  The paper presents a formula for the reclassification of multidimensional
data points (columns of real numbers, "objects", "vectors", etc.). This formula
describes the change in the total squared error caused by reclassification of
data points from one cluster into another and prompts the way to calculate the
sequence of optimal partitions, which are characterized by a minimum value of
the total squared error E (weighted sum of within-class variance,
within-cluster sum of squares WCSS etc.), i.e. the sum of squared distances
from each data point to its cluster center. At that source data points are
treated with repetitions allowed, and resulting clusters from different
partitions, in general case, overlap each other. The final partitions are
characterized by "equilibrium" stability with respect to the reclassification
of the data points, where the term "stability" means that any prescribed
reclassification of data points does not increase the total squared error E. It
is important that conventional K-means method, in general case, provides
generation of instable partitions with overstated values of the total squared
error E. The proposed method, based on the formula of reclassification, is more
efficient than K-means method owing to converting of any partition into stable
one, as well as involving into the process of reclassification of certain sets
of data points, in contrast to the classification of individual data points
according to K-means method.
