Figure 6.

Simulation results showing the variance of the combined estimate when the estimates undergo a random walk. The abscissa in each panel is the visual calibration proportion (p_V) and the ordinate is the visual combination weight (w_V). Color represents the variance of the combined estimate over time, dark red representing the smallest variance. The visual and haptic estimates were characterized by random walks with drifts of d_V and d_H. The recalibration rate was set with a Kalman filter. The estimator reliability ratio (r_V:r_H) was 1:3, as in the simulation in Figure 5. The top, middle, and bottom rows show the results when the drift magnitude, d_V + d_H, was large (${\sigma }_V^2+{\sigma }_H^2$), medium [${\sigma }_V^2+{\sigma }_H^2\times {10}^{-2}$], and small [${\sigma }_V^2+{\sigma }_H^2\times {10}^{-4}$], respectively. The two columns on the left show the results when the discrepancy introduced between the visual and haptic estimates was large($5{\sigma }_V^2$),and the two columns on the right show the results when the introduced discrepancy was small ($1∕5{\sigma }_V^2$). The first and third columns display the results when the drift ratio (d_V:d_H) was 3:1 and the second and fourth columns the results when the ratio was 1:3. The dashed yellow circles mark the predictions if minimum variance were determined by the reliability-based model (i.e., by the ratio of estimator reliabilities r_V:r_H); they are always at (0.75, 0.25) because the reliability ratio was always 1:3. The dashed blue circles represent the predictions if the minimum variance were determined only by the drift ratio d_V:d_H. The white circles indicate the calibration proportions and combination weights that yielded the combined estimate with the lowest variance.