# Improving Real-Time Position Estimation Using Correlated Noise Models +
## 2 Data Analysis
### Noise Models  
We studied and evaluated the performance of various noise models for GPS data [8]; specifically, we focused on uncorrelated noise, OU, autoregressive processes (\(x_{t}=-\sum_{k=1}^{p}h_{k}x_{t-k}+E_{t}\)), moving-average processes (\(x_{t}=\mu-\sum_{\ell=0}^{q}g_{\ell}E_{t-\ell}\)) and their combination, an ARMA(\(p=1\), \(q=1\)) model (\(x_{t}=-h_{1}x_{t-1}+g_{0}E_{t}+g_{1}E_{t-1}\)). Here, \(x_{t}\) represents the signal at time-step \(t\), while \(E_{t}\) denotes the noise at the same time-step. The coefficients \(h_{1}\), \(g_{0}\) and \(g_{1}\) are constants specific to the system under investigation. Using the standard Akaike Information Criterion (AIC) [18], we compared these model.  
One of the conclusions of the comparison was that all noise models proved to be better than the uncorrelated noise model. In this paper, we continue our investigations and compare techniques including or omitting correlation in noise models. An OU process is also included in these analyses, because it has the following beneficial properties: mean-reverting, continuous and having only a few parameters; hence, it lends itself to fast embedded computation and sensor fusion applications.  
The OU process is defined by the stochastic differential equation:  
\[dx_{t}=\theta\left(\mu-x_{t}\right)dt+\sigma dW_{t}, \tag{1}\]  
where \(W_{t}\) represents a Wiener process, while \(\theta\), \(\mu\) and \(\sigma\) are parameters to be fitted. The associated Fokker-Planck equation has an analytical solution [11]:  
\[x_{t}\sim\mathcal{N}\left(\mu+\left[x_{0}-\mu\right]e^{-\theta t},\frac{ \sigma^{2}}{2\theta}\left[1-e^{-2\theta t}\right]\right). \tag{2}\]  
Here, \(\mathcal{N}\left(\mu_{n},\sigma_{n}^{2}\right)\) is the Gaussian distribution with mean \(\mu_{n}\) and variance \(\sigma_{n}^{2}\). This probability distribution for \(x_{t}\) is initially a delta function located at \(x=x_{0}\), with the limit, \(\mathcal{N}(\mu,\sigma^{2}/(2\theta))\), as \(t\rightarrow\infty\). The process is mean-reverting for \(\theta>0\); i.e., its limit is a stationary solution to the Fokker-Planck equation with mean \(\mu\), as \(t\rightarrow\infty\). The time-discretised version of (1) for time-step \(\Delta t\) is given by:  
\[x_{t+1}=\mu+\left(x_{t}-\mu\right)\exp\left(-\theta\Delta t\right)+\epsilon_{ t}, \tag{3}\]  
where \(\epsilon_{t}\), the noise term, follows a Gaussian distribution with zero mean and variance \(\frac{\sigma^{2}}{2\theta}\left[1-e^{-2\theta\Delta t}\right]\).  
During data processing, we assumed that the long-term average represents the \"true\" value for a random variable; hence, after subtracting this long-term average, the residual data must have zero mean, \(\mu=0\). Hence, from each residual, we obtained maximum likelihood estimates \(\sigma^{2}\) and \(\theta\). For each coordinate, the logarithms \(\theta\) and \(\sigma^{2}\) were independently modelled, where the coordinate type (latitude, longitude and altitude) was assumed to have a fixed effect, while the geographic location and time have random effects. The geographic location seemed to have no significant effect on \(\theta\), but it weakly influenced \(\sigma^{2}\); though, we believe, that this effect has no practical significance. The type of coordinate, on the other hand, seems to affect both \(\theta\) and \(\sigma^{2}\). Parameter \(\theta\) was statistically significantly smaller for the altitude than for the other two coordinates; simultaneously, \(\sigma^{2}\) was significantly larger for altitude, than for latitude or longitude. These results essentially confirm the conclusions of a visual inspection of Figure 4. Other models: higher order Autoregressive (AR), Moving-Average (MA) and, their combination, ARMA processes [8] are outside the scope of the current paper, but are promising candidates for the extension of this study (see Section 6).