with Jl, 0 = r−1, where (t−Jl, k)+ = u+ is defined as u+ = u if u ≥ 0. A modified Akaike’s information criterion (AICc), derived by Hurvich and Tasi66 with c standing for the second order correction67, is employed to select an optimal numbers of break and join points and their locations, and the optimal model can be obtained by minimizing AICc.