## 2 A Novel Filtering Strategy for Reliable Navigation
### Student's t-Distribution for GNSS Measurement Errors  
In the Bayesian framework, the influence of measurement outliers on inferences for estimates, including population means and medians, can be reduced by replacing the normal distribution model by a heavy tailed distribution. Such heavy tailed distributions allow for the possibility of high noise and possibly biased observations. These distributions treat observations far from the regression line as high variance observations, yielding results similar to those obtained by deweighting the outliers [2]. This, to a certain extent, avoids having to reject measurements because of incorrect error modeling. Moreover, in the case of GNSS measurement errors occurring in environments like urban canyons where a significant proportion of measurements could be affected by multipath, a heavy tailed distribution is likely to be more representative of the true errors than the normal distribution, thus leading to more precise test statistics and hence a better FDE.  
The use of a heavy tailed distribution such as Student's _t_-distribution, or simply \(t\) distribution, however, makes the filtering problem intractable as the functional models (process and measurement model) that appear in the form of likelihood function and transition density in the Bayesian framework do not reduce to a closed form solution as in the case of normally distributed errors. Two techniques are possible to implement the filtering algorithm with such a distribution. One option is to use Monte Carlo techniques such as particle filters [3]. However, these techniques are computationally very demanding and are not preferred in most of the commercial products. Hence, a computationally effective option is chosen for the implementation of the filter with \(t\) distribution. This method uses Variational Bayes (VB) framework wherein the \(t\) distribution is approximated as a normal mixture. The VB algorithm, which follows from [4], introduces an auxiliary random variable \(\lambda\) and estimates certain parameters using fixed point iteration as depicted in Algorithm 1.  
```
begin \(m_{k}^{-}\leftarrow\int f(x_{k-1})N(x_{k-1}\mid m_{k-1},P_{k-1})dx_{k-1}\) \(P_{k}^{-}\leftarrow\int(f(x_{k-1})-m_{k}^{-})(f(x_{k-1})-m_{k}^{-})^{T}\,N(x_{ k-1}\mid m_{k-1},P_{k-1})dx_{k-1}+Q_{k}\) \(\overline{\lambda}\leftarrow 1\) for k from 1 to M do \(\mu_{k}\leftarrow\int h(x_{k})N(x_{k}\mid m_{k}^{-},P_{k}^{-})dx_{k}\) \(S_{k}\leftarrow\int(h(x_{k})-\mu_{k})(h(x_{k})-\mu_{k})^{T}\,N(x_{k}\mid m_{k} ^{-},P_{k}^{-})dx_{k}+\frac{1}{\lambda_{k}}R_{k}\) \(C_{k}\leftarrow\int(x_{k}-m_{k}^{-})(h(x_{k})-\mu_{k})^{T}\,N(x_{k}\mid m_{k} ^{-},P_{k}^{-})dx_{k}\) \(K_{k}\gets C_{k}S_{k}^{-1}\) \(m_{k}\gets m_{k}^{-}+K_{k}(y_{k}-\mu_{k})\) \(P_{k}\gets P_{k}^{-}-K_{k}S_{k}K_{k}^{T}\) \(D_{k}\leftarrow\int(y_{k}-h(x_{k}))(y_{k}-h(x_{k}))^{T}\,N(x_{k}\mid m_{k},P_{ k})dx_{k}\) \(\widetilde{\gamma}_{k}\gets tr(D_{k}R_{k}^{-1})\) \(\overline{\lambda}_{k}\leftarrow\frac{g+d}{g+\widetilde{\gamma}_{k}}\) end for end for
```  
**Algorithm 1**VB filter  
For this work, the numerical integrations in the algorithm are computed using the Cubature rule [5]. Despite using the \(t\) distribution for measurement errors, some form of FDE is still needed in order to account for gross errors. Hence, a FDE technique similar to the one used for normal distributed measurement errors has been used. Due to the symmetry of the \(t\) distribution, the test statistic for a local test assumes the same form as for the normal distribution [6], namely:\[\xi=\frac{r_{i}}{\sqrt{(C_{r})_{ii}}}. \tag{1}\]  
The threshold, however, is derived from the \(t\) distribution, \(Th=t(\vartheta,1-\alpha_{t})\), where \(\alpha_{t}\) is the probability of false alarm for the \(t\) distribution and is commonly a design parameter.