# Improving Real-Time Position Estimation Using Correlated Noise Models +
## 1 Introduction  
Global Positioning System receivers output a time-series of position measurements, but this signal suffers from errors due to many reasons, such as atmospheric and multipath effects [1, 2, 3] or insufficient coverage of satellite constellations. Rather than providing a full quantification of uncertainty, GPS devices generally provide Dilution-Of-Precision (DOP) variables, dimensionless multiplicative factors of the position error, designed to flag bad satellite coverage. In this paper, we developalgorithms that issue real-time position predictions with simultaneous uncertainty quantification and evaluate their performance. Embedding these algorithms in devices that use single frequency GPS will allow such devices to supply improved position estimates and real-time estimates of their uncertainty. Additionally, we will show in Section 5 that the DOP variables provide a negligible improvement to the error estimation. Kalman filters [4] are commonly used to estimate the state of a system from a time-series of measurements corrupted with uncorrelated Gaussian noise. Such filters are promising candidates to infer position from GPS measurements and to characterise the uncertainty of the position estimates. However, previous studies have shown GPS position errors to be temporally correlated, due to environmental processes such as multipath and atmospheric effects [1; 2]. These processes are too complicated either to model or to approximate in real time; thus, we opted to capture and then model the noise characteristics. A recent study found several correlated noise models [5; 6; 7], including the well-known Ornstein-Uhlenbeck (OU) model, to be a superior representation of GPS noise to the independent and identically distributed (iid) Gaussian noise model [8]. The correlated noise necessitates an extension to the ordinary KF. Kalman filters have been extended by shaping filters to deal with correlated noise [1; 2; 9; 10]. Such augmented filters have allowed improved measurement in the difference in position between two dual-frequency GPS receivers, one of which has an adjustable height [1; 2]. The differential signal studied in References [1; 2] still had correlated noise, but with a different character than that of a stand-alone single-frequency receiver, including a much reduced noise amplitude and shorter coherence time. The shaping filter included a noise variable governed by an Ornstein-Uhlenbeck (OU) process [11]. The authors found the inclusion of OU noise to be beneficial for detecting the changes in distance between the receivers; however, the study did not address the position uncertainty, which is the focus of the current paper. Since the KF is a class of sequential Bayesian inference algorithms, the uncertainty in the position measurement at each time-step is given by the shape of the posterior distribution. The quality of probability forecasts is commonly compared using scoring rules [12; 13; 14; 15; 16], which allow the quantification of probability distributions based on both their accuracy (how closely centred near the \"true\" value) and precision (how tightly they are distributed). In order to compare the posteriors from different algorithms, we use the log score [15; 16]. The log scores allow us to evaluate the effect of the noise model on uncertainty quantification of position inferences on GPS time-series datasets and quantify the improvement gained by using the OU noise model. We developed stochastic algorithms with different noise models to predict the position of a stand-alone single-frequency GPS receiver and quantified the uncertainty of its position. We evaluated the performance of the KF algorithms, as well as a basic, non-learning method, which uses no filtering. Based on a scoring method, developed for comparing probability distributions, we ranked the algorithms for performance and assessed the importance of the dilution-of-precision from incomplete satellite coverage. The OU noise-model augmented with a KF seemed superior than all other models discussed.