Midpoints for fuzzy sets and their application in medicine
ABSTRACT Using Kosko's hypercube, we identify a fuzzy set with a point in a unit hypercube. A nonfuzzy or crisp subset of a set is a vertex of the hypercube. We introduce some new ideas: the definition of the fuzzy segment joining two given fuzzy subsets of a set, the set of midpoints between those two fuzzy subsets, and the set of equidistant points from given points. We present some basic properties and relations between these concepts and provide a complete description of fuzzy segments and midpoints. In the majority of cases, there is no unique midpoint; one has an infinite set of possibilities to choose from. This situation is totally different from classical Euclidean geometry where, for two given points, there is a unique midpoint. We use the obtained results to study two sets of medical data and present two applications in medicine: the fuzzy degree of two concurrent food and drug addictions, and a fuzzy representation of concomitant causal mechanisms of stroke.

 "Some of these applications are based on the notions of segment joining two fuzzy subsets and set of midpoints between fuzzy subsets. The mentioned concepts were introduced for the first time by Nieto and Torres in [9]. The motivation for the study of these notions is given by their applications to analysis of real medical data. "
[Show abstract] [Hide abstract]
ABSTRACT: In 2003, Nieto and Torres introduced the notions of segment and midpoint between fuzzy sets with the aim of giving applications to medicine [J.J. Nieto, A. Torres, Midpoints for fuzzy sets and their application in medicine, Artif. Intell. Med. 27 (2003) 81101]. Since then the interest in the study of such concepts have grown significantly because of their applicability to model real problems where the solution can be associated with a range of “middle ways” between two given positions. Recently, J. Casasnovas and F. Roselló have generalized the previous work of Nieto and Torres, giving an explicit description of segments and midpoints between fuzzy sets for, among others, the wellknown weighted maximum distance [J. Casasnovas, F. Roselló, Averaging fuzzy biopolymers, Fuzzy Sets Syst. 152 (2005) 139158].Mathematical and Computer Modelling 05/2009; 49(91049):18521868. DOI:10.1016/j.mcm.2008.08.003 · 2.02 Impact Factor 
 "Remark. Also we use (see the section entitled Fuzzy clustering using the NTV distance function) the distance function of NTV (Nieto et al. 2003; Dress et al. 2004; Dress and Lokot 2003) via the membership function: "
[Show abstract] [Hide abstract]
ABSTRACT: In this paper we present a study of classification of the 20 amino acids via a fuzzy clustering technique. In order to calculate distances among the various elements we employ two different distance functions: the Minkowski distance function and the NTV metric. In the clustering procedure we take into account several physical properties of the amino acids. We examine the effect of the number and nature of properties taken into account to the clustering procedure as a function of the degree of similarity and the distance function used. It turns out that one should use the properties that determine in the more important way the behavior of the amino acids and that the use of the appropriate metric can help in defining the separation into groups.Journal of Theoretical Biology 12/2008; 257(1):1726. DOI:10.1016/j.jtbi.2008.11.003 · 2.30 Impact Factor 
 "then it is a simpler matter to see, from the classical Euclidean geometry, that the unique midpoint between µ and ν it is the fuzzy subset µ+ν 2 . Contrary to the Euclidean case J. Nieto and A. Torres showed that, for the Hamming distance, in general there is not a unique midpoint between two fuzzy sets ([17]). Let us recall that the Hamming distance on FP(X) is defined, for every µ, ν ∈ "
Article: A connection between Computer Science and Fuzzy Theory: Midpoints and running time of computing
[Show abstract] [Hide abstract]
ABSTRACT: Following the mathematical formalism introduced by M. Schellekens [Electronic Notes in Theoret. Comput. Sci. 1 (1995), 211232] in order to give a common foundation for Denotational Semantics and Complexity Analysis, we obtain an application of the theory of midpoints for asymmetric distances defined between fuzzy sets to the complexity analysis of algorithms and programs. In particular we show that the average running time for the algorithm known as Largetwo is exactly a midpoint between the best and the worst case running time of computing.01/2008; 15:251261.