FUZZINESS AND STATISTICS – MATHEMATICAL MODELS FOR UNCERTAINTY
Abstract
Real data from continuous quantities, considered under different models in economic theory, cannot be measured precisely. As a result, measurement results cannot be accurately represented by real numbers, as they contain different kinds of uncertainty. Beside errors and variability, individual measurement results are more or less fuzzy as well. Therefore, real data have to be described mathematically in an adequate way. The best up-to-date models for this are so-called fuzzy numbers, which are special fuzzy subsets of the set of real numbers. Based on this description, statistical analysis methods must be generalized to the situation of fuzzy data. This is possible and will be explained here for descriptive statistics, inferential statistics, objective statistics, and Bayesian inference.
References
[2] Filzmoser, P., and Viertl, R. 2004. Testing Hypotheses with Fuzzy Data: The Fuzzy p-Value, Metrika, 59: 21-29.
[3] Hansen, L. P. 2017. Uncertainty in Economic Analysis and the Economic Analysis of Uncertainty. Available at: http://larspeterhansen.org/lph_research/uncertainty-in-economic-analysis-and-the-economic-analysis-of-uncertainty/
[4] Klir, G., and Yuan, B. 1995. Fuzzy Sets and Fuzzy Logic–Theory and Applications, Upper Saddle River, NJ: Prentice-Hall.
[5] Kovářová, L., and Viertl, R. 2015. The Generation of Fuzzy Sets and the Construction of Characterizing Functions of Fuzzy Data, Iranian Journal of Fuzzy Systems, 12(6): 1-16.
[6] Krasker, W. S. et al. 1983. Chapter 11: Estimation for Dirty Data and Flawed Models, Handbook of Econometrics, 1: 651-698.
[7] Möller, B. et al. 2009. Fuzzy Random Process and their Application to Dynamic Analysis of Structures, Mathematical and Computer Modelling of Dynamical Systems, 15(6): 515-534.
[8] Möller, B., and Reuter, U. 2007. Uncertainty Forecasting in Engineering, Berlin: Springer-Verlag.
[9] Sunanta, O., and Viertl, R. 2016. On Fuzzy Bayesian Inference, in C. Kahraman and O. Kabak (Eds.): Fuzzy Statistical Decision-Making, Switzerland, Springer International, 55-64.
[10] Viertl, R. 2011. Statistical Methods for Fuzzy Data, Chichester: Wiley.
[11] Viertl, R. 2015. Measurement of Continuous Quantities and their Statistical Evaluation, Austrian Journal of Statistics, 44: 25-32.
[12] Viertl, R., and Sunanta, O. 2013. Fuzzy Bayesian inference, METRON, 71 (3): 207-216.
[13] Zadeh, L. A. 1965. Fuzzy Sets, Information and Control, 8(3): 338-353.
[14] Zadeh, L. A. 1975. The Concept of Linguistic Variable and Its Application to Approximate Reasoning, Information Science 8: 43-80.
Non-Exclusive License under Attribution 4.0 International Public License (CC BY 4.0):
This ‘Article’ is distributed under the terms of the license CC-BY 4.0., which lets others distribute, remix, adapt, and build upon this article, even commercially, as long as they credit this article for the original creation. ASERS Publishing will be acknowledged as the first publisher of the Article and a link to the appropriate bibliographic citation (authors, article title, volume issue, page numbers, DOI, and the link to the Published Article on ASERS Publishing’ Platform) must be maintained.