FIG. 9.

Hypnogram derived simultaneous state tracking and multiple parameter fitting. We chose three different parameters: gS,R, gA,LC, and gG,WR for fitting while observing FLC, FVLPO, and FR derived from the hypnogram. Parameter estimation is performed with the multiple-shooting method described in Sec. III C 3 over 20-min long windows, with a 6-min overlap. The trajectories for the short model generated (cyan), reconstructed (red), and true (black) FLC and FVLPO dynamics for different periods of the convergence of the three parameters are shown in the first two panels. We start with random initial values for each parameter and estimate its correct value at each step: gS,R (cyan), gA,LC (magenta), and gG,WR (green). Dashed lines indicate the correct value of the parameters. Note that initially, for parameter values significantly different from the true value, the short trajectories diverge quickly from the reconstructed values, and the reconstructed values of FLC and FVLPO are different from the true values. When the parameters approach their true values, both short model-generated and reconstructed trajectories approach their true values as well and the cost function (CF) approaches zero (black trace). Reconstruction metric ϵi2 is computed for each data assimilation window and is shown in the last two panels for FLC (green) and FVLPO (red). As a reference, the variance of the noisy observations of FLC and FVLPO is also shown in dashed green and red lines. ϵi2 decreases as the parameters and states approach their correct values.