A comparison of Gap statistic definitions with and without logarithm function

The Gap statistic is a standard method for determining the number of clusters in a set of data. The Gap statistic standardizes the graph of log(Wk), where Wk is the within-cluster dispersion, by comparing it to its expectation under an appropriate null reference distribution of the data. We suggest to use Wk instead of log(Wk), and to compare it to the expectation of Wk under a null reference distribution. In fact, whenever a number fulfills the original Gap statistic inequality, this number also fulfills the inequality of a Gap statistic using Wk, but not vice versa. The two definitions of the Gap function are evaluated on several simulated data set and on a real data of DCE-MR images.