# Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data
## 6 Discussion  
The use of Gaussian white noise models is very common in sensor fusion algorithms that incorporate GPS data ([PERSON], 1994). We have shown here that the assumption of uncorrelated noise does not capture important features of real-world GPS data. Fitting a Gaussian white noise model results in an artificially large variance and does not account for the observed autocorrelations. Analysis of location data shows that the error between the position measurement and the 'true' location shows strong autocorrelation up to approximately an hour. We proposed stochastic noise models that take into account such autocorrelation. Estimating the model parameters for the Ornstein-Uhlenbeck and \begin{table}
\begin{tabular}{l r r r r r} \hline \hline
**Thursday Island** & Gauss & AR(9) & MA(1) & MA(3) & ARMA(1, 1) & OU \\ \hline Latitude & 368,278 & \(-\)419,119 & 240,593 & & \(-\)391,014 & \(-\)390,082 \\ Longitude & 490,408 & \(-\)358,983 & 369,403 & & \(-\)201,441 & \(-\)212,528 \\ Altitude & 616,549 & \(-\)151,204 & 488,436 & & \(-\)106,982 & \(-\)107,027 \\ \hline
**Port Douglas** & & & & & ARMA(1, 1) & \\ \hline Latitude & 1,080,721 & \(-\)1,104,629 & 749,275 & & \(-\)1,048,908 & \(-\)1,031,940 \\ Longitude & 1,254,838 & \(-\)415,461 & 927,860 & & \(-\)887,968 & \(-\)1,016,929 \\ Altitude & 1,445,244 & \(-\)844,480 & 1,112,076 & & \(-\)924,883 & \(-\)659,902 \\ \hline
**North Stradbroke** & & & & ARMA(2,2) & \\ \hline Latitude & 616,991 & \(-\)846,881 & 417,772 & 160,425 & \(-\)54,584 & 4099 \\ Longitude & 552,469 & \(-\)847,589 & 369,864 & 138,410 & 48,340 & 5140 \\ Altitude & 991,469 & \(-\)638,245 & 783,784 & 549,365 & 280,719 & 401,359 \\ \hline
**Dunedin** & & & & ARMA(2,2) & OU \\ \hline Latitude & 2,827,919 & \(-\)4,855,973 & & & \(-\)4,762,660 \\ Longitude & 2,202,644 & \(-\)5,333,179 & 945,600 & & \(-\)5,343,304 & \(-\)5,292,592 \\ Altitude & 4,380,261 & \(-\)4,351,240 & 3,112,342 & & \(-\)4,360,823 & \(-\)4,114,385 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: AIC values (rounded) of noise models for GPS time series. Figure 7: The Akaike’s Information Criterion (AIC) score for autoregressive (AR) models of increasing order using the Dunedin GPS altitude dataset recorded on 15 August 2014, scaled relative to the AIC score for \(p=1\). AR models based on likelihood maximisation is straightforward where either the analytic solution is known or the numerical algorithm is well known and can be performed efficiently. Conversely, estimating the parameters for the MA or ARMA models is more involved and computationally expensive. If the stochastic process is known to be non-stationary, then the MA model is non-invertible, and further measures are required to estimate the model parameters. Here, we mention an alternative approach for supporting the suitability of novel noise models. One can calculate the second moment of the residuals, \(\langle E_{t}^{2}\rangle\), using single-path or multipath ensemble averaging for the observed time series and the one generated by the model. However, this approach is left for future analysis. Autoregressive processes of moderate order provide a sufficient noise model for the signal returned by the commercial GPS device, and the extra computational cost compared to that of a naive Gaussian white noise model is negligible. Increasing the order of the autoregressive process results in lower AIC values, indicating that, from the purely statistical perspective, the likelihood maximisation outweighs the increase in complexity. However, for the less accurate GPS units, the higher-order AR processes are only slightly more likely than the first-order AR(1) process. For an AR(1) process, we would advise a discretised Ornstein-Uhlenbeck process, which brings several advantages. First, the sampling period of the device is explicit in its formulation; thus, variable sampling periods can easily be fit into this description.