# Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data
## Appendix A Short Description of the QQ-Plot Method for Testing Normality  
Before constructing non-Gaussian noise models, it is desirable to show that the distribution of noise is significantly different from a Gaussian distribution ([PERSON], 1994). We thus carried out three normality tests for each coordinate (latitude, longitude, and altitude): the d'Agostino-Pearson test ([PERSON], 1994), the Lilliefors test ([PERSON], 1994), and the graphical QQ-plot method ([PERSON], 1994). In Section 2, we demonstrated the deviation of the residuals from a Gaussian distribution. Here, a short description of the QQ-plot method is provided.  
The central concept of the QQ-plot method is that of the quantiles, a measurable functional of the empirical distribution, such as the sample mean and sample median. The empirical distribution function, \(F_{n}\), of a sample of size \(n\), is defined as \(F_{n}(x)=P_{n}^{*}((-\infty,x))\), or, in words, the ratio of elements of the sample that are less than \(x\). The symbol \(*\) means that the probability is estimated by the observed frequency. Assuming that \(F_{n}(x)\) converges in probability to a distribution function \(F(x)\)as \(n\to\infty\), we can define the quantile, \(\zeta_{p}\), of order \(p\) as \(\zeta(p)=\sup(x\,|\,F(x)\leq p)\). Here, the usage of \(\sup\) guarantees that this definition remains meaningful even if \(F(x)\) has points of discontinuity. As a function of \(p\), the quantile \(\zeta(p)\) is the inverse function of \(F(x)\). A widely used example of a quantile is the median, which is simply \(\zeta(\frac{1}{2})\), i.e., the value \(\hat{x}\) of the sample for which half of the observations are smaller than \(\hat{x}\), and half of them are higher than \(\hat{x}\). Similarly, we may divide up the entire range of the random variable into segments in a way that the number of samples belonging into each segment are the same. In the sample quantile \(\zeta_{n}^{*}(p)=x_{\ell}\), where \(\ell=[np]+1\), \(x_{k}\) is equal to the \(k\)-th smallest value of the sample.  
The sample quantiles provide us a graphical approach to check whether the sample comes from a given probability distribution, \(G(x)\). Let us assume that \(X_{1}\), \(X_{2}\),...are independent, identically distributed random variables, with \(E[X]=0\) and \(\operatorname{Var}[X]=\sigma^{2}<\infty\). Even if the observed values do not satisfy these mild conditions, after normalisation, the two conditions can be fulfilled. Let \(G\) denote the distribution function of the standard normal distribution, while \(F_{n}\) stands for the empirical distribution function of \(S_{n}^{*}\), where  
\[S_{n}^{*}=\frac{1}{\sqrt{n\sigma^{2}}}\left(X_{1}+X_{2}+\dots\right).\]  
One measure of whether the sample \(\{x_{1},x_{2},\dots\}\) comes from a standard normal distribution is the 'closeness' of \(F_{n}\) and \(G\). The closeness is visually represented in a two-dimensional plot, where the abscissa is the \(t\)-th quantile, \(\zeta(t)\), while the ordinate is the \(t\)-th sample quantile, \(\zeta^{*}(t)\). The closer they are, the closer the points are to the diagonal. The plot of points \((\zeta(t),\zeta^{*}(t))\) for \(t\in\mathbb{R}\) is called the quantile-quantile plot, or, in short, the QQ-plot.