**2.4.2 The circumstance of combination of symmetric and asymmetric DTW**

The essence of DTW is to match the pairs of two trajectories on synchronization. At first, on symmetric DTW algorithm, the optimal path is reconstructed following above 3 constraints and Eq.5,6. Aligning points of *Bi* with *BREF* on asymmetric synchronization, some statuses would appear:

(a) Some point of *Bi* may be copied multiply, because it matches several points of *BREF*;

(b) Some point of *Bi* may be matched with the point in various time index of *BREF*, which means it will be transferred after synchronization;

(c) More than one point of *Bi* may be averaged to a point that will be aligned with the particular point of *BREF*, because they are aligned with only one point of *BREF* in symmetric DTW algorithm. Although some local feature of points may be smoothed, it is proved that ensure that all *Bi* after asymmetric operation have the same duration *bREF*.
