Distributed proximitybased granular clustering: towards a development of global structural relationships in data
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Abstract
The study is focused on a development of a global structure in a family of distributed data realized on a basis of locally discovered structures. The local structures are revealed by running fuzzy clustering (Fuzzy CMeans), whereas building a global view is realized by forming global proximity matrices on a basis of the local proximity matrices implied by the partition matrices formed for the individual data sets. To capture the diversity of local structures, a global perspective at the structure of the data is captured in terms of a granular proximity matrix, which is built by invoking a principle of justifiable granularity with regard to the aggregation of individual proximity matrices. The three main scenarios are investigated: (a) designing a global structure among the data through building a granular proximity matrix, (b) refining a local structure (expressed in the form of a partition matrix) by engaging structural knowledge conveyed at the higher level of the hierarchy and provided in the form of the granular proximity matrix, (c) forming a consensusbuilding scheme and updating all local structures with the aid of the proximity dependences available at the upper layer of the hierarchy. While the first scenario delivers a passive approach to the development of the global structure, the two others are of an active nature by facilitating a structural feedback between the local and global level of the hierarchy of the developed structures. The study is illustrated through a series of experiments carried out for synthetic and publicly available data sets.
Keywords
Fuzzy clustering Proximity matrix Global structure Granular proximity Granular clustering Distributed data Consensus formation1 Introduction
Clustering is about revealing a structure in a single data set. Distributed clustering is concerned with the same problem present in situations where there is a family of data sets for which clustering is carried out separately. The term distributed clustering is quite often encountered in the literature. Furthermore, this type of clustering quite often comes with a remarkable variety of terminology, methods, evaluation measures, extensions (Corsini et al. 2005; Pedrycz and Rai 2008; Pedrycz 2005) and applications (Coppi et al. 2010; Graves et al. 2012; Peters 2011). The results of clustering are conveniently interpreted as information granules in the sequel benefiting from a wealth of conceptual developments of Granular Computing (Apolloni et al. 2006; Pedrycz 2013, 2007). Likewise the distributed nature of the data may imply cases when objects (patterns) are described in different feature spaces or all data sets are in the same feature space (Mali and Mitra 2003; Pedrycz and Rai 2008). There are also combinations of these two alternatives (Pedrycz and Rai 2008).
 (i)
a way of communicating local findings (viz. the structure of the local data). The locally formed structures of the data (expressed in terms of partition matrices and prototypes) need to be compared. If we are concerned with different patterns, however, all of them are expressed in the same feature space, the structural signatures of the data that can be compared across the data come in the form of the prototypes of the clusters. For the same data (patterns), which are formed in different feature spaces, the local partition matrices are viable constructs using which we can build more abstract constructs (proximity matrices), which in the sequel can initiate some sharing findings across the sets of data. We stress that a direct comparison of partition matrices is not feasible as there are different numbers of clusters and there is no correspondence among the clusters constructed for the individual data sets.
 (ii)
a way of building a global view at the data. There are two variants. Depending whether locally available results are refined or left intact we distinguish between active and passive approaches.
The main objective of this study is to develop a general concept of distributed clustering based on the principles of Granular Computing (Pedrycz et al. 1998) and their constructs (Apolloni et al. 2006; Pedrycz et al. 2004). Here our focus is on the data described in different feature spaces, which implies that a communication vehicle is established in terms of proximity matrices. Based on this form of interaction, we discuss three main conceptual settings. The one is of a passive nature, which concentrates on a granular characterization of proximitybased structure with the invocation of granular proximity matrices of a global character. The two other activelike alternatives invoke some structural feedback to refine local structures on a basis of the global result (viz. a granular proximity matrix).
