Midpoints for fuzzy sets and their application in medicine.

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain.
Artificial Intelligence in Medicine (Impact Factor: 1.36). 02/2003; 27(1):81-101. DOI: 10.1016/S0933-3657(02)00080-5
Source: PubMed

ABSTRACT Using Kosko's hypercube, we identify a fuzzy set with a point in a unit hypercube. A non-fuzzy 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.

1 Bookmark
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Let d be a metric on the set FP(X) of fuzzy subsets of a set X. A mid- point of two fuzzy subsets µ, 2 FP(X) is any fuzzy subset 2 FP(X) such that d(,µ ) = d(, ) = 1 2d(µ, ). These midpoints can be used to describe "middle ways" or "compromises" between two situa- tions described by the fuzzy subsets µ and . In this work we explic- itly compute midpoints for weighted Hamming distances and for weighted maximum distances. The former is a generalization of a previous work by Nieto and Torres (Artif. Intell. Med. 27 (2003), 81-101). We also propose a new application of mid- points in medicine, based on their use as average representations of pa- tients of which we have available two descriptions as fuzzy subsets of a set of attribute variables.
  • [Show abstract] [Hide abstract]
    ABSTRACT: Although adaptive neuro-fuzzy inference system (ANFIS) has very fast convergence time, it is not suitable for classification problems because its outputs are not integer. In order to overcome this problem, this paper provides four adaptive neuro-fuzzy classifiers; adaptive neuro-fuzzy classifier with linguistic hedges (ANFCLH), linguistic hedges neuro-fuzzy classifier with selected features (LHNFCSF), conjugate gradient neuro-fuzzy classifier (SCGNFC) and speeding up scaled conjugate gradient neuro-fuzzy classifier (SSCGNFC). These classifiers are used to achieve very fast, simple and efficient breast cancer diagnosis. Both SCGNFC and SSCGNFC systems are optimized by scaled conjugate gradient algorithms. In these two systems, k-means algorithm is used to initialize the fuzzy rules. Also, Gaussian membership function is only used for fuzzy set descriptions, because of its simple derivative expressions. The other two systems are based on linguistic hedges (LH) tuned by scaled conjugate gradient. The classifiers performances are analyzed and compared by applying them to breast cancer diagnosis. The results indicated that SCGNFC, SSCGNFC and ANFCLH achieved the same accuracy of 97.6608 % in the training phase while LHNFCSF performed better than other methods in the training phase by achieving an accuracy of 100 %. In the testing phase, the overall accuracies of LHNFCSF achieved 97.8038 %, which is superior also to other methods. Applying LHNFCSF not only reduces the dimensions of the problem, but also improves classification performance by discarding redundant, noise-corrupted or unimportant features. Also, the k-means clustering algorithm was used to determine the membership functions of each feature. LHNFCSF achieved mean RMSE values of 0.0439 in the training phase after feature selection and gives the best testing recognition rates of 98.8304 and 98.0469 during training and testing phases, respectively using two clusters for each class. The results strongly suggest that ANFCLH can aid in the diagnosis of breast cancer and can be very helpful to the physicians for their final decision on their patients.
    Neural Computing and Applications 11/2012; · 1.76 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we study the numerical solution of fuzzy differential equations by using hybrid Euler method and hybrid predictor-corrector method. These methods are used to increase the accuracy and the computing speed. Also examples are presented to illustrate the computational aspects of the above methods.
    Fuzzy Information and Engineering 01/2012; 4(4).