# Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data
## 4 Modelling the Noise
### Ornstein-Uhlenbeck Model  
In this section, we consider a stochastic differential equation (SDE) rather than a discrete stochastic process. Stochastic partial differential equations are a natural choice to model dynamical systems subject to random fluctuation. For an introduction to SDEs, we point to [PERSON] [47] and [PERSON]'s [48] excellent monographs. The scatter-plots in Figure 1 resemble a random walk on a phenomenological level. However, in a Brownian motion, the variance of the position increases with time, which we do not observe here. On the contrary, we observe (and expect) that the GPS reading will return to the true location even if it deviates from it momentarily, i.e., it is mean-reverting. Furthermore, Figure 3 shows that the correlation function decreases exponentially. These properties suggest an Ornstein-Uhlenbeck (OU) process [49], defined by the SDE  
\[dX_{t}=\theta(\mu-X_{t})dt+\sigma dW_{t}, \tag{6}\]  
where \(\theta\), \(\mu\), and \(\sigma\) are real parameters, with \(\theta,\sigma>0\). For historical and physical reasons, the parameters \(\theta\) and \(\sigma\) are called the drift and diffusion coefficients, respectively. The fluctuation part, \(W_{t}\sim\mathcal{N}(0,t)\), is a Wiener process. The same process can also be represented as  
\[X_{t}=X_{0}e^{-\theta t}+\mu\left(1-e^{-\theta t}\right)+\sigma e^{-\theta t} W_{(e^{-2\theta t}-1)/2\theta}.\]  
The first term corresponds to the long-term but diminishing influence of the initial observation. The second reaches the steady value \(\mu\) exponentially fast, while the third describes a scaled and time-transformed random fluctuation. The expectation value at time \(t\) is \(\mathbb{E}(X_{t})=\mu\left(1-e^{-\theta t}\right)+e^{-\theta t}\mathbb{E}(X_ {0})\approx\mu\). One may simplify (6) by eliminating \(\mu\) with the \(X_{t}\mapsto X_{t}-\mu\) transformation, but we keep \(\mu\) explicit, as it is the 'true' position of the GPS device. The parameter \(\theta\) sets the scale of the autocorrelation function \(\rho(h)\), while \(\sigma\) determines the noise between measurements. The OU process is the continuous-time generalisation of an AR(1) process. It has several desirable properties: (a) It is mean-reverting for \(\theta>0\), (b) it has finite variance for all \(t\geq 0\) contrary to Brownian motion, (c) its analytic covariance \(\mathrm{cov}(X_{t},X_{u})\sim e^{-\theta|t-u|}\) permits a long autocorrelation time, (d) the autocorrelation function is insensitive to the sampling rate, and (e) it approaches Gaussian white noise as \(\theta\rightarrow\infty\). More precisely, the OU process is the only non-trivial, mean-reverting, Gauss-Markov process, i.e., whose transitions are memory-less, and it thus allows predictions of the future state based on thecurrent state [50]. This does not prohibit more sophisticated models for GPS time series, but these models will lack at least one of the properties listed above. To model our observations as an Ornstein-Uhlenbeck process, we must estimate \(\theta\), \(\mu\), and \(\sigma\) using maximum likelihood estimation [51]. [PERSON] and [PERSON] [52] provided a detailed derivation of estimators for these parameters, and [PERSON] and [PERSON] [53] showed that for time-equidistant observations, the maximum likelihood estimators are consistent and asymptotically normal, irrespective of the sampling time-step, \(\Delta t\). The estimators for \(\theta\), \(\mu\), and \(\sigma\) require \(\sim 2n\) operations each. Therefore, the overall parameter estimation takes \(O(6n)\) operations.