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Dealing with distributed data is one of the challenges for clustering, as most clustering techniques require the data to be centralized. One of them, k -means, has been elected as one of the most influential data mining algorithms for being simple, scalable, and easily modifiable to a variety of contexts and application domains. However, exact distributed versions of k -means are still sensitive to the selection of the initial cluster prototypes and require the number of clusters to be specified in advance. Additionally, preserving data privacy among repositories may be a complicating factor. In order to overcome k -means limitations, two different approaches were adopted in this paper: the first obtains a final model identical to the centralized version of the clustering algorithm and the second generates and selects clusters for each distributed data subset and combines them afterwards. It is also described how to apply the algorithms compared while preserving data privacy. The algorithms are compared experimentally from two perspectives: the theoretical one, through asymptotic complexity analyses, and the experimental one, through a comparative evaluation of results obtained from a collection of experiments and statistical tests. The results obtained indicate which algorithm is more suitable for each application scenario. [ABSTRACT FROM AUTHOR]

Abstract: There is a family of well-known external clustering validity indexes to measure the degree of compatibility or similarity between two hard partitions of a given data set, including partitions with different numbers of categories. A unified, fully equivalent set-theoretic formulation for an important class of such indexes was derived and extended to the fuzzy domain in a previous work by the author [Campello, R.J.G.B., 2007. A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognition Lett., 28, 833–841]. However, the proposed fuzzy set-theoretic formulation is not valid as a general approach for comparing two fuzzy partitions of data. Instead, it is an approach for comparing a fuzzy partition against a hard referential partition of the data into mutually disjoint categories. In this paper, generalized external indexes for comparing two data partitions with overlapping categories are introduced. These indexes can be used as general measures for comparing two partitions of the same data set into overlapping categories. An important issue that is seldom touched in the literature is also addressed in the paper, namely, how to compare two partitions of different subsamples of data. A number of pedagogical examples and three simulation experiments are presented and analyzed in details. A review of recent related work compiled from the literature is also provided. [Copyright &y& Elsevier]

A fuzzy extension of the Rand index [Rand, W.M., 1971. Objective criteria for the evaluation of clustering methods. J. Amer. Statist. Assoc. 846–850] is introduced in this paper. The Rand index is a traditional criterion for assessment and comparison of different results provided by classifiers and clustering algorithms. It is able to measure the quality of different hard partitions of a data set from a classification perspective, including partitions with different numbers of classes or clusters. The original Rand index is extended here by making it able to evaluate a fuzzy partition of a data set – provided by a fuzzy clustering algorithm or a classifier with fuzzy-like outputs – against a reference hard partition that encodes the actual (known) data classes. A theoretical formulation based on formal concepts from the fuzzy set theory is derived and used as a basis for the mathematical interpretation of the Fuzzy Rand Index proposed. The fuzzy counterparts of other (five) related indexes, namely, the Adjusted Rand Index of Hubert and Arabie, the Jaccard coefficient, the Minkowski measure, the Fowlkes–Mallows Index, and the Γ statistics, are also derived from this formulation. [Copyright &y& Elsevier]

This paper is concerned with both the problems of quantitative and qualitative modelling of complex systems by using fuzzy techniques. A unified approach for the identification and subsequent extraction of linguistic knowledge of systems using fuzzy relational models is addressed. This approach deals with the identification problem by means of optimal numerical solutions based on weighted least squares and quadratic programming formulations. The linguistic knowledge is extracted in the form of consistent fuzzy rules that describe linguistically the behaviour of the identified system. A new methodology for the simplification of the extracted rules is derived by using a pruning criterion based on the representability matrix concept introduced in previous work. Several numerical aspects concerning the proposed optimization schemes and a covering discussion about the linguistic interpretation of the resulting models are also included together with illustrative examples in the contexts of pattern classification and dynamic systems identification. The paper also provides an overview of fuzzy modelling techniques that intends to situate the relational models among other fuzzy model architectures typically adopted in the literature, highlighting their main advantages and drawbacks. [ABSTRACT FROM AUTHOR]

The recent growing size of datasets requires scalability of data mining algorithms, such as clustering algorithms. The MapReduce programing model provides the scalability needed, alongside with portability as well as automatic data safety and management. k -means is one of the most popular algorithms in data mining and can be easily adapted to the MapReduce model. Nevertheless, k -means has drawbacks, such as the need to provide the number of clusters ( k ) in advance and the sensitivity of the algorithm to the initial cluster prototypes. This paper presents two evolutionary scalable metaheuristics in MapReduce that automatically seek the solution with the optimal number of clusters and best clustering structure for scalable datasets. The first consists in an algorithm able to iteratively enhance k -means clusterings through evolutionary operators designed to handle distributed data. The second consists in applying evolutionary k -means to cluster each distributed portion of a dataset in an independent way, combining the obtained results into an ensemble afterwards. The proposed techniques are compared asymptotically and experimentally with other state-of-the-art clustering algorithms also developed in MapReduce. The results are analyzed by statistical tests and show that the first proposed metaheuristic yielded results with the best quality, while the second achieved the best computing times. [ABSTRACT FROM AUTHOR]