Algorithm 1 Penalized EM Algorithm for the Robust Absorbing-State HMM.

Require: Observed trajectories ${\left\{y_{it}\right\}}_{t=1}^{T_i}$ for $i=1,\dots ,N$; number of states K (state K absorbing); ridge penalties ${\left\{{\lambda }_k\right\}}_{k=1}^{K-1}$.   1:Initialize parameters ${\mathrm{\Theta }}^{\left(0\right)}=\{P^{\left(0\right)},({\eta }_{0,1}^{\left(0\right)},{\eta }_{1,1}^{\left(0\right)},{\sigma }_1^{2\left(0\right)}),\dots ,({\eta }_{0,K}^{\left(0\right)},{\eta }_{1,K}^{\left(0\right)},{\sigma }_K^{2\left(0\right)})\}$ (e.g., simple-failure $P^{\left(0\right)}$ and a global linear regression for emissions).   2:repeat   3:      E-step (forward–backward).   4:      Run the scaled forward–backward recursion to obtain ${\gamma }_{it}\left(k\right)$ for $t=1,\dots ,T_i$ and ${\xi }_{it}(k,\ell )$ for $t=1,\dots ,T_i-1$ (see (6)–(11)).   5:      M-step: transition matrix.   6:       Update $${\widehat{p}}_{k\ell }={\displaystyle \frac{{\displaystyle \sum _{i=1}^N\sum _{t=1}^{T_i-1}{\xi }_{it}(k,\ell )}}{{\displaystyle \sum _{i=1}^N\sum _{t=1}^{T_i-1}{\gamma }_{it}\left(k\right)}}},k,\ell =1,\dots ,K.$$   7:      M-step: emission parameters (ridge-regularized).   8:      Select ${\lambda }_k$ by weighted K-fold cross-validation under the current weights ${\gamma }_{it}\left(k\right)$.   9:      for each transient state $k=1,\dots ,K-1$ do 10:             Update $({\eta }_{0,k},{\eta }_{1,k})$ by slope-only ridge regression: $$({\eta }_{0,k},{\eta }_{1,k})=\arg \underset{{\eta }_0,{\eta }_1}{\min }\sum _{i=1}^N\sum _{t=1}^{T_i}{\gamma }_{it}\left(k\right){\left(y_{it}-{\eta }_0-{\eta }_{1}t\right)}^2+\frac{1}{2}{\lambda }_k{\eta }_1^2.$$ 11:            Update ${\sigma }_k^2$ using a Huber-type robust scale estimator based on the weighted residuals. 12:      end for 13:until convergence of the log-likelihood and parameter updates.