Fuzzy c-means clustering of incomplete data

The problem of clustering a real s-dimensional data set X = {x(1)..., x(n)} subset of R-s is considered. Usually, each observation (or datum) consists of numerical values for all s features (such as height, length, etc.), but sometimes data sets can contain vectors that are missing one or more of the feature values. For example, a particular datum xk might be incomplete, having the form x(k) = (254.3, ?, 333.2, 47.44, ?)(T), where the second and fifth feature values are missing. The fuzzy e-means (FCM) algorithm is a useful tool for clustering real s-dimensional data, but it is not directly applicable to the case of incomplete data. Four strategies for doing FCM clustering of incomplete data sets are given, three of which involve modified versions of the FCM algorithm. Numerical convergence properties of the new algorithms are discussed, and all approaches are tested using real and artificially generated incomplete data sets.