Soft clustering

Abstract

Clustering is one of the most used tools in data analysis. In the last decades, due to the increasing complexity of data, soft clustering has received a great deal of attention. There exist different approaches that can be considered as soft. The most known is the fuzzy approach that consists in assigning objects to clusters with membership degrees, depending on the dissimilarities between each object and all the prototypes, ranging in the unit interval. Closely related to the fuzzy approach, there is the possibilistic one that, differently from the previous one, relaxes some constraints on the membership degrees. In particular, the objects are assigned to clusters with degrees of typicalities, depending just on the dissimilarities between each object and the closest prototype. A further soft approach is the rough one. In this case, there are not degrees ranging between 0 and 1 but objects with intermediate features belong to the boundary region and are assigned to more than one cluster. Even if it is not universally recognized in the scientific community as an approach of soft clustering, from our point of view, the model-based approach can also be considered as such. Model-based clustering methods also produce a soft partition of the objects and the posterior probability of a component membership may play a role similar to the membership degree. The four approaches are critically described from a theoretical point of view and an empirical comparative analysis is carried out.

Diversity of global fisheries governance: Types and contexts

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“…In finite mixture modelling, the data are considered as coming from a mixture of probability density distributions (Stahl & Sallis, 2012). Each mixture component density represents a cluster or group (Ferraro & Giordani, 2019). Hence, the traits of the 68 governance indicators are assumed to have probability distributions that differ between the groups.…”

“…Hence, the traits of the 68 governance indicators are assumed to have probability distributions that differ between the groups. Finite mixture modelling does not perform a hard assignment of countries into groups (as you would get with, for example, k-means cluster analyses), but rather a probabilistic, or soft, assignment (Ferraro & Giordani, 2019). Polity is an autocracy-democracy index and ranges between −10 (strongly autocratic) and +10 (strongly democratic)…”

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“…Hard clustering (classical approach) assigns data point or object to just one cluster. On the other hand, the soft clustering approach assigns data points or object to more than one cluster [55]. The soft flat clustering method is based on the membership value where a document may be assigned to more than one cluster.…”

Cluster‐scaled principal component analysis

“…Then by defining an idea of fuzzy partition, an algorithm using a fuzzy objective function (Ruspini, 1969) has been proposed and developed for several extensions for data types, including more flexible cluster forms (Gath & Geva, 1989; Gustafson & Kessel, 1978) or feature of space of data (Ienco & Bordogna, 2018). In addition, clustering utilizing other representations of uncertainty based on the idea of fuzziness, such as clustering using the concept of possibility (Krishnapuram & Keller, 1993) or clustering using the idea of rough sets (Pawlak, 1982), have been developed and beyond the framework of the fuzzy clustering, categorization of clustering with uncertainty boundaries of clusters has been discussed (Ferraro & Giordani, 2020; Klawonn et al, 2015). Moreover, including adaptable fuzzy relations to clustering, several methods have been developed.…”