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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.

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.

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.

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.

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.

A fuzzy least squares estimator in the multiple with fuzzy-input&ndash, fuzzy-output linear regression model is considered. The paper provides a formula for the L2 estimator of the fuzzy regression model. This paper proposes several operations for fuzzy numbers and fuzzy matrices with fuzzy components and discussed some algebraic properties that are needed to use for proving theorems. Using the proposed operations, the formula for the variance, provided and this paper, proves that the estimators have several important optimal properties and asymptotic properties: they are Best Linear Unbiased Estimator (BLUE), asymptotic normality and strong consistency. The confidence regions of the coefficient parameters and the asymptotic relative efficiency (ARE) are also discussed. In addition, several examples are provided including a Monte Carlo simulation study showing the validity of the proposed theorems.

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.

The median two-sample test for the location problem is considered. We adopt this nonparametric test to interval-valued data perceived from the epistemic perspective, where the available observations are just interval-valued perceptions of the unknown true outcomes of the experiment. Unlike typical generalizations of statistical procedures into the interval-valued framework, the proposed test entails very low computational costs. However, the presence of interval-valued data results in set-valued p-value which leads no longer to a definite binary decision (reject or not reject the null hypothesis) but may indicate the abstention from making a final decision if the information is too vague.

The problem of comparing variability of two populations with fuzzy data is considered. A new permutation two-sample test for dispersion based on fuzzy random variables is proposed. A case-study illustrating the applicability of the suggested testing procedure is also presented.

In this paper we firstly review the definition of fuzzy semi-numbers and study some of their properties. Then, we consider some methods for converting fuzzy semi-numbers to fuzzy numbers in order to find the distance between fuzzy semi-numbers. By presenting a new distance function, we also find the distance between fuzzy semi-numbers directly without any change to their originality. Finally, we prove some properties of the presented distance and study a practical motivational medical case study along with some numerical examples.

Probabilistic implications and probabilistic S-implications, introduced recently by Grzegorzewski, have attracted attention of researchers engaged in fuzzy implications. Since these two families are based on copulas, they form a kind of bridge between probability theory and fuzzy logic. In this paper we generalize the aforementioned two classes and propose a new approach for constructing fuzzy implications which combines a given a priori fuzzy implication and a semicopula.

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.

International audience

1. XLVII National Conference on Applications of Mathematics, September 4-11, 2018, Zakopane-Kościelisko (Ł. Stettner) The 47th National Conference on Mathematics Applications was held on September 4-11, 2018 in Zakopane Kościelisko. Together with the conference, the XXIV National Conference on Mathematics Applications in Biology and Medicine (4-7 September 2018) was held simultaneously with two common first days of the meeting. The plenary lecture was delivered by Urszula Ledzewicz (Southern Illinois University Edwardsville, USA and Łodź University of Technology) and Heinz Schattler (Washington University, St. Louis, USA) with optimal control in biomedical problems. 2. GAMENET Conference Krakow 2018, 17 -- 21 września 2018, Krakow(F.Falniowski) On September 17-21, 2018 an international conference GAMENET Conference Krakow 2018 summarizing the first year of activity of the European network for game theoretic research GAMENET was held at the Cracow University of Economics. The main event of the first two days was a training school on the recent applications of Game Theory. The training school was build around three courses delivered by Olivier Baude (Electricite de France), Luiz DaSilva (Trinity College Dublin) and Vianney Perchet (CMLA, ENS Paris-Saclay & Criteo AI Lab). 3. Conference "Social sciences - mathematical or mathematizable?", 27 -- 28 września 2018, Krakow(Agnieszka Lipieta) The conference was held at Cracow University of Economics on September 27-28, 2018. It was devoted to the memory of outstanding mathematician, philosopher, and economist Professor Andrzej Malawski, who passed away in 2016. 4. VII Hurwicz Workshop on Mechanism Design Theory, 7--8 grudnia 2018 (Łukasz Woźny) IM PAN WarszawaThe 2018 Hurwicz Workshop on Mechanism Design Theory is a continuation of the initiative started in 2009 of holding an annual confernce to honour the 2007 Nobel Prize Laureate in Economics, professor Leonid Hurwicz. Leonid Hurwicz lived in Warsaw until 1938 and studied at the University of Warsaw. He frequently visited Poland in 1990's. In 1994 he obtained the Doctor Honoris Causa of Warsaw School of Economics. A special plaque commemorates his contacts with Warsaw School of Economics. Hurwicz is often credited with introducing rigorous mathematical approach to economic analysis. He received the Nobel Prize in Economic Sciences in 2007 for his fundamental contributions to the theory of the design of economic mechanisms. Theory of mechanism design relies heavily on mathematical methods of functional analysis, differential equations, differential topology, dynamical systems, etc. Hurwicz has made important contributions to mathematics as well as economics, in particular, to non-linear programming. 5. Conference “Mathematical Statistics”. Bedlewo, December 2--7, 2018 (Przemyslaw Grzegorzewski). On December 2--7, 2018, in Bedlewo, the XLIV Conference "Mathematical Statistics" was held, organized by the Banach Center of Institute of Mathematics Polish Accademy of Science, the Committee on Statistics of the Committee of Mathematics of the Polish Academy of Sciences, and the Faculty of Mathematics and Computer Science of the Nicolaus Copernicus University in Torun. During the 19 sessions, a total of 46 lectures and a series of lectures on "Bayesian Inference in Intractable Likelihood Models", i.e. on Bayesian modeling with a difficult likelihood function, were given. The lecture was delivered by Krzysztof Łatuszynski from the University of Warwick in Great Britain. 6. XLVIII Seminar on Applied Mathematics-XLVIII Seminarium Zastosowan Matematyki, 9-11.09.2018 Boguszow-Gorce (Mieczyslaw Chalfen)