Our investigations come with several wellarticulated aspects of originality. The formulation of the problem is original: although some facets of collaborative clustering have been investigated in the literature, those approaches focus on the passive mode meaning that the results of clustering are aggregated, however, an active facet is not considered at all meaning that no mechanisms adjusting local clustering findings were developed given some global findings. Let us recall that a passive mode implies that the locally available clustering results are provided and some aggregation mechanism is invoked, which gives rise to a general (global) view of the results. In this process, irrespectively of the result obtained at the global level, the local structures (clusters) are not modified (affected). In contrast, when talking about an active mode, a feedback loop is being formed so that in an iterative fashion the local results give rise to some global results. In the sequel, those are contrasted with the results available at the lower level and as a result the local results are modified following a certain adjustment strategy so that in the next iteration there is some improvement observed at the level of the global results.
It is worth noting that there have been some interesting earlier studies on intervalvalued clustering, cf. (Souza and Carvalho 2004; Gacek and Pedrycz 2013; Hathaway et al. 1996; Pedrycz et al. 1998; Hwang and Rhee 2007; Mali and Mitra 2003; Wong and Hu 2013; Zhang et al. 2014). There is, however, an important difference between the undertaken research and the previous line of investigation. Here we are concerned with numeric data whereas information granules come as a result of reconciliation of results and are reflective of the diversity of the local findings. The previous studies were focused on granular clustering, more specifically, intervalvalued data
The paper is structured as follows. In Sect. 2, we highlight an essence of the problem and identify a role of information granularity being played in this setting. In the sequel, we briefly outline the essence of the main classes of problems (Sect. 3). In Sect. 4, all associated optimization problems are formulated and solved. More specifically, we discuss a way of forming granular proximity matrices through the use of the principle of justifiable granularity (Pedrycz 2013), and look at the techniques of refining local partition matrices based on the gradientbased optimization and particle swarm optimization (PSO) as well as a hybrid of these two techniques. Two ways of characterization of granular proximity matrices are discussed. Numeric studies are covered in Sect. 5.
In the study, we adhere to the standard notation encountered in pattern recognition, clustering and system modeling. To emphasize the origin of the locally available data and the resulting constructs, we use indexes placed in square brackets, say \(c\)[ii], \(U\)[ii], \(u_{ik}\)[ii], etc.
2 The essence of the problem and underlying facet of information granularity
Let us consider a collection of \(p\) data sets \(\mathbf{D}_{1}, \mathbf{D}_{2}, {\ldots }, \mathbf{D}_{p}\) originating from a certain problem. For instance, those sets could be data describing a certain system for which formed are individual, local views associated with their local data. The data originated from different collections may be described in different feature spaces \(\mathbf{F}_{1}, \mathbf{F}_{2}, {\ldots ,} \mathbf{F}_{p}\). In general, we also assume that some data points are shared among the data sets meaning that an intersection of them is nonempty, namely \(\mathbf{D} = \mathbf{D}_{1} \cap \mathbf{D}_{2} \cap {\cdots }\cap \mathbf{D}_{p}\) card \((\mathbf{D}) =N\). Formally speaking, a selected object o\(_{k}\) belonging to the intersection of \(\mathbf{D}_{1}, \mathbf{D}_{2},{\ldots }\), and D \(_{p}\) comes with its own vectors of features x \(_{k}\)[1], x \(_{k}\)[2], ...,x \(_{k}\)[\(p\)] defined in the corresponding feature spaces. The data \({\{}\mathbf{x}_{k}[ii]{\}}\), \(k=\) 1, 2, ...\(N\) forming the data set D \(_{ii}\) are clustered in the corresponding feature space resulting in the corresponding partition matrix \(U\)[ii]. Clustering is completed for other data sets subsequently giving rise to partition matrices \(U\)[1], \(U\)[2],..., and \(U\)[\(p\)], respectively. In general, the number of clusters associated with these partition matrices, namely \(c\)[1], \(c\)[2],.., and \(c\)[\(p\)], could vary from one data set to another. The partition matrices formed in this way exhibit a local character, viz. they are concerned with the findings being confined to the given feature space and produced for the particular locally available data set. Our key objective of this study is to discover (or reconcile) a global structure in the data based on the reconciliation of the local views conveyed through the already constructed partition matrices.
