# Improving Real-Time Position Estimation Using Correlated Noise Models +
## 5 Results  
We analyse the predictions of the methods described in Section 3. Figure 5 shows the estimates for GPS data collected in Port Douglas. We note that this dataset is independent of the Dunedin datasets on which the model parameters were based. Port Douglas is a representative example, and the discussion below could be repeated for the other locations as well. The uncertainty of the unprocessed location data is well captured by the Gaussian prediction obtained from the OU stationary distribution. Taking into account the reported DOP parameters led to small modifications only. Although occasional large spikes did appear in the uncertainties, these did not seem to correspond to large deviations in the position data. The time-series of the log scores of these predictions with and without DOP are very similar; cf. Figure 6d-f. Figure 7a,b shows the time-averaged log scores, \(\overline{S}\), for both methods to be very similar, which holds for all the locations tested. The raw location data jump around the correct value, similarly to the marginals from the UKF inference obtained assuming a Brownian model. Due to the process noise, the Brownian model does not converge to the correct value. On the other hand, the position marginals from the UKF augmented with OU noise converge to the \"true\" value (see Figure 5d-f). The improvement in performance is supported by the log-scores. The posteriors produced from the OU noise score better by three orders of magnitude (see Figure 6). It is clear that the KF with the OU noise model is the superior method and the only method to produce negative scores, which continue to decrease with time. Figure 8 shows the scores for all the time-series from all locations averaged over the run-time. With respect to this average score, all KFs with OU noise outperform all other models using iid Gaussian noise models and all inferences using the non-learning method. The single exception is the altitude measurement for Thursday Island. This discrepancy is due to mischaracterised noise; \(\sigma^{2}\) in this run was much larger than the one calculated from the Dunedin dataset. We thus have doubled \(\sigma^{2}\) in the OU model for all the inferences. This manual intervention brought \(\sigma\) closer to the fitted value for the Thursday Island datasets, having negligible effect on the other inferences. We, therefore, suggest that extending the UKF to simultaneously estimate \(\sigma^{2}\) during the inference process, in an embedded calculation, could allow the algorithm to adjust automatically to the variation of \(\sigma^{2}\) with time and location. As expected based on Figures 3 and 4, the inferences of altitude are worse than those of the latitude and longitude. We believe that the otherwise well-characterised noise provides close to optimal quantification of uncertainty for the altitude coordinate, especially if one takes into account the intrinsic character of the altitude time-series due to the relative position of the satellite constellation. We include for completeness the \(\theta\) marginals in Figure 9. The \(\theta\) estimates from the Bayesian filtering have the same orders of magnitude as the fitted parameters (see Figure 4), with the \(\theta\) for the altitude coordinate smaller than the others, as expected. However, the \(\theta\) for the longitude is larger than that for the latitude, which is not expected from the fitted version--it is perhaps an artefact of the imperfect model fit. Figure 5: (**a–c**) The residuals are plotted for latitude (**a**), longitude (**b**) and (**c**) altitude as measured by a stationary receiver at Port Douglas. (**d–f**) The time-series of the inferred coordinates by an unscented Kalman filter algorithm equipped with OU noise model. The dataset plotted in (**a–c**) is used by this algorithm. Shaded regions represent the 90% credible intervals. (**g–i**) The same as the middle row, but using a Brownian model with process noise \(\Sigma_{d}=1\,\mathrm{m}^{2}\) within the UKF.