Michal Holčapek

A smoothing filter based on fuzzy transform

This paper is devoted to the smoothing of discrete functions using the fuzzy transform introduced by Perfilieva. We generalize a smoothing filter based on the fuzzy transform recently proposed by us to obtain a better control on the smoothed functions. For this purpose, a generalization of the concept of fuzzy partition is suggested and the smoothing filter is defined as a combination of the direct discrete fuzzy transform and a slightly modified inverse continuous fuzzy transform. An approximation behavior, total variation of smoothed functions and statistical properties including the description of the white noise reduction and the asymptotic expression of bias and variance are investigated and discussed. The proposed filter is compared with the Nadaraya-Watson estimator and the results are illustrated assuming financial data.

Filtering out high frequencies in time series using F-transform

Abstract In this paper, we focus on application of fuzzy transform (F-transform) to analysis of time series under the assumption that the latter can be additively decomposed into trend-cycle, seasonal component and noise. We prove that when setting properly width of the basic functions, the inverse F-transform of the time series closely approximates its trend-cycle. This means that the F-transform almost completely removes the seasonal component and noise. The obtained theoretical results are demonstrated on two artificial time series whose trend cycle is precisely known and on three real time series. At the same time, comparison with three classical methods, namely STL-method, SSA-method and low pass Butterworth filter is also provided.

Necessary and sufficient conditions for generalized uniform fuzzy partitions

Abstract The fundamental concept in the theory of fuzzy transform (F-transform) is the fuzzy partition, which is a generalization of the classical concept of partition. The original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1. This condition can be generalized by relaxing the assumption of normality for fuzzy sets. The result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse F-transform. A fuzzy partition with this property will be referred to as general. The problem is how a general fuzzy partition can be effectively constructed. If we use a generating function with special properties, then it is not immediately clear whether it defines a general fuzzy partition. In this paper, we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily, which is important in various practical applications of the F-transform, for example, image processing, time series analysis, and solving differential equations with boundary conditions.

Fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices

This paper presents basic notions about fuzzy measures over algebras of fuzzy subsets of a fuzzy set. It also presents basic ideas on fuzzy integrals defined using these fuzzy measures. Definitions of new types of fuzzy measures and integrals are motivated by our research on generalized quantifiers. Several useful properties of fuzzy measures and fuzzy integrals are stated and proved. Definitions presented in this paper and its results will be employed in subsequent papers on generalized quantifiers defined using this type of fuzzy integral.

Fuzzy Relational Compositions Based on Generalized Quantifiers

Fuzzy relational compositions have been extensively studied by many authors. Especially, we would like to highlight initial studies of the fuzzy relational compositions motivated by their applications to medical diagnosis by Willis Bandler and Ladislav Kohout. We revisit these types of compositions and introduce new definitions that directly employ generalized quantifiers. The motivation for this step is twofold: first, the application needs for filling a huge gap between the classical existential and universal quantifiers and second, the already existing successful implementation of generalized quantifiers in so called divisions of fuzzy relations, that constitute a database application counterpart of the theory of fuzzy relational compositions. Recall that the latter topic is studied within fuzzy relational databases and flexible querying systems for more than twenty years. This paper is an introductory study that should demonstrate a unifying theoretical framework and introduce that the properties typically valid for fuzzy relational compositions are valid also for the generalized ones, yet sometimes in a weaken form.

Discrete fuzzy transform of higher degree

In this paper, we reformulate the fuzzy transform of higher degree (Fm-transform) proposed originally for an approximation of continuous functions to the discrete case. We introduce two types of Fm-transform which components are defined using polynomials in the first case and using specific values of these polynomials in the second case. We provide an analysis of basic properties of Fm-transform.

Arithmetics of extensional fuzzy numbers - part I: Introduction

Up to our best knowledge, distinct so far existing arithmetics of fuzzy numbers, usually stemming from the Zadehs 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.

Arithmetics of extensional fuzzy numbers - part II: Algebraic framework

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.

Fuzzy objects in spaces with fuzzy partitions

A theory of fuzzy objects is derived in the category SpaceFP of spaces with fuzzy partitions, which generalize classical fuzzy sets and extensional maps in sets with similarity relations. It is proved that fuzzy objects in SpaceFP can be characterized by some morphisms in the category of sets with similarity relations. A powerset object functor