 (a)
It is apparent that any aggregation of the partition matrices is not feasible because of the fact that the number of clusters could vary from one partition matrix to another. To proceed with any comparison of partition matrices, this process cannot be realized directly but through comparing proximity matrices induced by the corresponding partition matrices.
 (b)
as the resulting proximity matrices exhibit an evident diversity, we may contemplate to use an aggregation mechanism that fully reflects and quantifies this diversity. This, in turn, brings a concept of granular proximity matrices as the constructs capturing this facet of the existing variety among the local proximity matrices.
The granular proximity matrix visualizes an emergence of groups of data that are kept close to each other. One can observe a jump in the values of the entries pointing at the occurrence of the wellformed clusters.
3 Main classes of problems
We can distinguish three general categories of problems where the fundamental ideas outlined so far can be fully exploited. The essence of these problems is visualized through a series of figures that help contrast different tasks being studied here.
3.1 Formation of a general description of data
The granular character of the proximity matrix formed at the upper level of hierarchy quantifies the diversity of the local structures. The entries of the granular proximity matrix \(G(P) =[p_{kl}^{}, p_{kl}^{+}], k, l =1, 2, {\ldots }, N\), which are intervals with the lower and upper bounds \(p_{kl}^{}\) and \(p_{kl}^{+}\), quantify a strength of linkage occurring between a pair of data, say \(k\) and \(l\).
3.2 Refinement of the locally discovered structure of data
The optimization mechanism is the one discussed in depth in Sect. 4.2.
3.3 Building consensus
The result formed at the global level is used to adjust the individual partition matrices \(U_{1}, U_{2}, {\ldots }, U_{p}\). The optimization is realized following the scheme outlined in the previous section. The updated partition matrices are used to build proximity matrices and those give rise to the granular proximity matrix. In turn, the new proximity matrix offers a navigation of optimization of the local partition matrices and the iterations are continued. The convergence of the process becomes critical and for this a suitable index needs to be established.
4 Associated optimization problems, their solutions and characterizations
The three categories of problems outlined in the previous section call for a certain way of formulating the ensuing optimization problem, building its solution and finally characterizing the quality of the obtained solutions. We come up with some design procedures and show their arrangement when solving the three categories of problems formulated above.
4.1 Development of granular proximity matrix
A formulation of a granular proximity matrix is a common task encountered in the three classes of problems discussed above. Given a collection of proximity matrices \(P[1], P[2], {\ldots }, P[p]\) we realize a granular proximity matrix \(G(P)\) so that the inherent granularity of this construct is captured and quantified. Proximity matrices exhibit an inherent diversity. Being cognizant of this fact, we can assume that the aggregation result is a granular proximity matrix \(G(P)\) whose entries are intervals located in the [0,1] interval. In other words, the granular proximity matrix comes with intervalvalues entries, \(G(P)= [g_{kl}^{}, g_{kl}^{+}], k, l=1, 2,{\ldots }, N\) The intervalvalued character of the construct is reflective of the variability in the local findings.
The construction of the granular proximity matrix \(G(P)\) realized for individual entries of the matrix is realized by invoking a principle of justifiable granularity (Pedrycz 2013). In a nutshell, this principle states that when aggregating some numeric experimental evidence, in the face of the diversity of the existing pieces of evidence, the result is a certain information granule (instead of another numeric outcome) such that it is supported enough by the experimental data while simultaneously demonstrating sufficient specificity thus coming with a wellconveyed semantics. The principle is applied to the individual entries of the proximity matrices. Let us consider the (\(k,l)\)th entry of the proximity matrices \(P[1], P[2],{\ldots }, P[p]\), namely consider a set \(\mathbf{P} = {\{}p_{kl}[1], p_{kl}[2],{\ldots }, p_{kl}[p]{\}}\). We also assume that some initial numeric representative (say mean or median) of P is provided. For the (\(k,l)\)th entry we denote it by \(m_{kl}\).
