# Analysis of Data Presented by Multisets Using a Linguistic Approach +
## 3 Analysis of Sets of Objects Represented by Multisets
### Ordering Objects in a Dataset  
When solving the problem of ordering objects represented by multisets, they usually work with some target cluster or class.  
The problem of ordering objects \(z_{i}\) (\(i=\overline{1,s}\)) represented by multisets is reduced to the problem of ordering the corresponding multisets \(Z_{i}\).  
The ordering of objects can be performed by proximity to the 'ideal' (best) object (2) or by distance from the 'anti-ideal' (worst) object (3).  
If it is necessary to order objects from worst to best [19; 52], then this can be done by calculating distances \(d(Z_{\min},Z_{i})\) using the Hamming metric  
\[d(Z_{min},Z_{i})=\sum\limits_{j=1}^{q}\chi_{j}\cdot\sum\limits_{j=1}^{u_{j}} \left|k_{Z_{min}}(p_{j}^{l_{j}})-k_{Z_{i}}(p_{j}^{l_{j}})\right|=2\cdot\sum \limits_{j=1}^{q}\chi_{j}\cdot\left|e-k_{Z_{j}}(p_{j}^{1})\right|, \tag{18}\]  
where \(\chi_{j}\) is the value of the coefficient of the relative importance of the \(j\)-th feature \((j=\overline{1,q})\); \(\chi_{j}>0\); \(\sum\limits_{j=1}^{q}\chi_{j}=1\).  
The values of the coefficients of the relative importance of the features \(\chi_{j}\) (\(j=\overline{1,q}\)) can be determined, for example, taking into account the conclusions about the significance of the features obtained during the formation of GDRCs.  
It should be noted that the features of objects can have different relative importance, but values \(p_{j}^{l_{j}}\) (\(j=\overline{1,q}\)) related to the same feature are equivalent. In the case when all features are equivalent, the values of all coefficients \(\chi_{j}\) (\(j=\overline{1,q}\)) are assumed to be equal to 1.  
The larger the number \(d(Z_{\min},Z_{i})\), the better the object \(z_{i}\) (\(i=\overline{1,s}\)).  
Object \(z_{h}\) is worse than object \(z_{g}\) (\(z_{h}\prec z_{g}\)) if \(d(Z_{min},Z_{h})<d(Z_{min},Z_{g})\).  
Objects \(z_{h}\) and \(z_{g}\) are equivalent, and the ordering of objects is not strict if \(d(Z_{min},Z_{h})=d(Z_{min},Z_{g})\).  
The problem of ordering objects \(z_{i}\) (\(i=\overline{1,s}\)) by distance from the 'anti-ideal' (worst) object is solved as follows [19; 52].  
First, the problem of comparing the weighted sums of the first (worst) values of the features of objects \(H_{Z_{i}}^{1}=\sum\limits_{j=1}^{s}x_{j}\cdot k_{Z_{i}}(p_{j}^{1})\) is solved. The object \(z_{i}\) with the largest sum \(H_{Z_{i}}^{1}\) will be the worst one. Objects are ordered from worst to best in descending order of \(H_{Z_{i}}^{1}\). If some objects \(z_{i}\) are equivalent, i.e., they 'occupy' the same place in the sum \(H_{Z_{i}}^{2}\) ordering list, then to order the equivalent objects having the same sums of the first assessments \(H_{Z_{i}}^{1}\), the problem of comparing the weighted sums of the second values of the features of objects \(H_{Z_{i}}^{2}=\sum\limits_{j=1}^{q}\chi_{j}\cdot k_{Z_{i}}(p_{j}^{2})\) is solved. Objects are ordered from worst to best in descending order of \(H_{Z_{i}}^{2}\).  
Calculation and comparison of the sums of the second, third, etc. values of features of objects is performed until complete ordering of all objects represented by multisets [19; 52].  
As a result, in the ordering list the worst objects will take the first places, and the best objects will take the last places. When ranking objects from best to worst based on such an ordering list, the rank of 1 should be given to the object that came last in the ordering list, and the highest rank \(s\) equal to the number of objects should be given to the object that was ranked first in the ordering list.  
The ordering of objects \(z_{i}\) (\(i=\overline{1,s}\)) in proximity to the 'ideal' (best) object can be done in a similar way. When doing this ordering, object \(z_{h}\) is better than object \(z_{g}\) if \(d(Z_{max},Z_{h})<d(Z_{max},Z_{g})\).