## 4 Fingerprinting-Based Positioning with Spatial Side Information
### Pattern-Matching Methods  
The pattern-matching methods selected in our study comprise first the traditional fingerprinting method without side information and then three pattern-matching methods exploiting spatial side information. We assign a mnemonic name to each one of the four pattern-matching methods selected, namely _1-SingFingIn-PosOut_, _2-SingFingIn-AggPosOut_, _3-MultFingIn-PosOut_, and _4-MultFingIn-ClassOut_. These are summarized in Table 6, shown in Figure 10, and explained below.  
The mnemonic names assigned to the pattern-matching methods listed in Table 6 compress the information described next. The mnemonic 1-SingFingIn-PosOut stands for pattern-matching method number one, taking a single vector of fingerprints as the input from a single sub-tile and producing a position as the output. The mnemonic 2-SingFingIn-AggPosOut stands for pattern-matching method number two, taking a single vector of fingerprints as the input for each one of \(N_{ST}\) sub-tiles independently and producing a position as the output. In this last method, the output results from aggregating \(N_{ST}\) independent position estimates. The mnemonic 3-MultFingIn-PosOut stands for pattern-matching method number three, taking multiple fingerprints as the input and producing a position as the output. The mnemonic 4-MultFingIn-ClassOut stands for pattern-matching method number four, taking multiple fingerprints as the input and producing a tile-class probability as the output.  
The positioning data aggregation domains listed in Table 6 and their relationships to the pattern-matching methods are consistent with those introduced in Section 2.4 and listed in Table 2.  
All of the pattern-matching methods take as their input the fingerprints collected through the 2D sensor array, shown in Figure 10a. The 2D sensor array returns a tensor with a set of fingerprints \(\mathcal{F}\). The tensor can be interpreted as a 2D arrangement of a total of \(N_{ST}\) fingerprint vectors, where each vector contains \(N_{F}\) elements. Each element of a vector is a fingerprint from one of the \(N_{F}\) fingerprint sources.  
\begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Fingerprinting Pattern-Matching Method** & **Side Information** & **Input Data to the Aggregation Domain** & **Output Data NN** & **Position Calculation from (Open. Type)** & **NN Type the NN Output** & **Implement. Structure)** \\ \hline
1-SingFingIn-PosOut & No side information [no aggregation] & Vector of fingerprints from center sub-tile & Position coordinates [regression] & None & Feedforward pattern-matching methods \\ \hline
2-SingFingIn-AggPosOut & Side information used at the output of the NN & Vector of fingerprints from one of the \(N_{ST}\) sub-tiles & Position coordinates [regression] & Aggregation of \(N_{ST}\) position estimates & Feedforward pattern-matching methods \\ \hline
3-MultFingIn-PosOut & Side information used at the input of the NN & Tensor of fingerprints from \(N_{ST}\) sub-tiles & Position coordinates [regression] & None & 1-SingFingIn-PosOut and 2-SingFingIn-AggPosOut) \\ \hline
4-MultFingIn-ClassOut & Side information used at the input of the NN & Tensor of fingerprints from \(N_{ST}\) sub-tiles (1D arrangement) & Class likelihood probability [regression] & Weighted sum of class probabilities multiplied by associated tile center & Feedforward pattern-matching method \\ \hline \hline \end{tabular}
\end{table}
Table 6: Pattern-matching methods.