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Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI
Fig 6
Model evaluation on experimental data.
A. Association between KL divergence measures on observation (KLx) vs. latent space (KLz) for the Lorenz system; y-axis displayed in log-scale. B. Association between 
(Eq 11; in log scale) and correlation between generated and inferred state series for models with inputs (top, displayed in shades of blue for M = 1…10), and models without inputs (bottom, displayed in shades of red for M = 1…10). C. Distributions of 
(y-axis) in an experimental sample of n = 26 subjects for different latent state dimensions (x-axis), for models including (top) or excluding (bottom) external inputs. D. Mean squared error (MSE) between generated and true observations for the PLRNN-BOLD-SSM (squares) and the LDS-BOLD-SSM (triangles) as a function of ahead-prediction step for models including (left) or excluding (right) external inputs. The PLRNN-BOLD-SSM starts to robustly outperform the LDS-BOLD-SSM for predictions of observations more than about 3 time steps ahead, the latter in contrast to the former exhibiting a strongly nonlinear rise in prediction errors from that time step onward. The LDS-BOLD-SSM also does not seem to profit as much from increasing the latent state dimensionality. E. Same as D for the MSE between generated and inferred states as a function of ahead-prediction step, showing that the comparatively sharp rise in prediction errors for the LDS-BOLD-SSM in contrast to the PLRNN-BOLD-SSM is accompanied by a sharp increase in the discrepancy between generated and inferred state trajectories after the 3rd prediction step. Globally unstable system estimates were removed from D and E.
doi: https://doi.org/10.1371/journal.pcbi.1007263.g006