Constraining least-square estimates with controlled dynamics enables robust parameter estimation in chaotic models.

A: Squared-difference between 10D Lorenz96 model trajectory x and measured data y, for various values of forcing parameter F, assuming that model was integrated forward from the true initial point. Only states x1, x4, x7, x10 were measured. The global minimum of C(F) (dotted line; F=8), resides in an extremely narrow basin of attraction within a highly nonconvex cost surface. B: Same as A, but for controlled dynamics (u = 4 and u = 15 for middle and right plots, respectively). Higher controls provide a far smoother cost function around the global minimum. C: 2D projection of the high-dimensional DSPE cost surface, along F in one dimension and the line Ul(tn)≡u, ∀l, n in the other. Here, xd(tn) were fixed at the values generated by integrating the model forward from its true initial state for the given F. D: Estimated trajectories of x8, using uncontrolled least squares (top; green) and DSPE (bottom; red). Black line: true trajectory. Shaded regions: SD of estimate across 100 random initializations of the optimization. E: Histogram of parameter estimates across 100 initializations of the optimization, for least squares (green) and DSPE (red).</p

Estimation of states and linear conductances in Morris-Lecar model.

A: Stimulating current used to estimate parameters and compare forward predictions. B: Most accurate (top) and least accurate (bottom) of 25 estimations using OC-DSPE, σ = 2 mV. Black: true voltage; blue: noisy observations; red: forward prediction using estimated parameters. C: Heatmap of voltage over time for all 25 runs, with the run giving lowest (highest) prediction error on top (bottom). D: Estimated conductances corresponding to state estimations in C. E: Analogue of B-D, for higher measurement noise σ = 10 mV. F: Estimations of the linear conductances among the Q = 25 runs using OC-DSPE (black), DSPE (red) and constrained least squares (blue), for 2 different levels of measurement noise (σ = 2, 10 mV, respectively). G: Top trace: Stimulus used for prediction (note timescale is from 100 to 200 ms) for the 3 methods in F. Bottom left 3 traces: predictions using optimally estimated parameters among the Q = 25 runs in F, using the 3 different estimation methods, for σ = 2 mV measurement noise. Black, red, blue: OC-DSPE, DSPE, and least squares predictions, respectively. Grey: Predicted trajectory using the true parameters (note this is visually indistinguishable from the accurate predictions). Bottom right 3 traces: Same, for σ = 10 mV measurement noise.</p