## 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  
Table 17 lists the 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) for the three side lengths of the sub-tiles \(S_{S}\) considered. This table complements Table 16, which details the number of NN parameters, the best \(N_{SS}\) obtained, and the side length of the tile \(S_{T}\) for each \(S_{S}\) considered. As was mentioned before, we observe in Table 17 that the error distances at the 95 th percentile decrease with the sub-tile sizes given by \(S_{S}\). The tile size, which is the size of the 2D sensor array, is correlated with this observation. That is, the smaller \(S_{S}\), the smaller the error distance and the size of the 2D sensor array, given by \(S_{T}\). So, from the results obtained, when \(S_{S}\) decreases, \(N_{SS}\) increases, but overall, the 2D sensor array size given by the side length of the tile, \(S_{T}\), decreases. When the tile size decreases, there are more tiles in the positioning area, with smaller inter-tile distances, which seems to contribute to a finer granularity for estimating the position. To complement the explanation above, it is noted that the size of the 2D sensor array in each case is calculated from the side length of the tile, \(S_{T}\), which is a function of \(S_{S}\) and the best \(N_{SS}\) obtained for each case. The side lengths of the tiles for each case are listed in Table 16. It follows from the results obtained that \(S_{T}\) decreases with the sub-tile size given by \(S_{S}\). In Sections 6.4 and 6.5, it was hypothesized that the bigger the tile, the coarser the resolution for estimating the position due to the increased inter-tile distance. A trade-off between increasing the amount of side information through increasing \(N_{SS}\) and the effects of increasing the inter-tile distance for a given fixed side length of the sub-tile \(S_{S}\) was also hypothesized. As it is observed that a reduction in the error distance is correlated with small side lengths of the sub-tiles \(S_{S}\) and a large number of sub-tiles \(N_{SS}\) but at the same time correlated with small tile sizes, a new hypothesis can be formulated. The new hypothesis can be stated as follows. For the given assumptions, the sub-tile should decrease in size through the parameter \(S_{S}\) and the number of sub-tiles in the tile should increase through the parameter \(N_{SS}\) such that the combination of parameters \(S_{S}\) and \(N_{SS}\) is constrained to result in a side length of the tile \(S_{T}\) that is small enough to produce a finer granularity or resolution for estimating the position. Actually, regarding the size of the 2D sensor array, we may expect a constraint to counteract how small the 2D sensor array can be from the actual physical arrangement of the antennas or receivers in the sensors array (as discussed in Section 7.2). However, such a constraint is not part of our assumptions. From an ideal assumption in which the 2D sensor array size or tile size can define the positioning granularity or resolution, we proceed as follows. We normalize the error distances obtained for each pattern-matching method and side length of the sub-tiles \(S_{S}\) by the corresponding side length of the 2D sensor array. In Table 17, we observe that for the case of pattern-matching method 3-MultFingIn-PosOut, the error distance normalized by the 2D sensor array's size is between 1.68 and 1.86 times the side length of the 2D sensor array. For the method 4-MultFingIn-ClassOut, the normalized error distance is approximately in the order of the side length of the 2D sensor array (0.94 to 1.06 times the side length of the 2D sensor array).