Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI
Fig 2
Illustration of DS reconstruction measures defined in state space () vs. on the time series (mean squared error; MSE).
A. Two noise-free time series from the Lorenz equations started from slightly different initial conditions. Although initially the two time series (blue and yellow) stay closely together (low MSE), they then quickly diverge yielding a very large discrepancy in terms of the MSE, although truly they come from the very same system with the very same parameters. These problems will be aggravated once noise is added to the system and initial conditions are not tightly matched (as almost impossible for systems observed empirically), rendering any measure based on direct matching between time series a relatively poor choice for assessing dynamical systems reconstruction except for a couple of initial time steps. B. Example time series and state spaces from trained PLRNN-SSMs which capture the chaotic structure of the Lorenz attractor quite well (left) or produce rather a simple limit cycle but not chaos (right). The dynamical reconstruction quality is correctly indicated by (low on the left but high on the right), while the MSE between true (grey) and generated (orange) time series, on the contrary, would wrongly suggest that the right reconstruction (MSE = 1.4) is better than the one on the left (MSE = 2.48).