This procedure is directly applicable to the construction of granular partition matrix \(G(P)\).
 (i)
formation of clusters and partition matrices for \(\mathbf{D}_{1}, \mathbf{D}_{2},\) \( {\ldots }\mathbf{D}_{p}\), viz. \(U\)[1], \(U\)[2] ,..., \(U\)[\(p\)].
 (ii)
building proximity matrices \(P(U[1]), P(U[2]), {\ldots }\) \( P(U[p])\). Note that they are produced for all pairs of the data belonging to D.
 (iii)
use of the principle of justifiable granularity to form the granular construct \(G(P)\). The intervalvalued proximity matrix is built for a certain predetermined value of \(\alpha \).
4.2 Refinement of local partition matrix
The crux of this scenario has been captured in Fig. 4. From the optimization perspective, we first form a granular proximity matrix \(G(P)\) and afterwards use it in the refinement of some locally constructed partition matrix.
For some given partition matrix \(U\)[ii], we proceed with its modifications (adjustments) in such a way that \(P(U\)[ii]) is “contained” in \(G(P)\) to the highest extent. The adjustments are made possible by engaging an idea of optimal allocation of information granularity. The underlying idea is to adjust the entries of \(U[ii], ii=1, 2, {\ldots },p\) in such away that the modified partition matrix produces a proximity matrix whose values are included in the intervalvalued entries of \(G(P)\).
The gradientbased mechanism can be considered as a standalone optimization scheme or could be considered in conjunction with more advanced populationbased optimization such as Particle Swarm Optimization (PSO) and establishes a hybrid optimization scheme in which both of these optimization mechanisms are arranged in a certain sequence with an ultimate intent of avoiding local minima.
With regard to the optimization, several hybridizations of the generic optimization mechanisms are worth investigating, namely, tandems of PSOgradient method and gradient optimization PSO where we capitalize on the key properties of these techniques. PSO as the populationbased technique is beneficial in realizing a globaloriented search, whereas the gradientoriented method comes with a very detailed search capabilities, however, it is also prone to being stuck in possible local minima. A hybrid scheme of the form of PSO followed by the gradientbased technique comes as a sound alternative emphasizing the advantages of the contributing methods.
4.3 A general scheme of consensus building
In contrast to the two previously outlined processes in which granular proximity matrices are involved, consensus building is an iterative process and its dynamics comes into play. We proceed in an iterative fashion by forming a granular proximity matrix \(G(P)\) on the basis of locally formed partition matrices (proximity matrices) and then update each \(U[1], U[2],{\ldots }, U[p]\) as discussed in the second scheme. Then these updated partition matrices lead to the proximity matrices and subsequently the new granular proximity matrix is produced. This complete iterative loop is repeated. The process is monitored with respect to its convergence. Some parameters of the method, especially the values of \(\alpha \) can impact the convergence process and their impact can be assessed in an experimental fashion.
4.4 Characterization of granular proximity matrices
There are several indicators that can be used as sound descriptors of the produced granular proximity matrix supporting also the quality of the convergence process encountered in consensus building.
4.4.1 Linkage analysis
4.4.2 Overall granularity of granular proximity matrix
5 Numeric studies
In this section, a series of experiments involving both synthetic and realworld data are presented to illustrate how different schemes discussed above operate and a form of the results formed. In all experiments, we consider Fuzzy CMeans (FCM) algorithm (Bezdek 1981) run with the fuzzification coefficient set to 2, \(m=\) 2.
