Polarization Degree of a Fuzzy Partition

It is then easy to prove that 5 / (F) 1 for each binary fuzzy partition. 

In what follows we show that a fuzzy partition P = A, A2 describes real clusters if and only if for each class / ,/= 1,2, there exists an x such that Aj(x) and R(P) t, where t is an appropriate threshold in the interval (0.5,1). We may interpret t as our confidence limit that the detected clusters are real. We consider now a data set X = x x, x, x and a fuzzy partition P = [A, A2 of X having the memberships specified as follows  

We may assume with a confidence limit of f = 0.63 that C, and C2 represent real clusters. 

Next we discuss a fuzzy hierarchical clustering procedure proposed by Dumitrescu. The procedure starts with the computation of a binary fuzzy partition A, A2] of the data set X. To do this, the FNM algorithm or other appropriate procedure may be used. If A j and A 2 do not describe real clusters, we may conjecture there is no structure in X or that the data comprise a single compact cluster. The process then ends. If the fuzzy classes A and A2 correspond to real clusters, we set = v4j,/I2. Assume the cluster structure of each class zl, / = 1,2, is given by a binary fuzzy partition of z4,. We may compute the cluster substructure of Aj using the GFNM algorithm or one of its relatives. There are now two possibilities  

For the unmarked classes of we follow the same procedure. In this way a new fuzzy partition P of X is obtained. The partitioning procedure is stopped when all the modules of the current partition P have been marked. Using this procedure we can detect a hierarchical structure in the data set X. The optimal number of clusters in X and the corresponding final classes are also obtained. 

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