Data mining [1–24] involves many tasks. One of the most important tasks in data mining is clustering [25–29]. According to different similarity criteria implemented by various clustering algorithms in the context of unsupervised learning, different objective functions are targeted [30]. According to “no free lunch” theory, there is no dominant clustering method [22,23]. Therefore, the idea of combining clustering, which is also called cluster (or clustering) ensemble, emerged. In a clustering ensemble, some of the basic partitions are combined to acquire a better solution capable of managing all objectives of different partitions which sometimes are contradictory [28–30]. Cluster ensemble offers many advantages, among which the following can be mentioned: robustness to noise [29], capability of producing novel results [31–33], quality enhancement [29], knowledge reusability [30], multi-view clustering [33], stability, parallel/distributed data processing [30], finding number of real clusters, adaptability and heterogeneous data clustering.

The process of clustering ensemble consists of two related phases as shown in Fig. 1a. The first phase includes production of the diverse base clusterings through the basic clustering algorithm (s), and the second phase involves extracting the final clustering from the primary basic partitions using a consensus function. To improve the performance of the clustering ensemble in the first phase, a special attention should be paid to the ensemble diversity which determines the quality of the combination. If the quality of each voter is greater than a random voter, the quality of the combination is improved by increasing the ensemble diversity, where the combination can be any community like the ensemble clustering [33]. There are many approaches, which are used in production of primary basic partitions in order to acquire the desired diversity, such as: Different initializations of parameters [30], heterogeneous ensemble clustering methods [28], different subsets of the features [30], different subsets of objects [28], projection of the object subsets [28] and hybrid methods. In our proposed method, a basic clustering algorithm is implemented with different parameters to achieve the desired diversity. In the second phase, a consensus function is utilized to achieve the final clustering. Selection of the clusters with high quality is an NP hard problem because clustering is an unsupervised problem [33]. Several consensus functions are proposed to face this issue, each of which uses a specific approach and different information from the primary basic partitions achieved from phase one, and sometimes considers the initial characteristics of the data. Consensus ensemble methods are divided into the following classes: 1. intermediate space clustering ensemble methods [28], 2. methods based on the co-association matrix [33], 3. hyper-graph based methods [30], 4. expectation maximization clustering ensemble methods [32], 5. mathematical modeling (median partition) approaches [34], and 6. voting-based methods [35].

There are two types of clustering algorithms: (a) First, hard clustering algorithms in which a data object either is definitely assigned to a cluster or is not assigned to it at all and (b) Second, fuzzy clustering algorithms in which there are some data points that are not allocated to a specific cluster but they are allocated to all clusters with different membership degrees (for a data point, the summation of all membership degrees to all clusters should be one). The basis of fuzzy clustering algorithms is the basic fuzzy c-means (FCM) clustering algorithm [36].

Although soft (or fuzzy) clustering has more generality than crisp clustering, researches in soft clustering are in early stages and fuzzy clustering ensemble approaches have not been widely developed. Fuzzy clustering ensembles are proposed by a few researchers, (e.g., Punera et al. [37] have proposed soft versions of Cluster-based Similarity Partitioning Algorithm (CSPA), Meta CLustering Algorithm (MCLA) and Hybrid Bipartite Graph Formulation (HBGF) which are respectively named soft CSPA (sCSPA), soft MCLA (sMCLA) and soft HBGF (sHBGF)) but crisp clustering has more maturity. Some of the existing fuzzy clustering ensemble methods convert fuzzy clusters to hard clusters through some existing simple methods and then the hard consensus functions are used to compute the final partition. This conversion causes the loss of uncertainty information. This loss of uncertainty information causes some challenges in suggesting an efficient fuzzy consensus clustering from multiple fuzzy basic partitions.

Consensus process is highly dependent on the quality of the primary partitions so that the low-quality or even noisy primary partitions could adversely affect it. Improving performance of the consensus functions through quality-evaluation and weighting of the primary partitions is attended to confront with low-quality primary partitions based on the implicit assumption that all clusters have the same reliability value in the same base clustering [38]. This is usually performed by assigning a weight to each primary partition, which is considered to be an individual as a whole, without caring about the quality of its clusters. We have to confront different reliability values of clusters in the same clustering because real-world datasets are inherently complex and noisy. In some methods, it is needed to have access to the data features such as what was proposed by Zhong et al. [33]. Their method has investigated the reliability values of the clusters using the Euclidean distances between objects of the clusters and its efficiency is highly dependent on the distribution of the data in the dataset. However, in a clustering ensemble in its general formulation, we assume that we have not access to the original data features.

