# Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data
## 3 Analysis of Observations
### Stochastic Processes  
The simplest non-trivial stochastic process can be formulated as the sum of the true position, \(\xi_{0}\), and an additive noise term:  
\[X_{t}=\xi_{0}+E_{t}. \tag{1}\]  
The measurement error, \(E_{t}\), is often considered to be a Gaussian white noise with zero mean and variance \(\sigma^{2}\). The problems of fitting Gaussian noise models to GPS measurements have been recognised by [27; 28], and two models were proposed to describe the measurements more accurately: a moving average process (MA) and an autoregressive model (AR). We check whether adjacent recordings can be considered independent random variables using the empirical autocorrelation function [29; 30; 31; 32; 33], defined as  
\[\widehat{\rho}(t,s)=\frac{\text{cov}(X_{t},X_{s})}{\sigma_{t}\sigma_{s}}=\frac {\mathbb{E}\left[\left(X_{t}-\mu_{t}\right)\left(X_{s}-\mu_{s}\right)\right]} {\sigma_{t}\sigma_{s}}, \tag{2}\]  
where \(\mu_{r}\) and \(\sigma_{r}\) are the expectation value and variance of the random process at step \(r\), respectively. The process \(\{X_{t}\}\) is stationary if \(\mu_{t}\) and \(\sigma_{t}\) are both time-independent; in such a case, the autocorrelation function depends only on the time difference \(h=s-t\) (_lag_) between its arguments and not explicitly on the arguments themselves. Henceforth, we assume stationarity; thus, \(\rho(t,s)=\rho(h)\) while \(\mu_{t}=\mu\)  
Figure 2: The main figure shows the QQ-plot of the residual of latitude data (measured in metres) collected in Dunedin. The inset depicts the histogram of the same data. The \(p\)-values of two normality tests are also given in the top left corner. Based on this evidence, the data do not support the assumption that the residual follows a Gaussian distribution. and \(\sigma_{t}=\sigma\) both become constants, which are calculated by time-averaging. Furthermore, we consider the empirical partial autocorrelation function calculated using the Yule-Walker method [32; 34] and defined as  
\[\widehat{\rho}_{\rm P}(h)=\frac{\text{cov}(X_{t},X_{t+h}|X_{t+1},X_{t+2},\cdots,X _{t+h-1})}{\sigma_{t}\sigma_{t+h}}, \tag{3}\]  
which provides the amount of correlation between a variable and a lag of itself that is not explained by correlations at all lower-order lags. As an example, Figure 3 shows the autocorrelation of a typical time series. The autocorrelation function decays slowly. Fitting an exponential decay to the first half-hour dataset provides a characteristic time estimate of \(\tau=20.98\pm 0.02\) minutes, i.e., it takes approximately an hour (\(\sim 3\tau\)) to reach statistical independence. Using the reported position for periods much shorter than \(\tau\) would lead to a biased estimate. We wish to draw the reader's attention to the fact that long-term correlations usually imply a power-law decay of correlation, while our fit assumes an exponential correlation. While exponential decay of correlation is relatively fast, compared to power-law decay, it seems to have an observable effect on location time series, and its inclusion in modelling may provide better positioning algorithms in the future. The theoretical autocorrelation function of Gaussian white noise is the Dirac-delta, \(\rho(t,s)\propto\delta(t-s)\). Assuming this model, one may test for statistical independence of \(X_{t}\) and \(X_{t+h}\) by checking whether the autocorrelation function is different from zero, i.e., whether it falls within the confidence interval around \(\widehat{\mu}=0\)[35; 36]. \[-\frac{z\big{(}1-\frac{\alpha}{2}\big{)}}{\sqrt{n}}\,\widehat{\sigma}\leq \widehat{\rho}(h)\leq\frac{z\big{(}1-\frac{\alpha}{2}\big{)}}{\sqrt{n}}\, \widehat{\sigma},\]  
where \(z(x)\) is the cumulative distribution function of the standard normal probability distribution, \(\alpha\) is the required level of significance, \(\widehat{\sigma}\) denotes the sample standard deviation, and \(n\) is the number of measurements.