The linguistic approximation of fuzzy models outputs

Abstract

Linguistic approximation is a process of assigning linguistic labels to various mathematical objects, frequently ones that are obtained as outputs of fuzzy models. Such an assignment cannot be arbitrary – the usual requirement on linguistic approximation is for the linguistic label to “represent the meaning” of the approximated object sufficiently, or at least to reflect its characteristics that are the most important for the given purpose. How to define such a sufficiency, or in other words how to recognize an appropriate method of linguistic approximation, however, remains an unresolved issue. Over the years several various approaches for linguistic approximation was introduced but almost no proper comparison of these approaches was made. This thesis strives to resolve this issue by suggesting a universal analytical framework that helps the designers (and also users) of the models to visualize the performance of linguistic approximation under different distance/similarity measure of fuzzy numbers and to use this visualization to compare their performance and identify the potential drawbacks of using selected distance/similarity measures. It therefore approaches the issue of sufficiency of the linguistic approximation from behind – mainly pointing out the problems and thus ruling out some of the notwell- functioning distance/similarity measures.

The contribution of the thesis lies in the proposal of a framework for the analysis of performance of different distance/similarity measures of fuzzy numbers such that its use is straightforward, it requires only limited knowledge from his potential user, it allows for a direct comparison of the performance of different distance/similarity measures in the given context and it provides results by means of graphical representation. Although most of the thesis focuses on the frequently used shapes of fuzzy numbers (triangular, trapezoidal; both symmetrical and asymmetrical), we also propose a modification of the framework which allows for the analysis and visualization of results for Mamdani-type fuzzy sets (outputs of Mamdani fuzzy inference). On this modification we also show the simplicity of the generalization of the framework for different conditions and contexts (represented by different approximated objects etc.). Another contribution is the proposal of a novel linguistic approximation method based on the idea of fuzzy 2-tuples in the thesis. This method differs from other methods in a way that it requires only small number of linguistic terms (i.e. the decision-maker’s vocabulary for the description of the results can remain reasonably limited, hence the requirements on understanding the meaning of the linguistic values can be kept to a reasonable minimum), but thanks to the 2-tuple concept it can results in an infinite n umber o f l inguistic l abels c omposed o f one of the known linguistic terms and a description of its “shift” which facilitates easy understandability of the results of the linguistic approximation.

Eleven publications on which the author participated are appended to the thesis. These publications summarize the mathematical results and provide closer insights into the issue of the investigation of the performance of linguistic approximation under different distance/similarity measures.