A comparison of Gap statistic definitions with and without logarithm function

Abstract

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 $\log(W_{k})$, where $W_{k}$ is the within-cluster dispersion, by comparing it to its expectation under an appropriate null reference distribution of the data. We suggest to use $W_{k}$ instead of $\log(W_{k})$, and to compare it to the expectation of $W_{k}$ 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 $W_{k}$, but not \textit{vice versa}. The two definitions of the Gap function are evaluated on several simulated data sets and on a real data of DCE-MR images.