In this work a method to make an easier pattern classification when the use of a classical membership function doesn't give satisfactory results due to the similarity between prototypes, is developed. A new formulation of the membership function itself is proposed for extreme cases of indistinguishability.
The object of this paper is to discuss how fuzziness of data is propagated when statistical inference for samples of non-precise data is carried out. The method of propagation of fuzziness as introduced by Schnatter (1989) is reviewed. This method may be applied to any statistical method which leads to a result that may be expressed as a function f(x1,...,x(n)) of the data x1,...,x(n). It is outlined that practical application of this method is equal to determining the images of a family of compact subsets of the sample space under the function f(.). An illustrative example from environmetrics is discussed. The general approach is used to formulate a fuzzy sample mean and a fuzzy valued empirical distribution function.
A number of hard clustering algorithms have been shown to be derivable from the maximum likelihood principle. The only corresponding fuzzy algorithm are the well known fuzzy k-means or FUZZY ISODATA of Dunn and its generalizations by Bezdek and by Gustafson and Kessel. The authors show how to generate two other fuzzy algorithms which are the analogous of known hard algorithms: the minimization of the fuzzy determinant and of the product of fuzzy determinants. By comparison between the hard and fuzzy methods it appears that the latter yield more often the global optimum, rather than merely a local optimum. This result and the comparison between the different algorithms, together with their specific domains of application, are illustrated by a few numerical examples.
Many criticisms of prototype theory and/or fuzzy-set theory are based on the assumption that category representativeness (or typicality) is identical with fuzzy membership. These criticisms also assume that conceptual combination and logical rules (all in the Aristotelian sense) are the appropriate criteria for the adequacy of the above "fuzzy typicality". The present paper discusses these assumptions following the line of their most explicit and most influential expression by Osheron and Smith (1981). Several arguments are made against the above identification, the most important being that representativeness in prototype theory is exclusively based on element-to-element similarity while fuzzy membership is inherently an element-to-category relationship. Also the above criteria for adequacy are criticized from the viewpoint of both prototype theory and fuzzy-set theory as well as from that of both conceptual and logical combination, and also from that of integration.
An extensive fuzzy boundary was identified within the adult extension of two exemplary object-words, 'dog' and 'ball'. The fuzzy boundaries were incorporated into the standard of adult extension against which the preschool child's use of 'dog' and 'ball' was compared, and the investigation provided a framework in which children's responses to fuzzy boundary referents could be accommodated. Responses to fuzzy boundary referents were considered as having precedents within adult extension and the data revealed the provision of precedents for a significant number of the young child's applications and non-applications of the words. The investigation concluded that, through adherence to an oversimplified concept of adult extension, previous research has underestimated the degree of correspondence that exists between child and adult extension.