Radek Valášek

Fuzzy Transforms in Removing Noise

The technique of fuzzy transform (F-transform for short) has been introduced in [6, 5]. It consists of two phases: direct and inverse. We have proved that the inverse F-transform has good approximation properties and is very simple to use.

Numerical solution of partial differential equations with help of fuzzy transform

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

Parallel mining of fuzzy association rules on dense data sets

The aim of this paper is to present a scalable parallel algorithm for fuzzy association rules mining that is suitable for dense data sets. Unlike most of other approaches, we have based the algorithm on the Webbs OPUS search algorithm [1]. Having adopted the master/slave architecture, we propose a simple recursion threshold technique to allow load-balancing for high scalability.

F-transform-based shooting method for nonlinear boundary value problems

We study a class of ordinary two-point boundary value problems for nonlinear second-order ordinary differential equations. We propose to use the framework of shooting method and apply the second-degree F-transform for solving the corresponding initial value problem. We make experiments that demonstrate effectiveness of the proposed F-transform-based shooting method in the class of second-order numerical methods. Moreover, we show stability of the proposed method and its ability to “ignore noise”.

GENERATING A FUZZY RULE BASE WITH AN ADDITIVE INTERPRETATION

Abstract Since 1965 the fuzzy set theory and its application have deep development especially in many disciplines close to the automatic control of processes. A fuzzy model has been shown to be able to approximate the behaviour of many complex processes. Very robust fuzzy controller can be constructed in various ways. One of them, learning algorithm, is focused in this paper while the approximation idea has been brought from the technique called F-transform..

Numerical solution of partial differential equations with the help of fuzzy transform technique

In this paper, we propose a numerical method based on the F-transform for solving a certain type of partial differential equations (PDEs) with Dirichlet boundary conditions and initial conditions. We show how the PDEs after the application of the Crank-Nicolson scheme for time discretization can be approximated by a system of linear equations with the direct F-transform components as their variables. The numerical solution of PDEs is obtained by solving the system of linear equations and the application of the inverse F-transform. The proposed method is then adjusted to the two-dimensional boundary value problem illustrated on two examples including the Black-Scholes equation well-known in financial modeling.

Specialized software for fuzzy natural logic and fuzzy transform applications

In this paper, three special soft computing software systems are presented. The systems are based on the original results in two areas: fuzzy natural logic and fuzzy transform. The first software is LFL Controller that is a universal SW system that can be used in fuzzy or linguistic control, and in decision-making. The system implements results of fuzzy natural logic, namely the theory of evaluative linguistic expressions and perception-based logical deduction. This means the control or decision strategy are defined directly in natural language. Additionally, it can also realize classical fuzzy control on the basis of relational interpretation of fuzzy IF-THEN rules. This system has wide range of applications and was already applied in control of real plants. The second system is LFL Forecaster that is a specialized SW for analysis and forecasting of time series. The analysis is realized using F-transform and forecasting using results of fuzzy natural logic. The third system is FT-Studio that is a specialized SW for computation of fuzzy transform of functions that can be defined either using a formula, or given by data.

Shooting method based on higher degree F-transform

We study the framework of the shooting method for nonlinear second-order ordinary differential equations with boundary conditions. We propose a new numerical scheme, based on the second degree F-transform for solving the corresponding initial value problem. We use the inverse F-transforms of the zero and first degrees to obtain approximate solutions to the initial second-order differential equation. We make experiments that demonstrate effectiveness of the proposed F-transform-based shooting method in the class of second order numerical methods.