Hierarchical structure invariance and optimal approximation for proximity data

Based on granular space, the hierarchical structure invariance and optimal approximation for proximity data are presented, and four results are obtained as follows. Firstly, an improved algorithm for computing the hierarchical structure and the related min-transitive closure of a fuzzy proximity relation are given, and the properties of key point sequence are studied by introducing the key points and key values of a fuzzy proximity relation. Secondly, two basic concepts, the hierarchical structure invariance and isomorphism of fuzzy proximity relations are introduced, and some examples are given to illustrate that the hierarchical structure of a fuzzy proximity relation is variant under three typical triangular norms. Thirdly, the hierarchical structure invariance theorem of fuzzy proximity relation under a mapping or transformation is given. Finally, for a given fuzzy proximity relation, a mathematical model for obtaining the optimal approximation in order to keep its hierarchical structure is established through the use of the linear combination of its minimum fuzzy proximity relation and min-transitive closure. These results help us understand the hierarchical structures, and provide theories and methodologies for the structural analysis and applications of proximity data.