## 4 Experimental Results
### Stop and Go Localisation Accuracy  
In this section, we present the analytic results that were calculated using the observations obtained during the training phase as inputs to the location system. We chose 500 scans randomly out of the crowdsourced training samples for each cell and excluded them from the training data. The remaining set was used to train the MVGMM model and get the probabilistic fingerprint for each cell. The 500 scans we removed from the training data were formed as the test set for each cell. Note the 500 test scans could be from any test device.  
We constructed the stop and go movements by including transition between cells; the observed RSS sequences were simulated by randomly choosing 50 scans from the 500 test samples of each cell.  
In the stop and go tests, nine different trajectories were designed to verify the proposed algorithm. The first six trajectories were designed to move only between adjacent cells, covering different parts of the test area. While Trajectory 7 was designed to repeat Trajectory 4 but miss transition data at three cells. Trajectory 8 was designed to repeat Trajectory 5 but miss more transition data at six cells. Trajectory 9 repeated Trajectory 6 with data missing for eight cell transitions. These latter three trajectories were selected to simulate the scenarios that continuous RSS measurements cannot be obtained for a period of time during the transition between cells. This is reasonable since one Wi-Fi scan can take around 3 s for some devices, while the user has already passed the transition cell.  
In the following analysis, the number of component \(K\) was set to \(K=7\). Table 5 gives the average matching accuracy for the designed nine trajectories. We performed the experiments 50 times for each trajectory, randomly choosing different test samples each time. The proposed system can still work properly when the system failed to get updated observation data for a certain time, referred to the tracking results of Trajectories 7-9. The accuracy decreased if the observation data in the transition cell is missing, however, the HMM based algorithm can still recover from the losing track of position with an average matching accuracy of 97.11%. Matching accuracy is defined as the percentage of the cells correctly determined:  
\[Accuracy=\frac{\sum\limits_{t=1}^{T}Equal(s_{t}^{HMM},s_{t}^{True})}{|T|} \tag{22}\]  
where  
\[Equal(a,b)=\left\{\begin{array}{ll}1&a=b\\ 0&otherwise\end{array}\right.\]  
With the aid of the proposed conditional likelihood observation function that utilises the information of the invisible APs, the system can achieve an average of 97.29% matching accuracy even when the observed data are not continuous. Figure 7 demonstrated efficiency of the distinct invisible AP set of different cells being a significant signature which helps to improves the localisation performance from an average 92.98% matching accuracy to an overall 97.29% matching accuracy.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Cell IDs**} & \multicolumn{2}{c}{**Training Data**} & \multirow{2}{*}{**Cell IDs**} & \multicolumn{2}{c}{**Training Data**} \\ \cline{2-5}  & **Sample Size** & **Visible AP Number** & & **Sample Size** & **Visible AP Number** \\ \hline
1 & 12,000 & 120 & 11 & 10,000 & 34 \\
2 & 12,000 & 94 & 12 & 12,000 & 38 \\
3 & 11,001 & 85 & 13 & 10,100 & 62 \\
4 & 12,004 & 67 & 14 & 4000 & 58 \\
5 & 8000 & 82 & 15 & 5473 & 42 \\
6 & 11,800 & 61 & 16 & 5500 & 42 \\
7 & 12,000 & 61 & 17 & 7368 & 45 \\
8 & 12,103 & 54 & 18 & 4500 & 53 \\
9 & 11,000 & 55 & 19 & 4499 & 49 \\
10 & 12,000 & 56 & 20 & 4800 & 47 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Training sample size and visible APs for each cell.