The necessary conditions for minimizing (1) with the constraint (2) result in the following iterative update formulas for the prototypes and the partition matrix [33]:
equation(3)vi=∑k=1nuikmxk∑k=1nuikmfori=1,2,…,c,and
equation(4)uik=[∑t=1c(∥xk−vi∥22∥xk−vt∥22)1m−1]−1fori=1,2,…,candk=1,2,…,n.
The iterations XL019 carried out until the changes in the values of the partition matrix reported in consecutive iterations are lower than a certain predetermined threshold.
2.2. Fuzzy C-Means clustering of incomplete data
In gestation subsection, four strategies based on the FCM algorithm, proposed by Hathaway and Bezdek, for handling incomplete data clustering are introduced. First, we introduce some required terminology and notation, which are consistent with those discussed in [1]. Let
equation(5)X= x1,x2,…,xn ⊂Rsbe an incomplete data set in s-dimensional real space,
equation(6)XW= xk∈X xkisacompletedatum be the whole-data subset of X,
equation(3)vi=∑k=1nuikmxk∑k=1nuikmfori=1,2,…,c,and
equation(4)uik=[∑t=1c(∥xk−vi∥22∥xk−vt∥22)1m−1]−1fori=1,2,…,candk=1,2,…,n.
The iterations XL019 carried out until the changes in the values of the partition matrix reported in consecutive iterations are lower than a certain predetermined threshold.
2.2. Fuzzy C-Means clustering of incomplete data
In gestation subsection, four strategies based on the FCM algorithm, proposed by Hathaway and Bezdek, for handling incomplete data clustering are introduced. First, we introduce some required terminology and notation, which are consistent with those discussed in [1]. Let
equation(5)X= x1,x2,…,xn ⊂Rsbe an incomplete data set in s-dimensional real space,
equation(6)XW= xk∈X xkisacompletedatum be the whole-data subset of X,