## 4 Discussion
The better performances of using \(d=5\) with respect to \(d=3\) seemed to be confirmed by the points-to-centroid distance within each cluster, which were lower for every gesture set (Figure 5) and also by the higher cluster sparsity (Figure 6). Thus, taking into account a higher number of possible permutations (3! versus 5!) could allow for a more detailed characterization of the sEMG signal. In some cases \(d\) was selected, depending on the number of samples, in order to comply with the above-mentioned constraint regarding the data length. [PERSON] et al. [31] used \(d=3\) on EEG time series in order to investigate the suitability of a modified version of the PEn to assess epileptic seizures by detecting abrupt changes in signal amplitude. This choice was mainly related to the width of the windows used in PEn computation (below 100 samples) and was adopted also in [35], where for heart rate data including 256 samples, an embedding dimension of three was employed. On the contrary, despite PEn being computed in some cases for large amounts of data with high \(d\) values, for instance, up to eight for \(N=10^{4}\)[36], other studies reported that even if a relatively large number of data samples were available, the selected embedding dimension did not exceed five for \(N=2000\) or even four for \(N=4097\)[34; 45]. Thus, the choice of the embedding dimension for PEn computation still represents an issue which deserves to be carefully investigated when dealing with biological time series. However, a possible even if partial explanation for the better performance of PEn with a relatively higher embedding dimension (\(d=5\)) observed in this study could lie in the frequency characteristics of sEMG signal. The greater bandwidth of the latter, with respect to EEG and heart-rate signals, leads to more complex patterns in terms of variations of signal amplitude in a given temporal epoch. Hence, a higher number of possible permutations (\(d!\)) could be more appropriate for describing the sEMG's fast and complex dynamics, even if the number of samples for each muscular activation during gesture performance (\(N\simeq 600\)) is comparable or lower than the numbers previously reported for EEG and heart-rate descriptions. On the contrary, small embedding dimensions encompass few ordinal schemes and could be not suitable for capturing time series patterns presenting fast dynamics [34]. Both complexity measures were thereby capable of properly characterizing hand gestures from forearm sEMG data, but a direct comparison between FEn and PEn is hardly feasible, due to their different ranges of values and also considering the different computational logic used for assessing complexity, the first being related to the geometrical distance between embedding vectors [16] and the second being based on the permutation patterns of ordinal symbolic schemes [30; 32]. A possible advantage of the PEn can be found in its dependence upon a single parameter, while for the FEn three parameters need to be set in advance. The latter aspect can be worthwhile from the experimenter's point of view, easing the initial settings of the measure, but it could potentially limit the application of PEn to biological time-series, where the heterogeneous nature of each different signal [16; 31; 46] may require more refined parameter choices, such as those provided by the FEn. It is worth noticing, however, that PEn could present a significant advantage in terms of computational load, which is far lower than that of the FEn: for a white noise time series of length \(N=10^{4}\) and an embedding dimension of three for both FEn and PEn, the former takes 4.61 s to be computed while the latter only 0.08 s. Despite this aspect depending also on the particular implementation and hardware characteristics, it could be valuable in view of using such a complexity measure for real time control of robotic arms or upper limb prostheses [11].