# The Impact of Random Models on Clustering Similarity

2018 Apr 10**The Impact of Random Models on Clustering Similarity**

* Abstract: Clustering is a central approach* for

*unsupervised learning. After clustering is applied, the most fundamental analysis is to quantitatively compare clusterings. Such comparisons are crucial for the evaluation of clustering methods as well as other tasks such as consensus clustering. It is often argued that, in order to establish a baseline, clustering similarity should be assessed in the context of a random ensemble of clusterings. The prevailing assumption for the random clustering ensemble is the permutation model in which the number and sizes of clusters are fixed. However, this assumption does not necessarily hold in practice; for example, multiple runs of K-means clustering*reurns

*clusterings with a fixed number of clusters, while the cluster size distribution varies greatly. Here, we derive corrected variants of two clustering similarity measures (the Rand index and Mutual Information) in the context of two random clustering ensembles in which the number and sizes of clusters vary. In addition, we study the impact of one-sided comparisons in the scenario with a reference clustering. The consequences of different random models are illustrated using synthetic examples, handwriting recognition, and gene expression data. We demonstrate that the choice of random model can have a drastic impact on the ranking of similar clustering pairs, and the evaluation of a clustering method with respect to a random baseline; thus, the choice of random clustering model should be carefully justified.*

**Discussion:** Given the prevalence of clustering methods for analyzing data, clustering comparison is*a fundamental problem that is pertinent to numerous areas of science. In particular, the* *correction of clustering similarity for chance serves to establish a baseline that facilitates* *comparisons between different clustering solutions. Expanding previous studies on the* *selection of an appropriate model for random clusterings (Meila, 2005; Vinh et al., 2009;* *Romano et al., 2016), our work provides an extensive summary of random models and clearly* *demonstrates the strong impact of the random model on the interpretation of clustering* *results.*

*Our results underpin the importance of selecting the appropriate random model for a*

*given context. To that end, we offer the following guidelines:* *1. Consider what is fixed by the clustering method: do all clusterings have a* user specified *number of clusters (use* Mnum), *or is the cluster size sequence fixed (use Mperm)?* *2. Is the comparison against a reference clustering (use a one-sided comparison), or are* *you comparing two derived clusterings (then use a two-sided comparison)?* *The specific comparisons studied here are not meant to establish the superiority of a* *particular clustering identification technique or a specific random clustering model, rather,* *they illustrate the importance of the choice of the random model. Crucially, conclusions* *based on corrected similarity measures can change depending on the random model for* *clusterings. Therefore, previous studies which did promote methods based on evidence* *from corrected similarity measures should be re-evaluated in the context of the appropriate* *random model for clusterings (Yeung et al., 2001; de Souto et al., 2008; Yeung and Ruzzo,* *2001; Thalamuthu et al., 2006; McNicholas and Murphy, 2010).*