In this work, according to estimation of ensemble driven cluster unreliability and local weighting strategy, a new approach of fuzzy clustering ensemble is proposed as shown in Fig. 1b. In order to increase the consensus performance, a locally weighted plan is achieved by integrating the validity and unreliability of the cluster. An entropic criterion is used to estimate unreliability of any fuzzy cluster based on relation of the cluster acceptability to the entire ensemble. The new metric is defined to estimate the fuzzy cluster unreliability and then the reliability values of the clusters are determined using an index named RDCI. Each cluster is assessed and weighted by its RDCI which evaluates the cluster using an effective indication provided by the crowd of diverse clusters in the ensemble. Then, a fuzzy weighted co-association matrix with the calculated weights based on the reliability, incorporates local acceptability into the conventional co-association (Co) matrix and is treated as a summary for the ensemble of diverse clusters. After that, new consensus algorithms are proposed to achieve the final clustering. The new consensus algorithms include: (a) RBFCM and (b) RBGP which consider cluster reliabilities, and (c) RBHC which considers cluster pairwise acceptability.

The main contributions of the paper are summarized as follows. (I) We estimate an unreliability value per a fuzzy cluster in relation to other clusterings through the proposed method utilizing an entropic criterion. The entropic criterion considers only distribution of membership degrees of all objects to each primary fuzzy cluster while the access to the original data features is not needed; it has also no assumptions on data distribution. (II) The primary fuzzy clusters in the ensemble are assessed and weighted by proposing an RDCI which results in providing a reliability indication at the cluster-level with a contribution to the local weighting plan. (III) A new approach for fuzzy co-association matrix computation in the fuzzy cluster ensemble is proposed. (IV) New consensus functions are proposed to construct the final clustering according to estimation of the reliability–driven fuzzy cluster unreliability and local weighting strategy. (V) Finally, the derived experimental results demonstrate the performance and robustness superiority of the proposed fuzzy clustering ensemble approach to the state-of-the-art approaches.

The rest of the paper is organized as follows. In Section 2, the related literature is reviewed. Section 3 provides the background knowledge about entropy and clustering ensemble. The proposed approach of fuzzy clustering ensemble is described in Section 4 based on the cluster unreliability estimation and local weighting strategy. The experimental results are provided in Section 5, and finally the paper is concluded in Section 6.

Some of the most important ensemble clusterings include k means-based consensus clustering [28], spectral ensemble clustering [39] and infinite ensemble clustering [40]. The following works are also considered to be important researches in fuzzy clustering ensemble: sCSPA and sMCLA introduced in [37] as the fuzzy extension versions of CSPA and MCLA are proposed in [30]. sHBGF is also proposed as a fuzzy version of HBGF [31].

To extract the final fuzzy clustering out of a fuzzy clustering ensemble, an explicit objective function is proposed in [41] based on the new contingency matrix. In order to accurately and efficiently extract the final clustering in this approach which is a parallelizable algorithm and capable of being used for big data clustering, a flexible utility function is employed to change fuzzy consensus clustering into a weighted piecewise FCM-like iterative.

Vote-based merging algorithm (VMA) as a fuzzy clustering ensemble method which was proposed in [42], calculated the final clustering based on the membership matrix averaging in each clustering. Since all the base clusters in this method have to be relabeled, VMAs would be among time-consuming algorithms. An information theoretic kmeans (ITK) was introduced by Dhillon et al. [43].

In [34], the clusters labels are shown as a 0–1 bit string where the goal is to obtain the final fuzzy clustering which is a membership matrix any of whose members shows the membership degree of a data object to a cluster. Accordingly, they introduce the Fuzzy String Objective Function (FSOF) which tries to minimize the summation of distances between centers of the initial clusters and the final clusters. Since the final clustering was fuzzy, some constraints were added to the objective function which created a non-linear optimization NP problem. They employed a genetic algorithm for its solution. Note that it is applicable while the base clusterings are crisp. Two consensus functions, i.e., Fuzzy String Cluster Ensemble Optimized by Genetic Algorithm numbers 1 and 2 (FSCEOGA1 and FSCEOGA2), were also proposed based on cross-over and mutation operators.

In order to enhance the stability of the fuzzy cluster analysis, a heterogeneous clustering ensemble has been proposed [44] in which basic fuzzy clustering algorithms are first utilized and then, final clustering is obtained using a FCM algorithm so that all clusters in the co-association matrix have equal participation weights. To achieve the consensus clustering from fuzzy clustering ensembles, a voting mechanism is used in [35]. The work includes disambiguation and voting procedures. Disambiguation is a phase in which Hungarian algorithm [45] with time complexity of O(n¯3) (n¯ is (average) number of clusters in each clustering) is used for re-labeling problem. The voting phase is implemented to achieve the final consensus clustering. There are many voting phases such as confidence-based voting methods (including sum voting rule and product voting rule) and positional-based voting methods (including Borda voting rule and Copeland voting rule). Time complexity of these voting rules is O(n¯Mβ) where n¯, M and β are (average) number of the clusters in each clustering, number of data objects and number of base clusterings respectively. There exist many different consensus functions depending on direct or repetitive combination of re-labeling and voting phases.

