Upper and lower values for the level of fuzziness in FCM

The level of fuzziness is a parameter in fuzzy system modeling which is a source of uncertainty. In order to explore the effect of this uncertainty, one needs to investigate and identify effective upper and lower boundaries of the level of fuzziness. For this purpose, Fuzzy c-means (FCM) clustering methodology is investigated to determine the effective upper and lower boundaries of the level of fuzziness in order to capture the uncertainty generated by this parameter. In this regard, we propose to expand the membership function around important information points of FCM. These important information points are, cluster centers and the mass center. At these points, it is known that, the level of fuzziness has no effect on the membership values. In this way, we identify the counter-intuitive behavior of membership function near these particular information points. It will be shown that the upper and lower values of the level of fuzziness can be identified. Hence the uncertainty generated by this parameter can be encapsulated.