Ground truth bias in external cluster validity indices

External cluster validity indices (CVIs) are used to quantify the quality of a clustering by comparing the similarity between the clustering and a ground truth partition. However, some external CVIs show a biased behaviour when selecting the most similar clustering. Users may consequently be misguided by such results. Recognizing and understanding the bias behaviour of CVIs is therefore crucial. It has been noticed that some external CVIs exhibit a preferential bias towards a larger or smaller number of clusters which is monotonic (directly or inversely) in the number of clusters in candidate partitions. This type of bias is caused by the functional form of the CVI model. For example, the popular Rand Index (RI) exhibits a monotone increasing (NCinc) bias, while the Jaccard Index (JI) index suffers from a monotone decreasing (NCdec) bias. This type of bias has been previously recognized in the literature. In this work, we identify a new type of bias arising from the distribution of the ground truth (reference) partition against which candidate partitions are compared. We call this new type of bias ground truth (GT) bias. This type of bias occurs if a change in the reference partition causes a change in the bias ∗Corresponding author Email addresses: [email protected] (Yang Lei), [email protected] (James C. Bezdek), [email protected] (Simone Romano), [email protected] (Nguyen Xuan Vinh), [email protected] (Jeffrey Chan), [email protected] (James Bailey) Preprint submitted to Elsevier June 20, 2016 ar X iv :1 60 6. 05 59 6v 1 [ st at .M L ] 1 7 Ju n 20 16 status (e.g., NCinc, NCdec) of a CVI. For example, NCinc bias in the RI can be changed to NCdec bias by skewing the distribution of clusters in the ground truth partition. It is important for users to be aware of this new type of biased behaviour, since it may affect the interpretations of CVI results. The objective of this article is to study the empirical and theoretical implications of GT bias. To the best of our knowledge, this is the first extensive study of such a property for external cluster validity indices. Our computational experiments show that 5 of 26 indices studied in this paper exhibit GT bias. Following the numerical examples, we provide a theoretical analysis of GT bias based on the relationship between the RI and quadratic entropy. Specifically, we prove that the quadratic entropy of the ground truth partition provides a computable test which predicts the NC bias status of the Rand Index.