According to the particle swarm optimization (PSO) which has the capability of finding fuzzy and crisp clusters [46,47], a method for construction of a fuzzy clustering ensemble is proposed in [46]. It creates initial clusters through parameter change; and then, using pruning process, n′ clusters are chosen from the β initial clusterings so that n′≪c where c is the number of the clusters in all of the β initial clusterings. For pruning process, one of the internal cluster validity indices such as Ball–Hall, Caliński et al. [48], Dunn index [49], Silhouette index [50] or Xie-Beni [51] is selected to evaluate the fitness of primary basic clusters; and then, one of the genetic selection mechanisms such as tournament or roulette wheel is used to choose the elite clusters. Final clustering is subsequently achieved by use of the consensus function through PSO algorithm. Unlike other PSO-based methods in which each particle represents a clustering, each particle represents a cluster in this method.

Through the following definitions, we introduce the general formulation of data, fuzzy clustering ensemble and entropy in this paper. Note that the notations implemented in this work are presented here. An object (or data point) is a tuple denoted by xi=(xi1,xi2,…xiN). It is also a vector where xij represents the j-th attribute from i-th data. Also, x: j is defined as j-th attribute of dataset, N=|x1: | denotes size of the dataset dimensions (i.e., quantity of the attributes), and M=|x: i| shows the dataset size (i.e., quantity of the data points). π(x) is a M×n matrix which represents the fuzzy partition defined on dataset x, where M=|x: 1| and n is an integer indicating the number of clusters, and we have Eq. (1).

(1) ∀j∈{1,…,n},i∈{1,…,M}:π(xi)j∈[0,1]∀i∈{1,…,M}:∑ j=1nπ(xi)j=1

where π(xi)j shows a real number indicating how much i-th data object belongs to the j-th cluster in partition π which is π={C1,C2,…,Cn}. Π denotes a clustering ensemble which includes β primary partitions, i.e., Π={π1,…,πβ} in which πm={C1m,…,Cnmm} where, πm represents the m-th primary partition in Π, Cim and n^m are the i-th cluster and the number of the clusters in πm respectively. The set of all clusters in the ensemble is denoted by C and defined as C={C11,…,Cn11,…,C1β…,Cnββ} where Cij is the i-th cluster of the partition πj. Therefore, the number of all clusters in the base clusterings is denoted by c and is defined as c=n1+…+nβ. Let’s assume that z is a discrete random variable. The entropy is a measure of the unreliability related to a random variable. For a discrete random variable z whose domain, i.e., the set of the values for z, is denoted by Z, it is defined according to Eq. (2).

(2) H(z)=-∑ z′ϵZρ(z=z′)log2ρ(z=z′)

where ρ(z=z′) is the probability mass function of z and 0log20 is always assumed to be 0 here. H(z,w) is the joint entropy as a measure of the unreliability associated with a set of two discrete random variables z and w. It is defined according to Eq. (3).

(3) H(z,w)=-∑ z′∈Z∑ w′∈Wρ(z=z′,w=w′)log2ρ(z=z′,w=w′)

where ρ(z,w) is the joint probability of two discrete random variables z and w. When z and w are independent random variables, the joint entropy is H(z,w)=H(z)+H(w). Thus, for q independent random variables z1,z2,…,zq, the joint entropy is defined according to H(z1;…;zq)=∑ q′=1qH(zq′).

Block diagram of the proposed method is presented in Fig. 1b in which a new approach of fuzzy clustering ensemble is proposed based on estimation of the ensemble driven cluster unreliability and local weighting strategy. The proposed algorithm includes the following stages. First, we compute the acceptability of each cluster. Then, the unreliability of each cluster is estimated and after that weight of each cluster is determined. Next, the weighted co-association matrix is produced and finally, the consensus clustering is achieved.

Based on fuzzy cluster unreliability estimation, a novel fuzzy clustering ensemble approach was proposed in this paper. Cluster unreliability was estimated according to an entropic criterion using the cluster labels in the entire ensemble. After that, based on the cluster unreliability and local weighting strategy, a new RDCI measure was proposed that is independent on features of the original data and is without any pre-assumption on data distribution. In order to promote the conventional co-association matrix, a local weighting scheme was proposed through the RDCI weight (WFCA) which determines how much amount each primary fuzzy cluster should contribute in construction of the co-association matrix; indeed, it should be with respect to its reliability in the ensemble. Three algorithms of RBFCM, RBGP and RBHC (suitable for large datasets) were proposed that respectively achieve the final clustering from WFCA matrix, the final clustering with respect to the reliability of each cluster and the final clustering by computing the acceptability of each cluster over other clusters of the ensemble. In comparison to other fuzzy clustering ensemble methods, the proposed approach has shown performance improvement and more robustness against noisy datasets through experimental results over fifteen datasets. We concluded that RBHC and RBGP algorithms are appropriate for large datasets because they are linear to the number of data points and have good performance comparison to other fuzzy clustering ensemble methods. It was shown in the paper that RBHC is the best method in terms of time complexity and performance, while RBGP is the best in terms of performance and robustness. RBFCM was also relatively the best in terms of performance. For future works, challenging high dimension datasets in RBFCM and missing values in a dataset can be considered to be problems to be solved. As other challenges for future studies, we can address the diversity impact of the proposed method and its performance in different sampling mechanisms.