In this paper, we concentrate on the usage of uncertainty associated with the level of fuzziness in determination of the number of clusters in FCM for any data set. We propose a MiniMax -stable cluster validity index based on the uncertainty associated with the level of fuzziness within the framework of interval valued Type 2 fuzziness. If the data have a clustered structure, the optimum number of clusters may be assumed to have minimum uncertainty under upper and lower levels of fuzziness. Upper and lower values of the level of fuzziness for Fuzzy -Mean (FCM) clustering methodology have been found as = 2.6 and 1.4, respectively, in our previous studies. Our investigation shows that the stability of cluster centers with respect to the level of fuzziness is sufficient for the determination of the number of clusters.
In this paper we describe a non-nested level-based representation of fuzziness, closely related to some existing models and concepts in the literature. Our objective is to motivate the use of this non-nested model by describing its theoretical possibilities, and illustrating them with some existing applications. From a theoretical point of view, we discuss the semantics of the representation, which goes beyond and has as a particular case fuzzy sets as represented by a collection of . In addition, the proposed operations on level-based representations, contrary to those of existing fuzzy set theories, satisfy all the properties of Boolean logic. We discuss the contributions of the representation and operation by levels to practical applications, in particular for extending crisp notions to the fuzzy case. In this respect, an important contribution of the proposal is that fuzzy mathematical objects (not only sets and the corresponding predicates) and operations are uniquely and easily defined as extensions of their crisp counterparts. In order to illustrate this claim, we recall level representations of quantities (gradual numbers) and their complementarity to fuzzy intervals (often inappropriately called fuzzy numbers). ► We provide a representation of fuzziness using a finite subset of levels in (0,1]. ► Fuzzy mathematical objects are an assignment of their crisp counterparts to levels. ► Contrary to fuzzy sets, crisp representatives are not necessarily nested between levels. ► Operations are performed on the representatives of the same level independently. ► Fuzzification of crisp objects/operations is unique and keep all the properties of the crisp case.
► Assessing landscape preference through photo-based questionnaires. First study with real photos, second one with manipulated photos. ► The Fuzziness of Mediterranean landscapes – mixed composition of land cover classes together with fuzzy borders make it difficult for the participants to respond to differences between images. ► Digital manipulation of photos – manipulation makes it possible for a specific landscape pattern to become more visible and understandable for the respondents. ► Testing the use of photo-based questionnaires in the selected case study. ► The results of the two studies presented have contributed to improved knowledge in methods to assess landscape preferences in Mediterranean fuzzy landscapes. Mediterranean landscapes reveal extremely adequate conditions for the development of other functions besides production (nature conservation, recreation, life quality, local identity). These functions support the provision of public goods and services increasingly recognized by society. With this goal, the production of knowledge that may support decision is highly needed. In Mediterranean extensively used areas, the analysis of landscape features and related public preferences is complex, as the landscape pattern is highly fuzzy and land cover classes are often mixed. Resulting from multiple research developments, this paper demonstrates how photo-based surveys can be a suitable tool for assessing landscape preferences by specific public groups. Landscape functions addressed are closely linked to land cover patterns, as resulting from land cover systems. Thus using photographs in landscape questionnaires is useful in focusing the discussion on specific aspects, related with the variations in land cover and in their combinations with other specific landscape features. But the photos shown need to be clear and easily perceivable by the respondents. In order to cope with the underlying fuzziness of these landscapes, manipulation of images has been developed as the best solution so that the variations shown to respondents are adequately controlled in the study and landscape features are easily recognized by the respondents. The methodological approach as well as the results of applied approaches, of two studies on the users preferences, applied to a case-study area in Alentejo region, Portugal, are presented. The issues concerned with photo manipulation are a particular focus of discussion.