# Analysis of Data Presented by Multisets Using a Linguistic Approach +
## 3 Analysis of Sets of Objects Represented by Multisets
### Classification of Datasets
The values \(k_{Y_{c}}(p_{j}^{l_{j}})\) and \(k_{Y_{c}}(w_{c})\) (\(j=\overline{1,q}\); \(j=\overline{1,u_{j}}\); \(c=\overline{1,2}\)) in the multiset \(\hat{Y}_{c}\) (\(c=\overline{1,2}\)) for the class \(Y_{c}\) can be calculated as sums of the corresponding values \(k_{z_{i}}(p_{j}^{l_{j}})\) and \(k_{z_{i}}(w_{c})\) for the objects \(z_{i}\) included in the class \(Y_{c}\) (\(c=\overline{1,2}\)) [17; 52]. Each class \(Y_{c}\) can be associated with a multiset of the form [17; 52]  
\[ \begin{split}\hat{Y}_{c}=\Big{\{}& k_{Y_{c}}(p_{1}^{1}) \bullet p_{1}^{1},\ldots,k_{Y_{c}}(p_{1}^{u_{1}})\bullet p_{1}^{u_{1}},\ldots, k_{Y_{c}}(p_{q}^{1})\bullet p_{q}^{1},\\ &\ldots,\ k_{Y_{c}}(p_{q}^{u_{q}})\bullet p_{q}^{u_{q}},k_{Y_{c} }(w_{1})\bullet w_{1},k_{Y_{c}}(w_{2})\bullet w_{2}\Big{\}}.\end{split}  \tag{14}\]  
Multiset \(\hat{Y}_{c}\) of class \(Y_{c}\) can be represented as \(\hat{Y}_{c}=\sum\limits_{j=1}^{q}P_{jc}+W_{c}\) (\(c=\overline{1,2}\); \(j=\overline{1,q}\)), where \(P_{jc}\) and \(W_{c}\) are multisets, elements of which are, respectively, sums of values of \(j\)-th features of objects \(z_{i}\) included in class \(Y_{c}\) (\(c=\overline{1,2}\)) and sums of belonging values of objects \(z_{i}\) included in the class \(Y_{c}\) (\(c=\overline{1,2}\)). The distance between multisets \(A\) and \(B\) can be calculated using the Hamming distance [17; 52]  
\[dist(A\Delta B)=\sum\limits_{j=1}^{q}\sum\limits_{l_{j}=1}^{u_{j}}\Big{|}k_{A }(p_{j}^{l_{j}})-k_{B}(p_{j}^{l_{j}})\Big{|}. \tag{15}\]  
Objects \(z_{i}\) (\(i=\overline{1,s}\)) in the decomposition \(\{W_{1},W_{2}\}\) based on the results of individual classifications of objects form the best possible decomposition of the set of objects \(Z=\{z_{1},\ldots,z_{s}\}\) into two classes. Distance between multisets \(d^{*}=d(W_{1},W_{2})\) is the maximum possible distance in the space of multisets between objects belonging to different classes. With ideal individual classifications of objects, that is, in the absence of contradictions, the distance can be calculated as \(d^{*}=s\cdot e\). The problem of searching for GDRCs of objects is reduced to the problems of optimization by features \(P_{j}\) (\(j=\overline{1,q}\)) [17; 52]  
\[d(P_{j1},P_{j2})\to max(d(P_{j1},P_{j2}))=d(P_{j1}^{*},P_{j2}^{*}). \tag{16}\]  
When solving problem (16), it is necessary to search for multisets \(P_{j1}^{*}\) and \(P_{j2}^{*}\), which will be located at the maximum possible distance and belong to different classes, for each feature \(P_{j}\) (\(j=\overline{1,q}\)),  
Multiset \(P_{jc}^{*}\) (\(j=\overline{1,q}z=\overline{1,2}\)) can be represented as a sum of two subsets: \(P_{jc}^{*1}\), \(P_{jc}^{*2}\), \(P_{jc}^{*}=P_{j1}^{*1}+P_{j2}^{*2}\). The solution to each of the problems (16) is expressed in terms of submultisets \(P_{jc}^{*1}\), \(P_{jc}^{*2}\), and determines the best binary decomposition \(\left\{P_{j1}^{*},P_{j2}^{*}\right\}\) of the set of objects \(Z=\left\{z_{1},\ldots,z_{s}\right\}\) for the feature \(P_{j}\) (\(j=\overline{1,q}\)). Let \(p_{j}^{*}\) be the boundary value of the assessment (the value of the feature), which determines the boundary of separation into pairs \(P_{jc}^{*1}\) and \(P_{jc}^{*2}\) in multiset \(P_{jc}^{*}\). Various combinations of boundary values \(\left\{p_{j}^{*}\right\}\) for different features \(P_{j}\) (\(j=\overline{1,q}\)) set the conditions for classifying object \(z_{i}\) and form all possible GDRCs of objects of the form (13). The boundary values of the characteristics \(p_{j}^{*}\) can be sorted in descending order of the distance values \(d(P_{j1}^{*},P_{j2}^{*})\). When forming GDRCs, it is advisable to use those boundary values \(p_{j}^{*}\) of the features that occupy the first places in the ordering list. The closer the value \(d(P_{j1}^{*},P_{j2}^{*})\) is to the value \(d^{*}=d(W_{1},W_{2})\), the more accurate the approximation of the individual classification of objects will be.