Similarity-based Partitions on Pre-ordered Sets

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Objective: To determine partitions of subsets where a preorder relation and a similarity relation coexist. Each partition-set represents a class of elements that reflects the variety of the whole set, so the similarity relation can be considered to reduce the size of large sets and to handle them more easily. We look at the minimal partitions originated by the similarity relation and compare them using well-established methods. Methods: A precise definition of the notion of similarity is proposed by unifying two pieces of relevant literature. The similarity-based partitions are mainly compared by using a method called the Rand index and a new method for the subset similarity index. Results: Looking at the minimal partitions originated by the similarity relation and comparing them using well-established methods, we observed that from an aseptic point of view, there are no essential differences between any of them. Conclusion: Comparing partitions produced by different methods is an important issue in clustering when comparing results obtained using different methods, parameters or initializations. When the sole objective is the variety offered by the sets in question, any smallest similarity-based partitions can be used to handle large sets. If it becomes necessary to consider the quality of the elements contained in those large sets, then one of them must be chosen specifically.