# Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion
## 4 Testing Results  
### Testing Results  
To demonstrate the analytical results derived in the previous section, we perform the simulations about the proposed approach under different noise perturbations, signal propagation models (see Table 1) and the number of calibrated locations and RSS samples in an indoor environment with the dimensions of 56 m \(\times\) 19.3 m, as shown in Figure 2. This environment is covered by five WLAN APs, notated as AP1, \(\cdots\), AP5, and meanwhile, there are 93 labeled locations (with \"\(\bullet\)\"), which are evenly distributed with 2-m intervals. Here, the simulation parameters are set as follows. The standard deviation of noise ranges from 2 dB to 5 dB; the ratio of calibrated fingerprints ranges from 30% to 100%; and the number of RSS samples at each RP ranges from 150 to 200. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Types of Models** & **Models Formulation** & **Parameters Setting** \\ \hline Log-distance[33] & \(P(d)=P(d_{0})-10n\log(\frac{d}{d_{0}})-\chi_{\sigma}\) & \(P(d_{0})=-12\,\text{dB}\text{m};n=2\); \\  & \(d_{0}=1\,\text{m};\chi_{\sigma_{\sigma}}=2\,\text{dB}\) \\ \hline Multi-wall[34] & \(P(d)=P(d_{0})-10n\log(\frac{d}{d_{0}})-W_{\text{a}}N_{\text{w}}\) & \(P(d_{0})=-12\,\text{dB}\text{m};n=2\) \\  & \(d_{0}=1\,\text{m};W_{\text{a}}=10\,\text{dB}\) \\ \hline Supachai[35] & \(P(d)=P(d_{0})-10n_{\text{SF}}\log(\frac{d}{d_{0}})-FAF-\chi_{\sigma}\) & \(P(d_{0})=-12\,\text{dB}\text{m};n_{\text{SF}}=2\); \\  & \(d_{0}=1\,\text{m};\chi_{\sigma_{\sigma}}=2\,\text{dB};FAF=5\,\text{dB}\) \\ \hline Breakpoint[36] & \(P(d)=P(d_{0})-10n_{1}\log(\frac{d}{d_{0}})u(d_{\text{bp}}-d)-\) & \(P(d_{0})=-12\,\text{dB}\text{m};n_{1}=2\); \\  & \(10(n_{1}\log(\frac{d}{d_{0}})+n_{2}\log(\frac{d}{d_{0}}))u(d-d_{\text{bp}})\) & \(n_{2}=1.5;d_{0}=1\,\text{m};d_{\text{bp}}=5\,\text{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in different signal propagation models. Figure 3a shows the sum of elements in \(\mathbf{P}_{x}\) and \(\mathbf{P}_{y}\) under different values \(\boldsymbol{\Delta}\). The purple and green lines stand for the fitted curves with respect to \(\mathbf{P}_{x}\) and \(\mathbf{P}_{y}\), respectively. From this figure, we can find that as the value \(\boldsymbol{\Delta}\) increases, the sum of elements in both \(\mathbf{P}_{x}\) and \(\mathbf{P}_{y}\) decreases, which demonstrates that the value \(\mathbf{P}\) will decrease with the increase of \(\boldsymbol{\Delta}\). Figure 3b shows the Cumulative Distribution Functions (CDFs) of errors by using the proposed approach under different standard deviations of noise. From this figure, we can find that a slighter noise perturbation (or smaller standard deviation of noise) results in a higher localization accuracy as expected. In Figure 3c, we show the localization performance of the proposed approach when the signal attenuation property satisfies the log-distance [34], multi-walls [35], Supachai [36] and breakpoint [37] models, respectively. From this figure, we can find that the breakpoint model performs best in localization accuracy due to the fact that most of the labeled locations are distributed in the straight corridors, which is applicable to the breakpoint model [38]. In the results that follow, we set the initial parameters as follows. The standard deviation of noise is 2 dB; the ratio of calibrated fingerprints is 100%; the number of RSS samples at each RP is 200; the signal attenuation property satisfies the breakpoint [37] model in the simulation; and the augmentation dimension is \(m\), where \(m\) is the number of APs. Based on this initial parameter setting, we use the control variates method to perform each testing. Specifically, when the value of a parameter varies, the initial values are distributed to the other parameters. Figure 3d shows the CDFs of errors by using the proposed approach under different ratios between the number of calibrated and labeled locations (or called different ratios of calibrated fingerprints).