5.1 Synthetic data
Statistical characteristics of synthetic two and threedimensional data
Set  \(c\)[\(p\)]  

D \(_{1}\)  2  m \(=\) [\(\)3 9] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.2} &{} 0 \\ 0 &{} {0.6} \\ \end{array} }} \right] \)  m \(=\) [\(\)2 4] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.4} &{} 0 \\ 0 &{} {0.8} \\ \end{array} }} \right] \)  
D \(_{2}\)  2  m \(=\) [2 \(\)9 1] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.1} &{} 0 &{} 0 \\ 0 &{} {1.7} &{} 0 \\ 0 &{} 0 &{} {0.4} \\ \end{array} }} \right] \)  m \(=\) [8 6 4] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.2} &{} 0 &{} 0 \\ 0 &{} {1.7} &{} 0 \\ 0 &{} 0 &{} {0.7} \\ \end{array} }} \right] \)  
D \(_{3}\)  3  m \(=\) [9 1] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.3} &{} 0 \\ 0 &{} {1.9} \\ \end{array} }} \right] \)  m \(=\) [\(\)10 \(\)6] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.0} &{} 0 \\ 0 &{} {0.1} \\ \end{array} }} \right] \)  m \(=\) [\(\)1 \(\)6] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.8} &{} 0 \\ 0 &{} 1 \\ \end{array} }} \right] \)  
D \(_{4}\)  3  m \(=\) [2 4] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.2} &{} 0 \\ 0 &{} {0.1} \\ \end{array} }} \right] \)  m \(=\) [\(\)2 \(\)3] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.8} &{} 0 \\ 0 &{} {0.9} \\ \end{array} }} \right] \)  m \(=\) [10 2] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.6} &{} 0 \\ 0 &{} {0.5} \\ \end{array} }} \right] \)  
D \(_{5}\)  4  m \(=\) [8 9] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.0} &{} 0 \\ 0 &{} {0.5} \\ \end{array} }} \right] \)  m \(=\) [6 \(\)3] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.4} &{} 0 \\ 0 &{} {0.6} \\ \end{array} }} \right] \)  m \(=\) [2 2] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {0.9} &{} 0 \\ 0 &{} {0.1} \\ \end{array} }} \right] \)  m \(=\) [\(\)3 \(\)1] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.9} &{} 0 \\ 0 &{} {1.9} \\ \end{array} }} \right] \) 
D \(_{6}\)  2  m \(=\) [3 0 \(\)10] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {1.9} &{} 0 &{} 0 \\ 0 &{} {0.4} &{} 0 \\ 0 &{} 0 &{} {0.1} \\ \end{array} }} \right] \)  m \(=\) [\(\)10 1 5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.6} &{} 0 &{} 0 \\ 0 &{} {1.8} &{} 0 \\ 0 &{} 0 &{} {0.2} \\ \end{array} }} \right] \)  
D \(_{7}\)  4  m \(=\) [10 \(\)4] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.0} &{} 0 \\ 0 &{} {1.2} \\ \end{array} }} \right] \)  m \(=\) [\(\)9 \(\)9] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.5} &{} 0 \\ 0 &{} {1.8} \\ \end{array} }} \right] \)  m \(=\) [\(\)9 7] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.4} &{} 0 \\ 0 &{} {1.6} \\ \end{array} }} \right] \)  m \(=\) [\(\)2 \(\)4] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {0.1} &{} 0 \\ 0 &{} {0.4} \\ \end{array} }} \right] \) 
D \(_{8}\)  2  m \(=\) [6 10 \(\)5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {0.5} &{} 0 &{} 0 \\ 0 &{} {0.3} &{} 0 \\ 0 &{} 0 &{} {0.9} \\ \end{array} }} \right] \)  m \(=\) [1 8 \(\)5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.9} &{} 0 &{} 0 \\ 0 &{} {0.3} &{} 0 \\ 0 &{} 0 &{} {0.6} \\ \end{array} }} \right] \)  
D \(_{9}\)  2  m \(=\) [\(\)8 1 0] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.1} &{} 0 &{} 0 \\ 0 &{} {0.1} &{} 0 \\ 0 &{} 0 &{} {0.8} \\ \end{array} }} \right] \)  m \(=\) [8 \(\)7 0] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {1.7} &{} 0 &{} 0 \\ 0 &{} {1.7} &{} 0 \\ 0 &{} 0 &{} {0.1} \\ \end{array} }} \right] \)  
