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Reconstructing Mammalian Sleep Dynamics with Data Assimilation
Figure 5
Parameter estimation with multiple shooting method for reconstruction of DB model from measurement of 
and 
with unknown value for parameter 
.
Parameter estimation is performed by minimizing the divergence between the UKF reconstructed dynamics and short model-generated trajectories that originate on the reconstructed trajectories. To sample the full state space, each step of this minimization averages this divergence over time windows longer than the cycle time of the dynamics. Here we use half hour windows, with 80% overlap. A) Convergence of the estimated parameter 
to the true value. B) Trajectories for the short model generated (magenta), reconstructed (red), and true (black) 
dynamics for different periods of the convergence of 
. Note that initially, for 
significantly different than the true value, the short trajectories diverge quickly from the reconstructed values, and the reconstructed values of of 
are different from the true values. When 
approaches the true value, both short model-generated and reconstructed trajectories approach the true values. C) Reconstruction metric 
computed for each data assimilation window for three of the variables. As a reference point, the reconstruction metric for the original noisy observation of 
is shown in blue. Note that although the parameter estimation essentially optimizes short model generated forecasts, it has the effect of optimizing hidden variable reconstruction.
doi: https://doi.org/10.1371/journal.pcbi.1002788.g005