## 4 Discussion

Table 6: Experimental data concerning the elements (in seconds) of the sequences \(I\) and \(II\) in (3) and their adjoint versions \(I^{\text{adj}}\), \(II^{\text{adj}}\) in (4) for the healthy subject 3 (\(\text{DS}_{\text{r}}=0.11733\) s, \(\text{DS}_{\text{y}}=0.12133\) s). \begin{table}
\begin{tabular}{c c} \hline \hline AT patient A. & \(\mathcal{Y}_{\Phi}=0.37838\) \\ \hline AT patient F. & \(\mathcal{Y}_{\Phi}=6.0926\) \\ \hline AT patient G. & \(\mathcal{Y}_{\Phi}=0.72389\) \\ \hline Healthy subject 1. & \(\mathcal{Y}_{\Phi}=0.22046\) \\ \hline Healthy subject 2. & \(\mathcal{Y}_{\Phi}=0.17933\) \\ \hline Healthy subject 3. & \(\mathcal{Y}_{\Phi}=0.1811\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The values of the computed \(\Phi\)-bonacci gait number in (6) (all the gains within it are set equal to 1). pathology. Among them, patient A--with the mildest AT phenotype--exhibits the smallest number, despite it being approximately twice as large as the healthy subjects'. Patient G--with an intermediate AT phenotype--exhibits an index that is, in turn, approximately twice as large as patient A.'s. Lastly, a very high value of this index is computed for patient F, who has the most severe AT phenotype. It is worth noticing that the mere analysis of the double support phase durations does not allow us to highlight all these quantitative considerations. In fact, as shown in the related tables, the durations of the double support phases for patient A are similar to those corresponding to the healthy subjects, whereas the same durations for patient G are even lower than those corresponding to healthy subjects. Only for patient F, are these durations are meaningfully large. These results, which comply with the relevant qualitative evaluations provided by the clinicians, support the idea that the \(\Phi\)-bonacci gait number can be successfully used in a rare disease such as AT. In more detail, despite a study on a wider sample being needed, these data are sufficient to find statistically significant differences between the patients and the healthy controls (\(p=0.05\), Mann-Whitney one tailed U test). The data for the most severe patient (F) exhibit the highest value of the \(\Phi\)-bonacci gait number, confirming not only the accuracy of this index in differentiating the patients from the healthy subjects, but also to objectively quantify the deficit among the patients. Therefore, the use of objective measures of motor abilities in AT patients, such as the ones described in this paper, might facilitate the standardization of rehabilitative treatments and the evaluation of outcomes. These can help, first, in identifying predictors of the progression of motor and movement dysfunctions, and, next, facilitate patient selection and stratification in clinical trials, potentially enabling studies with direct clinical outcomes, which can be hindered, mostly, by inter- and intrasubject variability regarding severity and progression of the disease. Concurrently, in home evaluation through objective measures of motor abilities might minimize difficulties in accessing specialized clinical care for disabled AT people and those living in underserved areas, highlighting the need for the latest strategies involving telemedicine.