify whether two samples are drawn from the same populations against the general alternative that their distributions just differ. Sometimes we are interested in testing more direct alternatives that two populations differ in location or in dispersion. Both parametric and nonparametric tests designed to solve the aforementioned problems belong to a basic statistical toolbox. However, testing potential difference in location or dispersion do not exhaust the topic of the two-sample comparison in fuzzy data analysis. Actually, fuzzy data may describe vague notions, imprecise perceptions, fuzzy concepts, etc. Hence, in statistical reasoning with fuzzy samples besides the traditionally considered problems an interesting issue could be to make a comparison of uncertainty in both populations. To meet these needs a two-sample test for ambiguity is proposed in this contribution. Besides the test construction we provide practitioners with algorithms ready to work.

A new statistical goodness-of-fit for comparing distributions of two or more populations and based on fuzzy data is proposed. Its idea goes back to the k-nearest neighbor technique applied in pattern recognition, where it simply consists in classifying an object by the majority vote of its neighbors. In our paper we show that by an appropriate test statistic construction which counts the number of nearest neighbors between and within samples it is possible to check whether available fuzzy samples come or not from the same distribution. It is worth underlying that the suggested testing procedure is completely distribution-free which seems to be of extreme importance in statistical reasoning with fuzzy data. Our test proposal is completed with a study of its properties and a case study related to quality assessment.

Hydroforming is a relatively new technology of forming and profiling. So far, the application of this method has been limited by the costs of die production. The cost of the dies and the long production start-up time made this method economically viable for the production of hundreds of products. The approach change to the tool design for profile shaping techniques has allowed to develop the new hydroforming method perfectly suited to low-volume or even unit production. In traditional solutions, the die is rigid and does not deform during the expansion of the profile. In the newly patented RTH (Rapid Tube Hydroforming) method, the die undergoes controlled deformation during the process. The specificity of the granular materials used for the production of the dies makes modeling the behavior of the die during the expansion of the profile a remarkable problem. This contribution presents considerations on the fuzzy inference method used to model the technological process. As a result, it was possible to more accurately determine the importance of individual die parameters (geometry and material properties), and thus better predict the final shape of the formed profile. The main goal is to understand the effect of shaped profile on the matrix and to recognize the influence of granular material in the matrix under the compaction conditions of the expanded profile on its final geometry.

The problem of testing goodness-of-fit for k distributions based on fuzzy data is considered. A new permutation test for fuzzy random variables is proposed. Besides the general constrution of the test an algorithm ready for the practical use is delivered. A case-study illustrating the applicability of the suggested testing procedure is also presented.

Two versions of the generalized sign test and the signed-rank test for interval-valued data (both for one-sample and paired two-sample problem) are proposed. These two versions correspond to different possible views on the interval outcomes of the experiment—either the epistemic or the ontic one. Each view yields its own approach to data analysis which results in different test construction and the way of carrying on the statistical inference.

The problem of the piecewise linear approximation of fuzzy numbers giving outputs nearest to the inputs with respect to the Euclidean metric is discussed. The results given in Coroianu et al. (Fuzzy Sets Syst 233:26–51, 2013) for the 1-knot fuzzy numbers are generalized for arbitrary n-knot ( $$n\ge 2$$ ) piecewise linear fuzzy numbers. Some results on the existence and properties of the approximation operator are proved. Then, the stability of some fuzzy number characteristics under approximation as the number of knots tends to infinity is considered. Finally, a simulation study concerning the computer implementations of arithmetic operations on fuzzy numbers is provided. Suggested concepts are illustrated by examples and algorithms ready for the practical use. This way, we throw a bridge between theory and applications as the latter ones are so desired in real-world problems.

The generalized Kolmogorov goodness-of-fit test for interval-valued data is proposed. Two versions of the test are considered - each corresponding to a different view on the outcomes of the experiment, i.e. either the epistemic or ontic one. It is shown that each view on interval-valued data yield different approaches to data analysis and statistical inference.

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.

Trapezoidal approximation of fuzzy numbers preserving the expected interval is considered. A general problem of the trapezoidal approximation of fuzzy numbers with respect to the distance based on bi-symmetrical weighted functions is solved. A practical algorithm for constructing approximation operator is given.

Classical statistical tests may be sensitive to violations of the fundamental model assumptions inherent in the derivation and construction of these tests. It is obvious that such violations are much more probable in the presence of vague data. Thus nonparametric tests seem to be promising statistical tools. In the present paper, a distribution-free statistical test for the so-called “many-one problem” with vague data is suggested. This test is a generalization of the k-sample median test. In our approach, we utilize the necessity index of strict dominance, suggested by Dubois and Prade. © 2009 Wiley Periodicals, Inc.

The traditional query languages used in database management systems require precise and unambiguous queries only. Fuzzy querying was introduced to relax this rigidity and allow the user more natural information retrieval. In this article we suggest how to enrich fuzzy querying by the use of IF-sets. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 587–597, 2007.

The problem of measuring association between preference systems in situations with missing information or noncomparable outputs is discussed. New correlation coefficient, which generalizes Kendall's rank correlation coefficients used traditionally in statistics, is suggested. The construction utilizes intuitionistic fuzzy sets.

The problem of measuring degree of inclusion between intuitionistic fuzzy sets is discussed. A simple subsethood measure based on the Hamming distance between intuitionistic fuzzy sets is suggested.

The problem of the interval defuzzification of fuzzy numbers is discussed. A new continuous operator is suggested and investigated.