D \(_{10}\)  2  m \(=\) [1 \(\)4] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.2} &{} 0 \\ 0 &{} {0.1} \\ \end{array} }} \right] \)  m \(=\) [\(\)7 8] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.7} &{} 0 \\ 0 &{} {0.6} \\ \end{array} }} \right] \)  
D \(_{11}\)  2  m \(=\) [\(\)8 7] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.8} &{} 0 \\ 0 &{} {1.7} \\ \end{array} }} \right] \)  m \(=\) [\(\)6 \(\)1] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.1} &{} 0 \\ 0 &{} {0.4} \\ \end{array} }} \right] \)  
D \(_{12}\)  4  m \(=\) [\(\)5 5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {2.9} &{} 0 \\ 0 &{} {0.6} \\ \end{array} }} \right] \)  m \(=\) [\(\)5 6] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {0.6} &{} 0 \\ 0 &{} {0.4} \\ \end{array} }} \right] \)  m \(=\) [8 \(\)5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {0.5} &{} 0 \\ 0 &{} {1.7} \\ \end{array} }} \right] \)  m \(=\) [\(\)9 10] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l} {1.2} &{} 0 \\ 0 &{} {1.6} \\ \end{array} }} \right] \) 
D \(_{13}\)  2  m \(=\) [6 1 8] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {0.9} &{} 0 &{} 0 \\ 0 &{} {1.8} &{} 0 \\ 0 &{} 0 &{} {0.5} \\ \end{array} }} \right] \)  m \(=\) [\(\)3 8 8] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.9} &{} 0 &{} 0 \\ 0 &{} {1.3} &{} 0 \\ 0 &{} 0 &{} {0.8} \\ \end{array} }} \right] \)  
D \(_{14}\)  4  m \(=\) [6 10 \(\)5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {0.5} &{} 0 &{} 0 \\ 0 &{} {0.3} &{} 0 \\ 0 &{} 0 &{} {0.9} \\ \end{array} }} \right] \)  m \(=\) [1 8 \(\)5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.9} &{} 0 &{} 0 \\ 0 &{} {0.3} &{} 0 \\ 0 &{} 0 &{} {0.6} \\ \end{array} }} \right] \)  m \(=\) [\(\)9 \(\)6 5] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.0} &{} 0 &{} 0 \\ 0 &{} {0.1} &{} 0 \\ 0 &{} 0 &{} {0.2} \\ \end{array} }} \right] \)  m \(=\) [\(\)9 1 8] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {1.7} &{} 0 &{} 0 \\ 0 &{} {1.9} &{} 0 \\ 0 &{} 0 &{} {0.9} \\ \end{array} }} \right] \) 
D \(_{15}\)  2  m \(=\) [2 6 \(\)6] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {1.5} &{} 0 &{} 0 \\ 0 &{} {0.2} &{} 0 \\ 0 &{} 0 &{} {0.7} \\ \end{array} }} \right] \)  m \(=\) [\(\)3 \(\)7 9] \(\Sigma =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {2.9} &{} 0 &{} 0 \\ 0 &{} {2.0} &{} 0 \\ 0 &{} 0 &{} {0.3} \\ \end{array}}} \right] \) 
In the sequel, we elaborate on the three models of building global structures of data or reconciling structural characteristics of the local nature.
Formation of granular proximity matrix
Figure 6 Granular proximity matrix for D \(_{1}\). A sudden jump in the levels of brightness (resemblance values) occurring along the 30th data point is a result of the occurence of two wellformed clusters, see Fig. 8. The first one involves the first 30 data points while the rest of the data (from 31 to 50) form the second cluster.
The granular proximity matrix visualizes an emergence of groups of data that are kept close to each other. One can observe a jump in the values of the entries pointing at the occurrence of the wellformed clusters.
These results are reported for several values of \(\alpha \) (namely, 0.3, 0.6, and 1). There is a visible tendency of a stronger and visible revealing of the outliers when the levels of binding locally available structures are made stronger. For instance, as shown in Fig. 7c, it is apparent that there are some collections of data (those indexed as 1–5 and 35–50) which are different with regard to the locally present structures.
