## 6 Results of the Simulations
### Observations for Cases with Spatial Side Information and Aggregation in the Fingerprint Domain
#### 6.8.2 Two-Dimensional Sensor Array Sizes and Positioning Errors for the Best Results
Under the ideal assumptions, the latter observation means that at least 95 percent of the measurements have an error distance equal to or below the sampling resolution used by the 2D sensor array (where the sampling resolution is equivalent to the tile sizes used to sample the whole positioning area). Table 17 also lists the differences between the error distance obtained using the pattern-matching method without side information, 1-SingFingIn-PosOut, and the error distances \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Pattern-Matching Method**} & \multicolumn{3}{c}{**Difference in Error**} \\ \cline{2-9}  & \multicolumn{3}{c}{**3-MulfFingIn-PosOut**} & \multicolumn{3}{c}{**4-MultFingIn-ClassOut**} & \multicolumn{1}{c}{**Distance Between**} \\ \cline{2-9} \(S_{S}\) & **Error** & **Error** & **Error Dist.** & **Error** & **Error Dist.** & **Methods 3-MultFingIn-PosOut** \\
**[mm]** & **Distance** & **Distance** & **Difference** & **Distance** & **Distance** & **Difference** & **FingIn-PosOut** \\  & **at 95 th** & **Norm. by** & **from Method** & **at 95 th** & **Norm. by** & **from Method** & **and 4-MultFingIn-** \\  & **Percentile** & **Side Length** & **1-SingFingIn-** & **Percentile** & **Side Length** & **1-SingFingIn-** & **ClassOut at 95 th** \\  & **[m]** & **of Tile Tiles** & **PosOut** [m]** & **[m]** & **of Tile Tiles** & **PosOut** [m]** & **Percentile** [m]** \\ \hline
30 & 0.5594 & 1.86 & 1.8406 & 0.4117 & 1.06 & 1.9883 & 0.1477 \\
50 & 0.6364 & 1.59 & 1.7636 & 0.4970 & 0.99 & 1.9030 & 0.1394 \\
70 & 0.7062 & 1.68 & 1.6938 & 0.5905 & 0.94 & 1.8095 & 0.1157 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Error distances at the 95 th percentile in the CDF of the error distances for the best results obtained using pattern-matching methods 3-MultFingIn-PosOut and 4-MultFingIn-ClassOut (Figures 22 and 26), normalization of the error distances at the 95 th percentile to the side length of the tiles (2D sensor array), and the difference in the error distances between these methods and the method 1-SingFingIn-PosOut. Results for the three side lengths of the sub-tiles \(S_{S}\) considered. obtained using the methods 3-MultFingIn-PosOut and 4-MultFingIn-ClassOut, measured at the 95 th percentile of the CDFs of the error distance. In addition, we listed the difference in the error distance observed between the methods 3-MultFingIn-PosOut and 4-MultFingIn-ClassOut measured at the 95 th percentile of the CDFs of the error distance. * Discussion About the Reduction in the Size of the 2D Sensor Array and Constraints The observation mentioned above, discussing that at least 95 percent of the measurements have an error distance equal to or below the sampling resolution used by the 2D sensor array opens many research questions. For example, one line of thought suggests that the smaller the 2D sensor array, the smaller the positioning error. The reduction in the size of the 2D sensor array or tile is actually controlled by the combined result of reducing the side lengths of the sub-tiles, \(S_{S}\), and increasing the number of sub-tiles per side of the tile, \(N_{SS}\). Thus, if this trend continues, near the limit, we would have a very small 2D sensor array with a large number of sensors. Decreasing the 2D sensor array further to the limit so that it converges to what is ideally a point on a 2D positioning plane, it can be thought that the minimization of the positioning error is not due to the availability of adjacent samples but due to the high number of samples collected at that point. However, note that this observation is associated with the assumed collection of training samples from every position delimited by fixed tiles, which in this case would be assumed for every single point on the positioning plane (and which at the limit will be an infinite number of points).