Data can be perceived as a staring point for any modeling and further scientific analysis. Here we discuss the specifity of industrial data and its impact on scientific modeling. Following some general remarks on mathematical modeling the idea of a hard modeling and soft modeling in engineering is developed.

Each scientific modeling starts from data. However, even most sophisticated mathematical methods cannot produce a satisfying model if the data is of low quality. Before concluding about the quality of available data it is worth realizing the difference between datum quality or database quality. Moreover, most of data mining algorithms deal with the data in the form of an appropriately prepared single matrix. Unfortunately, the raw data is rarely stored in such form but is scattered over several databases, may contain observations which differ in formats or units, may abound with “garbage”, etc. Thus an adequate data preparation is an inevitable stage that should precede any modeling and further analysis. Both problems of data quality and data preparation are discussed in this chapter.

Data appear at the beginning and at the end of any reasonable modeling. Indeed, data deliver a motivation and a starting point for a model construction. But data are also necessary to validate a resulting model. Data bring information on a considered phenomenon. Gathering information enables to widen our knowledge. But, on the other hand, without some knowledge one would not be able to extract information from data and interpret the received information adequately. Such terms like data and information are widely used in the context of scientific modeling and applications. Although sometimes treated interchangeably they are not synonyms. Another closely related concepts are knowledge, uncertainty, reasoning, etc. The main goal of the present chapter is to discuss and clarify the meaning of the aforementioned notions, to indicate their interrelations and set them in the broad framework of the cognition oriented activity. Finally, three basic types of reasoning used both in science as well as in practice is briefly characterized.

We propose an axiomatic definition of a dispersion measure that could be applied for any finite sample of k-dimensional real observations. Next we introduce a taxonomy of the dispersion measures based on the possible behavior of these measures with respect to new upcoming observations. This way we get two classes of unstable and absorptive dispersion measures. We examine their properties and illustrate them by examples. We also consider a relationship between multidimensional dispersion measures and multidistances. Moreover, we examine new interesting properties of some well-known dispersion measures for one-dimensional data like the interquartile range and a sample variance.

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.

Measuring similarity between preferences is a crucial problem for recommender systems. This task becomes significantly harder when preferences are incomplete or somehow vague. In this paper we show how to model vague preferences using IF-sets and then how to quantify similarity between preferences in a way that might be useful in collaborative filtering. We consider some comparison measures between IF-sets to find those possessing properties desirable in recommender systems. Then we construct some measures that might be useful in finding other customers somehow similar to our new user of a recommender system and in promoting those customers who have an extensive knowledge on many products not yet familiar to this new user. We also suggest how to combine the aforementioned methodology with some new entropy-based analytical and graphical tools to create recommendations and support customer’s decisions. The proposed graphical method for comparing possible recommendations due to several aspects enables to choose a recommendation that fits best to individual decision-making strategy of each user.

