Before revision:  Compared to the traditional optimization method used in  LPPLS model,  the covariance matrix adaptation evolution strategy (CMA-ES) may have a significantly lower computation cost . The CMA-ES is recommended as an alternative algorithm in the LPPLS model . Moreover, the exponent m does not show a remarkable feature of change when the start day t1 is fixed while the end day t2 is moved towards the actual critical time t_c in the expanding windows. In the LPPLS fitting with expanding windows, the gap (tc -t2) shows a significant decrease when the end day t2 approaches the actual bubble crash time.
After revision:  Compared to the traditional optimization method used in LPPLS model, the covariance matrix adaptation evolution strategy (CMA-ES) may have a significantly lower computation cost , and thus recommend this as a better alternative algorithm for LPPLS model fit. Furthermore, in the LPPLS fitting with expanding windows, the gap (tc -t2) shows a significant decrease when the end day t2 approaches the actual bubble crash time.