The article stems from distinct studies on arithmetics of fuzzy numbers, orderings of fuzzy numbers, and metrics on fuzzy numbers. Trying to capture the existing knowledge in the mentioned areas and putting them together, we motivate the construction of metric-like spaces on fuzzy numbers by desirable connection to their arithmetics. The desirable "metric" should be mapping pairs of fuzzy numbers again to fuzzy numbers and thus, reflecting the vagueness of operation on fuzzy numbers. This leads to developing all such areas under the joint umbrella and connecting such basic notions as orderings of fuzzy numbers to arithmetics and elementary "metrics" such as the absolute value of the difference of two fuzzy numbers. This article focuses mainly on the orderings and investigation of the preservation of their most natural properties. However, links to further studies going towards applications are also foreshadowed and referred to.

The notion of the metric space that allows to measure a distance between objects of the given space, has a crucial importance for distinct parts of mathematics, for instance, for the approximation theory, interpolation methods, data analysis, optimization etc. In fuzzy mathematics, the same areas of applications have an analogous importance and thus, not surprisingly measuring the distance between objects possesses a desirable importance. In many cases, e.g., in fuzzy clustering, the use of the standard metric spaces is absolutely sufficient. However, if we deal with vague quantities represented by fuzzy numbers, though the application of a standard metric to fuzzy numbers is mathematically correct, it may lead to counterintuitive and undesirable results. Our investigation constructs the "metric-like" spaces enabling to measure the distance between two fuzzy numbers in a way that is not disconnected from the used arithmetic of fuzzy numbers. Following the analogy from the classical math where the most natural distance between two numbers is the absolute value of their difference, in the case of fuzzy numbers and under the assumption that the distance is connected to the arithmetic, the most natural distance of two fuzzy numbers is the absolute values of their difference too. But then, naturally, the distance should map fuzzy numbers again to fuzzy numbers, not to crisp numbers. This article is a contribution to this area that guides readers from the fundamental notions to the final construction supported by some theoretical results.

This article gathers several topics together, namely fuzzy relational compositions, partial fuzzy logics, generalized quantifiers and, finally, classification as a problem serving for the demonstrative purposes. Fuzzy relational compositions as one of the most fundamental areas from the fuzzy set theory are being extended and investigated from several perspectives in the last decades. One of such perspective was to employ generalized quantifiers replacing the original existential and universal quantifiers in the construction of the compositions. Recently, the compositions have been also re-designed to deal with undefined values and thus, the topic has been joined to the fuzzy partial logics. One of such approaches proposed a specific algebra for handling missing values. This article puts all the mentioned approaches together in order to design fuzzy relational compositions dealing with missing values with help of appropriate partial operations. After a brief investigation of properties of the proposed compositions, we provide readers with an example demonstrating that the construction allows to minimize negative impact of the missing values.

This article follows the arithmetics of extensional fuzzy numbers that are based on the transitive closure of the used similarity relations and that form particular MI-prefields. The direction of the contribution goes towards defining the ordering of extensional fuzzy numbers in order to construct further notions from the mathematical analysis in the environment of extensional fuzzy numbers, especially the metric spaces.

Fuzzy relational compositions based on fuzzy quantifiers naturally do not preserve all the properties that are preserved for “standard” fuzzy relational compositions and, in many cases, the property is preserved only in a weaker form. For example, the associativity, that is preserved in the standard case derived from the universal and the existential quantifiers, generally does not hold for the case of compositions based on fuzzy quantifiers. However, is it the case that only the standard quantifiers lead to the preservation of such properties? Without any restriction on the shape of the fuzzy relations, the answer is positive.

The aim of this paper is to present the results of the competition of time series forecasting using soft computing methods that was organized within the IEEE WCCI 2016 congress.

In [1], B. Moser and M. Navara defined three “interpolation” axioms for fuzzy inference systems and showed, that usually neither very popular Mamdani-Assilian nor logically motivated implicative fuzzy systems, do satisfy them all simultaneously. The investigated inference mechanism was the compositional rule of inference (abb. CRI). Therefore, the authors introduced so-called conditionally firing rules and proved, that under very mild and practically very feasible conditions, all three axioms may be satisfied simultaneously. Note, that the conditionally firing rules were stemming from the Mamdani-Assilian rule interpretation and from the CRI. In [2], M. Štěpnička and S. Mandal showed that if the original axioms are supposed to express the intended meaning for the implicative interpretation of rules, they have to be re-defined. Furthermore, after such a re-definition, very similar results were obtained. Here, the so-called Bandler-Kohout subproduct (abb. BKS) was used as the inference mechanism. Therefore, analogously to [1], conditionally firing implicative rules were proposed and in order to satisfy the re-defined axioms simultaneously. However, as we know from the field of solvability of fuzzy relational equations, the advantageous composition/ image, which is directly related to the inference, is the direct image (related to CRI) when dealing with the implicative interpretation of fuzzy rules, and the subdirect image (related to BKS) when dealing with Mamdani-Assilian interpretation of fuzzy rules, not vice-versa. This article aims directly at this point and investigates the satisfaction of Moser-Navara axioms by the preferable combinations of rule interpretations and inference mechanisms.

The aim of this paper is to present the results of the time series forecasting competition that was organized within the IFSA-EUSFLAT 2015 conference.

