# Kinematic Determination of the Aerial Phase in Ski Jumping
## 3 Results and Discussion
### Filtering Analysis  
The filter settings were analyzed to highlight the effect filters have on such data. Ten different filters (F1-F10) were applied to the data, where the cut-off frequency for position, and first and second position-time derivatives varied from 10 to 0.5 Hz. The settings for F1-F10, along with the average \(LD\)-ratio and rate-of-change of the \(LD\)-ratio for all filtered and unfiltered data (F0), are shown in Appendix A. The glide phase detection algorithm was applied to all filter settings (F1-F10) and analyzed for \(\tau\) between 0.005 s\({}^{-1}\) and 0.015 s\({}^{-1}\). After an initial check, the unfiltered data and the data from the filters F1-F4 were discarded from further analysis, as the algorithm was unable to detect any steady glide phase of \(\geq\)5 m in the majority of the jumps (seen in Appendix A). Filters F9 and F10 were also discarded after visual inspection (shown in Figure A1), since the signal clearly deteriorated for these filter settings; i.e., not only noise, but also substantial aspects of the signal were removed. Hence, the filters analyzed were F5-F8 and the settings for these are shown in Table 2.  
Filter F6 is the setting used in the first part of this study and also in [PERSON] et al. (2015). Compared to F6, F5 highlights the effect of a high cut-off frequency on the position data. The application of F7 leads to an increase in position error, while decreasing the first and second derivatives. F8 is a filter where the effect of the cut-off frequency on position data is similar to F6 with an increase in the first and second derivative filter. The average data of F6-F8 are displayed in Figure 7.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter** & \(f_{c}\)**\%f(x,y,x)** & \(f_{c}\)**\%f\({}^{\prime}\)(x,y,x)** & \(f_{c}\)**\%f\({}^{\prime\prime}\)(x,y,x)** \\
**(HZ)** & **(HZ)** & **(HZ)** & **(HZ)** \\ \hline F5 & 4 & 3 & 2 \\ F6 & 2 & 3 & 2 \\ F7 & 3 & 2 & 1.5 \\ F8 & 2 & 1.5 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cut-off frequency (\(f_{c}\)) on position, first derivative, and second derivative for the four filters analyzed.  
The effects of filter settings were analyzed by assessing SD\({}_{1-4}\), when different \(\tau\) were applied in the algorithm for the HS106 data, as shown in Figure 8. The variation in rate-of-change is large in F5, as shown in Figure 7a. This will cause the algorithm to not identify a steady glide phase in many jumps; i.e., start and end will be defined at p\({}_{m}\), which is also shown in Appendix A (Table A2). This explains the small SD\({}_{1,2}\) for small \(\tau\) for F5 in Figure 8a,b. SD\({}_{1,2}\) increased rapidly for increasing \(\tau\) and were in total much larger than for the other filters; i.e., the data were considered as not sufficiently filtered. Filter setting F8 showed a lower SD\({}_{1,2}\) than the other filters. However, SD\({}_{3,4}\) was large, indicating that the algorithm had defined parts of the preparation phases as steady, i.e., the data were considered as too heavily filtered.  
To investigate F6 and F7 further, the cost function of the standard deviation was analyzed to find the \(\tau_{opt}\), as in Figure 6, shown in Figure 9.  
Both filter settings showed similar trends with \(\tau_{opt}\) = 0.01 s\({}^{-1}\). Both filter settings were considered adequate for this data collection, but F6 was chosen as the average standard deviation, the aforementioned parameters were \(\sim\)2% lower on average over all thresholds, and \(\sim\)1.2% was lower at 0.01 s\({}^{-1}\).  
Figure 7: Average data from HS106 with filter setting (**a**) F5, (**b**) F6, (**c**) F7, and (**d**) F8. The upper figure shows the average \(LD\)-ratio and the lower shows the rate-of-change. Gray shaded error band displays the standard deviation with \(n\) = 38, and the horizontal distance from the jump edge on the \(x\)-axis.