Involvement of global granular results in the enhancements of the local partition matrix. Here, as discussed earlier, the optimization strategy involves plain gradientbased method and PSO as well as several of their hybrid approaches combining these generic methods.
The results are reported in terms of the performance index \(V\) whereas the initial learning rate was set to \(\xi \) = 0.01.
The hybrid optimization method PSOgradient, Fig. 8d, outperforms other optimization schemes. The partition matrix \(U\) generated with the use of the PSO method, Fig. 8b, serves as a sound initial condition for the gradientbased method, which is helpful in carrying out finetuning of the entries of \(U\). For the other hybrid method (gradientPSO) Fig. 8c, it is clearly shown that PSO does not produce further improvement for \(V\) as it is eventually stuck in some local maxima.
It is noted that the value of the learning rate \(\xi \) was made quite low with intent of making the process stable when it comes to the finetuning phase of the entries of \(G(P)\). Furthermore, we have adopted some dynamic changes of the values of the learning rate. If the value of \(V\) decreases in a certain iteration, this iteration is ignored and the value of \(\xi \) is decreased, say being made 0.75*\(\xi \). Next we continue the learning with this new value of \(\xi \) until \(V\) decreases again, etc.
While the local structure is quite apparent, the refinements guided by the globally produced structure lead to some changes of the structure. This is not surprising as some global structure is considered and its impacts become clear.
Figure 9 reveals some interesting relationships. When only local data are considered, there is a welldelineated structure, which points at two clusters. For the increasing values of \(\alpha , \alpha \) = 0.3 and 0.6, we witness an increasingly influential impact of the global structure so the clusters are not as distinct as in the first case. Obviously, this is not surprising, as now we have started accommodating a global view (structure), which might not be in full agreement with the local topology of the data. Furthermore, the partition matrices displayed in this figure identify data points, which are mostly impacted by the global structure. This is a useful insight into the nature of the individual data, which helps pinpoint the elements, which are the least compatible with the global structure revealed at the higher level of the hierarchy.
Experiments with realworld data
 1.
I0 Impedivity (ohm) at zero frequency
 2.
PA500 phase angle at 500 KHz
 3.
HFS highfrequency slope of phase angle
 4.
DA impedance distance between spectral ends
 5.
AREA area under spectrum
 6.
A/DA area normalized by DA
 7.
MAX IP maximum of the spectrum
 8.
DR distance between I0 and real part of the maximum frequency point
 9.
P length of the spectral curve

D \(_{1}\): 1, 7

D \(_{2}\): 4, 8

D \(_{3}\): 2, 3, 9

D \(_{4}\): 5, 6
Number of clusters based on the inspection of the objective function
D \(_{1}\)  D \(_{2}\)  D \(_{3}\)  D \(_{4}\)  

\(c\)  5  7  5  4 
6 Conclusions
In this study, we have conceptualized, developed the algorithmic setting, and experimented with granular proximity matrices. It has been demonstrated that granularity of these matrices plays an important role in the realization of collaborative processes of forming views at the global structures not only facilitating this process, but also quantifying the diversity of locally available structures through the associated level of information granules of the granular proximity matrix. The guidance offered by global granular proximity matrices is an example of a realization of a structural feedback loop which augments the clustering processes by auxiliary sources of knowledge.
There are two open directions, which are worth further investigations:
Formation of structures exhibiting a higher type of granularity Higher level structures such as granular\(^{2}\) proximity matrices (if a hierarchy having three levels is present) can be discussed.
Exploration of various formal ways of realizations of granular proximity matrices. While in this study, we are concerned with intervalvalued proximity matrices (and this has been done for illustrative purposes), detailed considerations could involve other formalisms such as, e.g., fuzzy sets, rough sets and shadowed sets.
Notes
Acknowledgments
This study was funded by King Abdulaziz University (KAU), under Grant No. (41351434/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
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