As there are many various methods for time series prediction developed but none of them generally outperforms all the others, there always exists a danger of choosing a method that is inappropriate for a given time series. To overcome such a problem, distinct ensemble techniques, that combine more individual forecasts, are being proposed. In this contribution, we employ the so called fuzzy rule-based ensemble. This method is constructed as a linear combination of a small number of forecasting methods where the weights of the combination are determined by fuzzy rule bases based on time series features such as trend, seasonality, or stationarity. For identification of fuzzy rule base, we use linguistic association mining. An exhaustive experimental justification is provided.

Up to our best knowledge, distinct so far existing arithmetics of fuzzy numbers, usually stemming from the Zadeh's extensional principle, do not preserve some of the important properties of the standard arithmetics of classical (real) numbers. Obviously, although we cannot expect that a generalization of standard arithmetic will preserve precisely all its properties however, at least the most important ones should be preserved. We present a novel framework of arithmetics of extensional fuzzy numbers that preserves more or less all the important (algebraic) properties of the arithmetic of real numbers and thus, seems to be an important seed for further investigations on this topic. The suggested approach arithmetics of extensional fuzzy numbers is demonstrated on many examples and besides the algebraic properties, it is also shown that it carries some desirable practical properties.

In the first part of this contribution, we proposed extensional fuzzy numbers and a working arithmetic for them that may be abstracted to so-called many identities algebras (MI-algebras, for short). In this second part, we show that the proposed MI-algebras give a framework not only for the arithmetic of extensional fuzzy numbers, but also for other arithmetics of fuzzy numbers and even more general sets of real vectors used in mathematical morphology. This entitles us to develop a theory of MI-algebras to study general properties of structures for which the standard algebras are not appropriate. Some of the basic concepts and properties are presented here.

Frequent practical problems from decision-making as well as automatic control lead to intuitively monotone fuzzy rule bases. let us assume an appropriate ordering of fuzzy sets is defined. Then by the monotone fuzzy rule base we mean a rule base consisting of such fuzzy rules expressing the monotone dependence of consequent fuzzy sets on antecedent fuzzy sets. In other words the “bigger” antecedent fuzzy is present in a fuzzy rule the “bigger” consequent fuzzy set appears on the right hand side of the same fuzzy rule. Very often real-world applications require some defuzzification to be employed at the end of the inference process. The problem is that after the defuzzification we obtain a crisp input-output function which is not necessarily monotone anymore. Obviously, such behavior is not only counterintuitive but also dangerous. Most of the attention has been paid to the Mamdani-Assilian conjunctive kind of models of fuzzy rule bases built with help of particular t-norms. This paper focuses on the implicative approach for arbitrary residual implication.

Linguistic approach of time series analysis is suggested. It adopts aspects of the decomposition and autoregression. The linguistic, i.e., interpretable and transparent, nature of the approach is emphasized. Precision of the suggested approach is demonstrated on real time series.

It is well known that a fuzzy rule base can be interpreted in different ways. From a logical point of view, the conjunctive interpretation is preferred, while from a practical point of view, the disjunctive interpretation has been dominantly present. Each of these interpretations results in a specific fuzzy relation that models the fuzzy rule base. Basic interpolation requirements naturally suggest a corresponding inference mechanism: the direct image for the conjunctive interpretation and the subdirect image for the disjunctive interpretation. Interpolation then corresponds to solvability of some system of fuzzy relational equations. In this paper, we show that other types of fuzzy relations, which are closely related to Takagi–Sugeno (T–S) models, are of major interest as well. These fuzzy relations are based on addition and multiplication only, from which we get the name arithmetic fuzzy models. Under some mild requirements, these fuzzy relations turn out to be solutions of the same systems of fuzzy relational equations. The impact of these results is both theoretical and practical: There exist simple solutions to systems of fuzzy relational equations, other than the extremal solutions that have received all the attention so far, which are, moreover, easy to implement.

Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler–Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence offirst-infer-then-aggregate (FITA)andfirst-aggregate-then-infer (FATI)inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself.

The compositional rule of inference (CRI) is widely used in approximate reasoning schemes using fuzzy sets. In this work we discuss the suitability of the Bandler-Kohout subproduct for an alternative inference mechanism from the computational point of view.

A new methodology for forecasting of time series is proposed. It is based on combination of two techniques: fuzzy transform and perception-based logical deduction on the basis of learned linguistic description.

This contribution is intended as a position paper that favors the viewpoint that inference based on deductive rules (i.e., the rules are interpreted using fuzzy implication) can indeed be considered as a valuable inference scheme in real-world applications. For this purpose, we highlight the basic concepts behind the most common fuzzy inference schemes and demonstrate their interpretation by means of illustrative examples. We conclude that, under some reasonable conditions, deductive inference is able to compete with or even outperform the well-known Mamdani-Assilian inference.

A fuzzy if-then rule base may be viewed as a partial function between universes of fuzzy sets. For the construction of a fuzzy inference module, this partial function needs to be extended to a total one. Here, we propose a new method how to do so, making use of the method of smoothing splines. To this end, we identify the fuzzy sets with elements of a finite-dimensional real parameter space in an approximate way, using Perfilieva's fuzzy transforms. We then determine a function between two such parameter spaces by requiring that it reproduces the rule base as precise as possible and that it minimizes a parameter depending on its smoothness.

The paper is devoted to a fuzzy approach to numerical solutions of partial differential equations. Three main types of partial differential equations have been considered to demonstrate the algorithms with help of the fuzzy transform. We have introduced an example of a reasonable application of the fuzzy transform in this area. The justification of our approach including the convergence theorem has been presented as well