# Device-Free Crowd Size Estimation Using Wireless Sensing on Subway Platforms
## 5 Results and Discussion
### Crowd Size Estimation  
As a benchmark, we trained the single regression model depicted in Equation (2) on the complete dataset regardless of whether a rail vehicle was present or not. Equation (4) shows the resulting regression model:  
\[ \begin{array}{ll}\hat{p}&=w_{2}\mathbf{\hat{a}}^{2}+w_{1}\mathbf{\hat{a}}+b\\ \hat{p}&=-0.757\cdot\mathbf{\hat{a}}^{2}+12.171\cdot\mathbf{\hat{a}}+0.311\end{array}  \tag{4}\]  
We evaluated the significance of the estimated parameters \(w_{2}\), \(w_{1}\), and \(b\) by calculating the corresponding \(p\)-values, which were \(38.84\cdot 10^{-15}\), \(1.475\cdot 10^{-78}\), and \(0.451\), respectively. The low parameter value and large \(p\)-value for the intercept term show that it does not contribute to the function. The function passes through the origin, which is expected given that we use attenuation values relative to the empty environment. To test for autocorrelation, we calculated the Durbin-Watson statistic, which was 0.903. This value indicates significant autocorrelation between the residuals. This can be explained by the fact that consecutive ground-truth samples are often taken in very similar situations; more precisely, the locations at which people are standing will be similar to the previous ground-truth sample. The location at which a person is standing will impact the amount of attenuation they cause on the links. This difference in attenuation can result in an error that remains consistent across multiple evaluation samples. We trained the model in Equation (2) for both environment states using the corresponding training sets. Equation (5) shows the resulting regression model when no rail vehicle was present in the environment. Equation (6) shows the resulting regression model when a vehicle is present in the environment. \[\hat{p} =w_{2}\mathbf{\hat{a}}^{2}+w_{1}\mathbf{\hat{a}}+b\] \[\hat{p} =-0.033\cdot\mathbf{\hat{a}}^{2}+11.735\cdot\mathbf{\hat{a}}+0.417 \tag{5}\] \[\hat{p} =-0.021\cdot\mathbf{\hat{a}}^{2}+7.082\cdot\mathbf{\hat{a}}-5.155 \tag{6}\]  
We evaluated the significance of the estimated parameters \(w_{2}\), \(w_{1}\), and \(b\) for both models by calculating the corresponding \(p\)-values. For Equation (5), we obtain \(p\)-values of 0.834, \(0.295\cdot 10^{-63}\), and 0.151, respectively. For Equation (6), we obtain \(p\)-values of 0.941, 0.016, and 0.350, respectively. For both models, we observe low values and high \(p\)-values for the second-order parameter, indicating that the data mainly fit a first-order trend. The \(p\)-values for the intercept of both models indicate they are not significant. To test for autocorrelation, we calculated the Durbin-Watson statistic. For Equation (5), we obtain a value of 0.786, indicating a strong positive autocorrelation between the residuals. This can be explained, as above, by consecutive samples representing fairly similar environment states with their corresponding error. For Equation (6), we obtain a value of 2.063, indicating practically no autocorrelation. This can be explained by the fact that we only took a single sample when a rail vehicle stopped; this causes more time and differences between consecutive samples. Figures 6 and 7 show the correlation between the used training samples for both approaches, together with the resulting polynomial regression models. These graphs visually show the advantage of using separate models depending on the environment's state. When evaluating the crowd size estimation with no vehicle present, a median error of 2.769 people is observed, along with an MAE of 3.342 people and an RMSE of 4.211. The CDF plot is shown in Figure 8 as a green dash-dotted line. Figure 6: This correlation plot shows the resulting regression model and the used training data for the baseline approach. This model uses all data regardless of whether a vehicle is present or not. Figure 7: This correlation plot shows the resulting regression models and the used training data for both models, i.e., when no rail vehicle is present (in green) and when a vehicle is present (in orange).