Figure 1

Multilevel splitting applied to CE phase of MCEM². Using${\widehat{\mathit{\theta }}}^{\left(0\right)}$, we first simulate an ensemble of K trajectories from the initial system state (black circle at t = 0) until time t₁ (red traces). The ending states of the ⌈ρK⌉ trajectories closest to the first observed data point (bold red traces) are sampled with replacement to provide starting states for the next simulation interval. We then simulate a second ensemble of K trajectories starting at time t₁ until reaching t₂. Here, we select the ⌈ρK⌉ trajectories spanning the interval [0, t₂] that are closest to the first and second data points (black circles at times t₁ and t₂) and use them to initiate the third simulation ensemble. We repeat this process until reaching t₄, at which time we compute the first set of parameter estimates${\widehat{\mathit{\theta }}}^{\left(1\right)}$ using the ⌈ρK⌉ trajectories closest to all data points (full length bold red traces). Using${\widehat{\mathit{\theta }}}^{\left(1\right)}$, we begin the process again at t = 0, producing the green traces. Finally, using${\widehat{\mathit{\theta }}}^{\left(2\right)}$ to generate the blue traces, we obtain ⌈ρK⌉ trajectories coinciding exactly with all data points, which we use to compute${\widehat{\mathit{\theta }}}^{\mathrm{CE}}\equiv {\widehat{\mathit{\theta }}}^{\left(3\right)}$.