Algorithm 1.

CT-HMM Parameter learning (Soft/Hard)

1:	Input: data O = (o0, …, oV ) and T = (t0, …, tV ), state set S, edge set L, initial guess of Q
2:	Output: transition rate matrix Q = (qij)
3:	Find all distinct time intervals tΔ, Δ = 1, …, r, from T
4:	Compute P(tΔ) = eQtΔ for each tΔ
5:	repeat
6:	Compute p(v, k, l) = p(s(tv) = k, s(tv+1) = l∣O, T,Q) for all v, and the complete/state-  optimized data likelihood l by using Forward-Backward (soft) or Viterbi (hard)
7:	Create soft count table C(Δ, k, l) from p(v, k, l) by summing prob. from visits of same tΔ
8:	Use Expm, Unif or Eigen method to compute $\mathbb{E}[n_{\mathit{ij}}\mid O,T,Q]$ and $\mathbb{E}[{\tau }_i\mid O,T,Q]$
9:	Update $q_{\mathit{ij}}=\frac{\mathbb{E}[n_{\mathit{ij}}\mid O,T,Q]}{\mathbb{E}[{\tau }_i\mid O,T,Q]}$, and qii = −Σi≠j qij
10:	until likelihood l converges