## 4 Discussion
In fact, our results represent the highest classification accuracy that has ever been reported based on the PaHaW database (see Table 1). We hypothesize that the improvement was caused by the inclusion of the state-of-the-art XGBoost algorithm into our machine learning pipelines. As already mentioned, the result is based on one in-air feature: median horizontal velocity of a sentence. In comparison with the HC cohort, the PD patients exhibited much lower values of this measure, i.e., while writing the sentence, the PD patients were not able to perform horizontal transitions (movement between neighboring letters or words) as quickly as the HC could. This finding is in line with the work of [PERSON] et al. [53], who observed that wrist extension stiffness in PD patients makes the handwriting in the horizontal direction more problematic. Therefore, scientists started to use the term _horizontal  
Figure 3: Handwriting samples of the repetitive loop task for HC and PD patients are on the left, and the resulting velocity profiles are on the right. dysgraphia_[13]. Generally, vertical or horizontal dysgraphia may be considered a presymptomatic neurobehavioral biomarker of PD with possible significance in early PD diagnosis [13]. In [20], we proved that the FD features improved the accuracy of PD dysgraphia diagnosis in the Archimedean spiral drawing task by 10%. Contrary to our pilot results, in the frame of this work, these features did not lead to any improvements. After a deeper analysis, we found that this was caused by a combined task approach. Performance of the Archimedean spiral is a quasiparticle and continuous task with some repetitive patterns. It looks as though the FD features work especially well in these specific cases. Nevertheless, when combining these tasks with a complex handwriting task (such as a sentence), the measures quantifying in-air movement tend to be more discriminative (in our case, the median in-air horizontal velocity of a sentence). This brings us to the same conclusion that was given during the correlation analysis--the FD features advance the PD dysgraphia diagnosis only in some specific cases. The best regression model, estimating the UPDRS V score with a 12.51% error, is based only on the standard deviation of on-surface vertical velocity (\(\alpha=0.1\)) extracted from the sentence. This FD-based parameter was selected from the feature set combining all on-surface measures; therefore, we can confirm the positive influence of FC on the regression analysis performance. In fact, the FD features outperformed the conventional ones in all scenarios. To better understand this result, we plotted vertical velocity patterns of the sentence task for different orders of FD (see Figure 4). We can observe a big difference between \(\alpha=0.1\) and the rest of the orders, including the full derivative. This large distance means that we are working with completely new information that is far from that contained in the full derivative. Although it is difficult to clinically interpret this information, it is clear that FC opens new possibilities for monitoring PD severity. Regarding the PD duration estimation results, the most successful model (\(\text{EER}=23.46\%\)) consists of 16 conventional on-surface/in-air features (all features' importance values can be found in Supplementary Table S1). The most frequent feature with the highest feature importance is the jerk extracted from several handwriting tasks. This probably means that as PD progresses, handwriting becomes more jerky and  
Figure 4: Vertical velocity patterns of the sentence task for different orders of fractional-order derivatives (FD). irregular. Vertical velocity is the second most frequent feature involved in the models, which is probably linked with micrographia. Generally, in the case of PD duration estimation, the FD-based features did not yield any improvement. In conclusion, the FD-based features are better for modeling PD severity (in terms of UPDRS V score estimation), but they do not lead to an improvement in PD duration modeling.