Multi-dimensional Fuzzy Set identification using Persistent Homology

In this paper we introduce a new way to determine a threshold for making a multidimensional fuzzy set from sample data set using persistent homology on these data. A multi dimensional fuzzy set is a fuzzy set that has arbitrary shape on the support parameter space. In previous paper, network distance of data is used to make a multi dimensional fuzzy set from sampling data keeping its topological properties. There are some control parameters to make sample data network and they are ad-hoc adjusted parameters with trial and errors. Persistent homology is a new mathematical methodology to make a feasible network structure from sample data set. We employ persistent homology and introduce a new algorithm and shown a feasibility of proposed method through numerical examples.