Math: pyhasse.fuzzy

1. The relation between two objects x and y is replaced by a fuzzy subsethood, and the degree of subsethood of x fuzzy y, and that of y fuzzy x will be calculated. The idea is that by defuzzification a limit can be introduced whether or not a subsethood is considered as a
2. There may be different measures of the subsethood degree. Kerber and Brüggemann [2015] suggest a generalized approach within the framework of fuzzy set algebra. Here, the measure is taken from ideas of Kosko [1992] which were first time transferred into the fuzzy framework by De Walle et al., [De Walle, 1995]. A matrix, where the subsethood of each object versus each other object, is obtained.
3. The crisp transitivity demand as the essential axiom to declare a relational matrix like the subsethood matrix representing a partial order cannot applied. Instead a fuzzy transitivity requirement is needed: The fuzzy relation R(x,y), R(y,z) fulfills the fuzzy transitivity if equation (4) holds:

(4)

Here R (Relation) is specialized for the subsethood measure SH, as obtained in step 2.According to Kosko SH(x,y) is calculated as follows:

(5)

If the denominator = 0, then SH(x,y) = 1.
4. By an iterative procedure the matrix SH is modified, until the requirement (4) is fulfilled. In PyHasse the matrix method, [De Baets, DeMeyer, 2003] is applied. It results a matrix , which obeys (3) and w from which one can speak of a representation of a fuzzy partial order.
5. A defuzzification is needed, to obtain relations, which can be drawn as Hasse diagram. A crisp matrix is obtained, following equation (6).

(6)

The parameter (acut in PyHasse) serves as a threshold. The matrix may still contain equivalence relations. After extracting them, a description of a poset by the resulting matrix, is obtained.

Examples

See Bruggemann et al., 2011, Altschuh et al, 2015 and Bruggemann, Patil, 2011.

References

Kosko, B. 1992. Neural Networks and Fuzzy Systems - A dynamical Systems approach to Machine Learning. Pentice Hall, London.

Van de Walle, B., B. De Baets, and K. C. Kersebaum. 1995. Fuzzy multi-criteria analysis of cutting techniques in a nuclear dismantling project. Fuzzy sets and Systems 74:115-126.