## 3 HELIOS Concept in the Real Estate Market--Methodology and Implementation
### Verification of Models Outputs  
Using the Agile method for selecting optimal parameters for both the size of the grid created by SOM and the number of iterations and the learning rate and sigma size, values were identified that yielded optimal results. The efficiency parameters of the analyses conducted in Agile mode are presented in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **The Lower Value** & **The Highest Value** & **Optimal Value** \\ \hline Iterations & 1000 & 10,000 & 4000 \\ \hline Sigma & 3.1 & 0.1 & 1.5 \\ \hline Learning rate & 3.1 & 0.1 & 1.1 \\ \hline Average of mean distance & 31,841 & 8653 & 10,887 \\ \hline Grid resolution & 2 \(\times\) 2 & 10 \(\times\) 10 & 6 \(\times\) 6 \\ \hline \hline \end{tabular} Source: Own elaboration.  
\end{table} 
Table 1: Ranges of parameters obtained in Agile mode. The consistency of the model was further verified by the substantive interpretation of the similarities among the obtained objects, as reflected in the homogeneous groups. \begin{table}
\begin{tabular}{c c c c} \hline
**Node X** & **Node Y** & **Mean Distance** & **Quantity Vectors** \\ \hline  
\multicolumn{3}{c}{Average Mean Distance 10,887} \\ \hline \end{tabular}
\begin{tabular}{c} Sum 4372 \\ \end{tabular}
\end{table} 
Table 2: Nodes characteristics in SOM grid: mean distance, quantity of classified vectors. An analysis of the average intra-cluster distances for all clusters is presented in Table 2. The analysis of selected created homogenous groups confirms the effectiveness of the applied method from a substantive point of view. This analysis shows that the SOM clusters are consistent and align with the initial assumptions. The group related to node (X, Y) = (5, 2) in Table 2 serves as the test base, and all these real estates are grouped in one cluster and have one of the smallest average distances of 3087, indicating that they are very similar real estates, aligning to introduce this base for testing. Another example is within a node (X, Y) = (2, 0), which consists of the largest group of homogenous real estate representing features most found in this market. The smallest group (excluding the test group) is defined by node (X, Y) = (4, 1), consisting of 37 real estate properties with exceptional characteristics, such as units with an area exceeding 215 m\({}^{2}\), luxury apartments valued over PLN 1 million, including those with land, which may be considered outliers or discrepancies in the databases. Including artificially generated real estate transaction information in one group was crucial for evaluating the solution. However, this needed to occur alongside stabilizing the minimization of gradient changes in the position of nodes within the SOM solution (see Figure 7). Simply put, this means that the solution can only be deemed final and subject to validation when, during the iterative process, the migration of real estate transaction records between individual nodes ceases and the positions of the nodes become stable from a numerical accuracy standpoint. Additionally, the most numerous groups were formed almost identically to how they were without introducing the test cluster. The spatial arrangement of the groups within the SOM was different, but their composition remained nearly identical. In summary, the introduction of test data was effectively detected within the SOM space, grouping them into a single node and thereby almost entirely excluding the introduction of noise in other formed groups--confirming the stability of the solution. The entire substantive analysis of the created groups is based on the concept of a real estate fingerprint, which represents a synergistic combination of standardized tokens used to capture and analyze unique characteristics and features (see Table 3).