# Analysis of Data Presented by Multisets Using a Linguistic Approach +
## 3 Analysis of Sets of Objects Represented by Multisets
### Clustering of Datasets  
When solving the problem of clustering a set of objects represented by multisets, multisets of the form (1) are grouped into clusters in accordance with the principles laid down in the applied clustering algorithm [53]. In particular, clustering algorithms that implement the formation of a hierarchy of clusters with the construction of dendrograms, or clustering algorithms that search for cluster centroids, for example, the k-means algorithm or the fuzzy-c-means algorithm (FCM), can be used. Therefore, when deciding on the assignment of an object to a certain cluster, it is possible to take into account all, even non-coinciding (contradictory) values of assessments (values of features) of objects. Just as when working with ordinary data sets, the optimal number of clusters can be determined using one or another indicator of the clustering quality, for example, using the cluster silhouette index (in the general case), which should be maximized, or the Xie-Beni index (which is typical of the FCM algorithm) that should be minimized. For example, let the problem of clustering objects represented by multisets be solved using the FCM algorithm [8; 9]. Ideally, the resulting clusters should be compact and well separable from each other. Objects represented by multisets that fall into the same cluster can be considered similar to each other. Objects represented by multisets that fall into different clusters can be considered significantly different. Let the sought-for fuzzy clusters \(S_{k}\) form a fuzzy cover of the set \(Z\) containing objects represented by multisets: \(\bigcup\limits_{k=1}^{C}S_{k}=Z\). Then, we can write the following [8; 9]  
\[\sum\limits_{k=1}^{c}u_{k}(Z_{i})=1\ \big{(}i=\overline{1,s}\big{)}, \tag{4}\]  
where \(c\) is the number of fuzzy clusters \(S_{k}\) (\(k=\overline{1,c}\)), which is considered to be predetermined (\(c\in N\), \(c>1\)), \(u_{k}(Z_{i})\) is the membership function, which determines the fuzzy degree of multiset membership to the cluster. For objects represented by multisets, the FCM algorithm implements the minimization of the objective function of the form [8; 9]  
\[J(U,V)=\sum\limits_{k=1}^{c}\sum\limits_{i=1}^{s}\left(u_{k}(Z_{i})\right)^{m }\cdot d_{ki}^{2}, \tag{5}\]  
where \(U=[u_{k}(Z_{i})]\) is the fuzzy-\(c\)-partition of the set of objects represented by multisets based on membership functions \(u_{k}(Z_{i})\); \(V=(v_{1},\ldots,v_{c})\) are the centroids of clusters; \(d_{ki}\) is the distance between the cluster centroid \(v_{k}\) and the multiset \(Z_{i}\); \(m\) is the fuzzifier (\(m\in R\), \(m>1\)); \(c\) is the number of fuzzy clusters \(S_{k}\); \(s\) is the number of objects (multisets); \(k=\overline{1,c}\); \(i=\overline{1,s}\). Let each of the cluster centroids be a vector \(v_{k}=(v_{k,1}^{1},\ldots,v_{k,1}^{u_{1}},\ldots,v_{k,q^{\prime}}^{1},\ldots,v _{k,q}^{u_{q}})\). The distance between the cluster centroid \(v_{k}\) (\(k=\overline{1,c}\)) and the multiset \(Z_{i}\) (\(i=\overline{1,s}\)) can be determined based on Euclidean metric as [8; 9]  
\[d_{ki}=\left(\sum\limits_{j=1}^{q}\quad\sum\limits_{l_{j}=1}^{u_{j}}\Big{(}k_{ z_{i}}(p_{j}^{l_{j}})-v_{k,j}^{l_{j}}\Big{)}^{2}\right)^{\frac{1}{2}}, \tag{6}\]  
where \(k_{z_{i}}(p_{j}^{l_{j}})\) is the number of experts who gave the individual value of the assessment (the value of the feature) \(p_{j}^{l_{j}}\); \(v_{k,j}^{l_{j}}\) is the coordinate of the center of the \(k\)-th cluster, corresponding to the \(l_{j}\)-th assessment by the \(j\)-th feature; \(u_{j}\) is the number of different assessments on the \(j\)-th feature; \(i=\overline{1,s}\); \(k=\overline{1,c}\); \(j=\overline{1,q}\); \(l_{j}=\overline{1,u_{j}}\).