# Analysis of the Motion Sickness and the Lack of Comfort in Car Passengers
## 5 Probability Density Statistical Analysis  
The statistical analysis performed in the previous section has brought some overall conclusions about the drivers, the cars and the sections. However, in order to extract more meaningful information, next, a different statistical approach to several comfort parameters is provided. With this purpose, the kernel density estimation (KDE) is used. The KDE technique, unlike histogram, produces a smooth estimate of the probability density function (PDF) and is able to suggest multimodality [54]. It is useful to estimate the PDF of datasets which are difficult to be modeled by parametric density functions. The formal definition of KDE is a function defined by equation  
\[f_{h}(x)=\frac{1}{nh}\sum_{i=1}^{n}K(\frac{x-x_{i}}{h}) \tag{8}\]  
where \(K(x)\) is called the kernel function that is generally a smooth, symmetric function such as a Gaussian, n number of samples, \(x\) a given point and \(h>0\) the smoothing bandwidth that controls the amount of smoothing [54].  
In this section, the variables have been calculated approximately every 10 s, that is, we have selected a window size of 1024 points which, taking into account the sampling frequency of the device, corresponds to 10.24 s.  
It has been observed that \(N_{x}\) and \(N_{y}\) are scattered variables and, for a high number of windows, there are no situations in which the accelerations surpass the threshold; so, kernel density analysis makes no sense. On the other hand, as it has already been concluded, when \(A_{y}\) is considered, in some of the cases, there are no significant differences among cars, drivers and sections, being its value very similar. Finally, when \(MSDV\) variable's magnitude is analyzed, it can be concluded that its value in the vertical axis is much smaller than the longitudinal and lateral ones.  
Thus, Figure 9b depicts the bivariate kernel density function of the \(MSDV_{x}\) and \(MSDV_{y}\) corresponding to the four sections and the three drivers (data collected with both vehicles have been considered). The color represents the probability density, from red (improbable) to pink (probable) through yellow, green and blue. In addition, the closer the lines, the more regular the conduction is.  
As we already know, this figure shows that D1 produces more motion sickness than D2 and that both of them generate far higher values than D3. Besides, this analysis demonstrates that D3 presents a rather regular and comfortable driving style and that D1 can be considered to be a less regular driver. It is worth mentioning that, although D2 drives most of the time with lower \(MSDV\) than D1, she shows an irregular driving comparing with D3, which increases her average \(MSDV\).  
With regards to the sections, the results are consistent with what has already been explained in the previous section: S3 and S1 are the ones with the least motion sickness and S2 and S4 the ones with the most. This analysis also demonstrates that all drivers exhibit a more spread kernel in the latter situations.  
Figure 9a resumes the differences when vehicle analysis is considered. As it has already been demonstrated that driver D3 is the most regular one when the driving is considered, the figure focuses on his driving, but the conclusions can be extrapolated for the rest of the drivers. As can be seen, concerning the regular driving, during the most comfortable sections (S1 and S3), the driving seems to be more regular. Besides, car C1 presents greater values for \(MSDV_{y}\) than C2 and smaller values for \(MSDV_{x}\) than C1, especially in the less comfortable sections.