Description

FuzzyNumbers is an open source (LGPL 3) package for R. It provides S4 classes and methods to deal with fuzzy numbers. The package may be used by researchers in fuzzy numbers theory (e.g., for testing new algorithms, generating numerical examples, preparing figures).

Details

Fuzzy set theory gives one of many ways (in particular, see Bayesian probabilities) to represent imprecise information. Fuzzy numbers form a particular subclass of fuzzy sets of the real line. The main idea behind this concept is motivated by the observation that people tend to describe their knowledge about objects through vague numbers, e.g., "I'm about 180 cm tall" or "The event happened between 2 and 3 p.m.".

For the formal definition of a fuzzy number please refer to the FuzzyNumber man page. Note that this package also deals with particular types of fuzzy numbers like trapezoidal, piecewise linear, or “parametric” FNs (see TrapezoidalFuzzyNumber PiecewiseLinearFuzzyNumber, PowerFuzzyNumber, and *EXPERIMENTAL* DiscontinuousFuzzyNumber)

The package aims to provide the following functionality:

1. Representation of arbitrary fuzzy numbers (including FNs with discontinuous side functions and/or alpha-cuts), as well as their particular types, e.g. trapezoidal and piecewise linear fuzzy numbers,

2. Defuzzification and approximation by triangular and piecewise linear FNs (see e.g. expectedValue, value, trapezoidalApproximation, piecewiseLinearApproximation),

3. Visualization of FNs (see plot, as.character),

4. Basic operations on FNs (see e.g. fapply and Arithmetic),

5. etc.

For a complete list of classes and methods call help(package="FuzzyNumbers"). Moreover, you will surely be interested in a step-by-step guide to the package usage and features which is available at the project's webpage.

Keywords: Fuzzy Numbers, Fuzzy Sets, Shadowed Sets, Trapezoidal Approximation, Piecewise Linear Approximation, Approximate Reasoning, Imprecision, Vagueness, Randomness.

Acknowledgments: Many thanks to Jan Caha, Przemyslaw Grzegorzewski, Lucian Coroianu, and Pablo Villacorta Iglesias for stimulating discussion.

The development of the package in March-June 2013 was partially supported by the European Union from resources of the European Social Fund, Project PO KL “Information technologies: Research and their interdisciplinary applications”, agreement UDA-POKL.04.01.01-00-051/10-00.

Author(s)

Marek Gagolewski, with contributions from Jan Caha

References

FuzzyNumbers Homepage, http://www.gagolewski.com/software/.

Ban A.I. (2008), Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy Sets and Systems 159, pp. 1327-1344.

Ban A.I. (2009), On the nearest parametric approximation of a fuzzy number - Revisited, Fuzzy Sets and Systems 160, pp. 3027-3047.

Bodjanova S. (2005), Median value and median interval of a fuzzy number, Information Sciences 172, pp. 73-89.

Chanas S. (2001), On the interval approximation of a fuzzy number, Fuzzy Sets and Systems 122, pp. 353-356.

Coroianu L., Gagolewski M., Grzegorzewski P. (2013), Nearest Piecewise Linear Approximation of Fuzzy Numbers, Fuzzy Sets and Systems 233, pp. 26-51.

Coroianu L., Gagolewski M., Grzegorzewski P., Adabitabar Firozja M., Houlari T. (2014), Piecewise linear approximation of fuzzy numbers preserving the support and core, In: Laurent A. et al. (Eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems, Part II (CCIS 443), Springer, pp. 244-254.

Delgado M., Vila M.A., Voxman W. (1998), On a canonical representation of a fuzzy number, Fuzzy Sets and Systems 93, pp. 125-135.

Dubois D., Prade H. (1978), Operations on fuzzy numbers, Int. J. Syst. Sci. 9, pp. 613-626.

Dubois D., Prade H. (1987a), The mean value of a fuzzy number, Fuzzy Sets and Systems 24, pp. 279-300.

Dubois D., Prade H. (1987b), Fuzzy numbers: An overview, In: Analysis of Fuzzy Information. Mathematical Logic, vol. I, CRC Press, pp. 3-39.

Grzegorzewski P. (2010), Algorithms for trapezoidal approximations of fuzzy numbers preserving the expected interval, In: Bouchon-Meunier B. et al. (Eds.), Foundations of Reasoning Under Uncertainty, Springer, pp. 85-98.

Grzegorzewski P. (1998), Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems 97, pp. 83-94.

Grzegorzewski P,. Pasternak-Winiarska K. (2011), Trapezoidal approximations of fuzzy numbers with restrictions on the support and core, Proc. EUSFLAT/LFA 2011, Atlantis Press, pp. 749-756.

Klir G.J., Yuan B. (1995), Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall, New Jersey.

Stefanini L., Sorini L. (2009), Fuzzy arithmetic with parametric LR fuzzy numbers, In: Proc. IFSA/EUSFLAT 2009, pp. 600-605.

Yeh C.-T. (2008), Trapezoidal and triangular approximations preserving the expected interval, Fuzzy Sets and Systems 159, pp. 1345-1353.