Fig. 3. Filtering with quasi-continuous time angular observations.

a) Single quasi-continuous time update step (68) with angular observation z_t, where the length of the vector indicates observation reliability $\alpha \left({\kappa }_{z}dt\right)$. The update step for Bayesian inference on the circle is equivalent to a vector addition in the 2D plane. The lower panel demonstrates that a conflicting observation leads to a decreased certainty of the estimate directly after the update, corresponding to a shorter vector. b) Empirical precision ${\widehat{r}}_t$ (upper panel) and estimated precision r_t of the circular Kalman filter (circKF) and a Gaussian projection filter (Gauss filter), when compared to a particle filter, for different values of the observation precision κ_z at time $T=10{\kappa }_{\phi }^{-1}$. Parameters were ${\kappa }_{\phi }=1$ and ${\kappa }_u=1$, times are in units of ${\kappa }_{\phi }^{-1}$. c) Estimated versus empirical precision up to $T=10{\kappa }_{\phi }^{-1}$ for the different filters at ${\kappa }_z=10$. The precisions shown in b) and c) are averages across 5000 simulation runs.