Several new resampling methods for generating bootstrap samples of fuzzy numbers are proposed. To avoid undesired repetitions in the secondary samples we do not draw randomly directly observations from the primary samples but construct them allowing for some modifications in their membership functions, however only such which do not disturb the canonical representation of the initial fuzzy numbers. We consider both two-parameter and three-parameter canonical representations, as well as the triangular and trapezoidal outputs in the secondary samples. Numerical experiments concerning some statistical tests based on fuzzy samples show that the suggested methods may appear helpful in statistical reasoning with imprecise data.

Almost all experiments reveal variability of their results. In this contribution we consider the measures of dispersion for sample of random intervals. In particular, we suggest a generalization of two well-known classical measures of dispersion, i.e. the range and the interquartile range, for interval-valued samples.

A new methodology for processing non-normal fuzzy sets is proposed. To break the predominant constraint on normality of fuzzy numbers the concept of fuzzy semi-numbers is introduced Then it is shown how to generalize operations defined on fuzzy numbers onto a family of fuzzy semi-numbers with possibly different heights.

The Kolmogorov–Smirnov goodness-of-fit test for equality of two distributions is considered. Two generalizations of this test for interval-valued data are proposed. Each version correspond to a different view on the interval outcomes of the experiment – either the epistemic or the ontic one. Each view yield its own approaches to data analysis and statistical inference.

The two-sample dispersion testing problem is considered. Two generalizations of the Sukhatme test for interval-valued data are proposed. These two versions correspond to different possible views on the interval outcomes of the experiment: the epistemic or the ontic one. Each view yields its own approach to data analysis which results in a different test construction and the way of carrying on the statistical inference.

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.

The Mann-Whitney test for the two-sample location problem is considered. We adopt this nonparametric test to interval-valued data perceived from the epistemic perspective, where the available observations are just interval-valued perceptions of the unknown true outcomes of the experiment. Unlike typical generalizations of statistical procedures into the interval-valued framework, the proposed test entails very low computational costs. However, the presence of interval-valued data results in set-valued p-value which leads no longer to a definite binary decision (reject or not reject the null hypothesis) but may indicate the abstention from making a final decision if the information is too vague.

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.

This proceedings volume is a collection of peer reviewed papers presented at the 8th International Conference on Soft Methods in Probability and Statistics (SMPS 2016) held in Rome (Italy). The book is dedicated to Data science which aims at developing automated methods to analyze massive amounts of data and to extract knowledge from them. It shows how Data science employs various programming techniques and methods of data wrangling, data visualization, machine learning, probability and statistics. The soft methods proposed in this volume represent a collection of tools in these fields that can also be useful for data science.

Fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is considered. New operators that fulfill additional requirements for the core and support of the fuzzy number are suggested. These supplementary conditions guarantee the proper interpretation of the solution even for very skew input fuzzy numbers. Moreover, the trapezoidal approximation without expected interval invariance is also examined and the relationship between these two approximations is discussed.

A novel approach for detecting patterns in price time series is shown. The proposed system for identifying consolidation phases is based on fuzzy geometric protoforms and classification trees. Promising results of the empirical studies prove that the suggested fuzzy geometric protoforms are very useful for identifying patterns in graphical visualizations of data. Moreover, the architecture of the system enables successful incorporation of genetic optimization what enables capturing various data sets structure and unstable conditions on financial markets.

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 multiple fuzzy linear regression model with fuzzy input–fuzzy output is considered. Assuming that fuzzy inputs and fuzzy outputs are modeled by triangular fuzzy numbers, we prove the consistency and asymptotic normality of the least squares estimators.

Two versions of the generalized sign test for interval-valued data are proposed. Each version correspond to a different view on the interval outcomes of the experiment—either the epistemic or the ontic one. As it is shown, each view yield different approaches to data analysis and statistical inference.