Algorithm 1 Discrete curvature estimation for time series

Initialize a maximum length of the searching vector $k_m$ and a small angle variation $\Delta$, within the range of which a sequence of data points can be considered as a DSS; Find the maximum length of the backward searching vector for data point $p_i$. For the data points from $p_{i-k}$ to $p_{i-1}$, compute and check if all their centered slope angle variations satisfy $-\Delta \le {\delta }_{j,2}\le \Delta (i-k\le j\le i-1)$. If the condition does not meet, $k_b=k$. If $k\ge k_m$, $k_b=k_m$; Find the maximum length of the forward searching vector for data point $p_i$. For the data points from $p_{i+1}$ to $p_{i+k}$, compute and check if all their centered slope angle variations satisfy $-\Delta \le {\delta }_{j,2}\le \Delta (i+1\le j\le i+k)$. If the condition does not meet, $k_f=k$. If $k\ge k_m$, $k_b=k_m$; Compute the length of the backward and forward DSS, $L_b$, $L_f$, and the slope angles of the two DSS, ${\theta }_b$, ${\theta }_f$. Compute the curvature as $C_i=\frac{(L_b+L_f)({\theta }_b+{\theta }_f)}{4L_b{L}_f}$.