Before revision:  Furthermore, for various fictitious ` present' times t2 before the crashes, we employ a clustering method to group the predicted critical times tc of the LPPLS fits over different time scales, where tc is the most probable time for the ending of the bubble.
After revision:  Furthermore, for various fictitious ` present' times t2 before the crashes, we employ a clustering method to group the predicted critical times tc of the LPPLS fits over different time scales, where t_c is the most probable time for the ending of the bubble.