# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 3 Camouflage Effect of an RB RAIM Detector in Multi-Outlier Mode
### Camouflage Effect of Residuals  
The residual vector is the projection of the error vector under matrix \(\mathbf{S}\). Equation (6) can be expressed as the following equations:  
\[\left[ \begin{array}{c}r_{1}\\ r_{2}\\ \vdots\\ r_{n}\end{array} \right]=\left[ \begin{array}{cccc}S_{11}&S_{12}&\cdots&S_{1n }\\ S_{21}&S_{22}&\cdots&S_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ S_{n1}&S_{n2}&\cdots&S_{nn}\end{array} \right]\cdot\left(\left[ \begin{array}[] {c}\varepsilon_{1}\\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{array} \right]+\left[ \begin{array}{c}b_{1}\\ b_{2}\\ \vdots\\ b_{n}\end{array} \right]\right). \tag{9}\]  
As the residuals depend on the state parameters estimation, the residual vector \(\mathbf{r}\) and the linear observation matrix \(\mathbf{H}\) are also polluted when there are some outliers in observations. An outlier affects the residuals of all observations and the impacts of different outliers are different. The residual of each observation could consist of two parts: one caused by itself, \(r_{i,self}\) and the other one caused by other observations, \(r_{i,others}\). In the case of no outliers, the bias vector \(\mathbf{b}\) is a zero vector and the residual of each pseudo-range could be expressed as:  
\[r_{i}=\mathbf{S}_{i,\cdot}\cdot\epsilon=\mathbf{S}\cdot\epsilon+\sum_{j=1,j \
eq i}^{n}S_{ij}\cdot\varepsilon_{j}=r_{i,self}+r_{i,others} \tag{10}\]  
where \(r_{i,self}=S_{ii}\cdot\varepsilon_{i}\) and \(r_{i,others}=\sum\limits_{j=1,j\
eq i}^{n}S_{ij}\cdot\varepsilon_{j}\). In general, \(r_{i,self}>r_{i,others}\) as \(S_{ii}>S_{ij}\), \(j\
eq i\) in matrix \(\mathbf{S}\). In this case, the presence of \(r_{i,others}\) does not have an obvious effect on the FDE performance of an RB RAIM detector. There are differences when different pseudo-ranges are outliers, which is also known as the leverage effect. Therefore, the studentized residuals are used to replace the residuals. Leverage observations are pseudo-ranges that have a high potential to influence the estimated parameters. The corresponding residual may be unobvious, which may cause masking and swamping under poor geometric conditions. In the case of a single outlier, let the \(g\)th observation be the outlier and the bias vector can be expressed as \(b=\left[ \begin{array}{cccccc}0&\cdots&0&b_{g}&0&\cdots&0\end{array} \right]^{T}\). The residual of each observation can also consist of two parts: one caused by noise, \(r_{i,noise}\) and the other one caused by bias, \(r_{i,bias}\). The residual of the normal pseudo-range \(r_{i}\) could be expressed as:  
\[r_{i}=\sum_{j=1}^{n}S_{ij}\cdot\varepsilon_{j}+S_{ig}\cdot b_{g}=r_{i,noise}+ r_{i,bias} \tag{11}\]The residual of outlier \(r_{g}\) can be expressed as  
\[r_{g}=\sum\limits_{j=1}^{n}S_{gj}\cdot\varepsilon_{j}+S_{gg}\cdot b_{g}=r_{g, noise}+r_{g,bias} \tag{12}\]  
Matrix \(\mathbf{S}\) has the following properties:  
* It is a symmetric matrix, \(\mathbf{S}^{\mathrm{T}}=\mathbf{S}\);
* It is an idempotent matrix, \(\mathbf{S}^{2}=\mathbf{S}\) and \(\mathbf{S}_{i,}\cdot\mathbf{S}_{,i}=S_{ii}\); and
* The diagonal elements are \(S_{ii}\in(0,1)\) and \(S_{ii}>S_{ij},i\
eq j\). The influence of an outlier on its own residual is greater than that of the residuals of normal observations \(\left|r_{g,bias}\right|>\left|r_{i,bias}\right|\); therefore, the outlier may be identified relatively easily by the RB RAIM detector especially when using the studentized residuals. However, in a multiple outlier mode, let the \(g\)th and \(k\)th pseudo-range be the two outliers and the residual of the normal pseudo-range \(r_{i}\) could be expressed as:  
\[r_{i}=\sum\limits_{j=1}^{n}S_{ij}\cdot\varepsilon_{j}+S_{ig}\cdot b_{g}+S_{ ik}\cdot b_{k} \tag{13}\]  
The residuals of outliers \(r_{g}\) and \(r_{k}\) can be expressed as:  
\[\left\{ \begin{array}{l}r_{g}=\sum\limits_{j=1}^{n}S_{gj}\cdot\varepsilon_{j} +S_{gg}\cdot b_{g}+S_{gk}\cdot b_{k}\\ r_{k}=\sum\limits_{j=1}^{n}S_{kj}\cdot\varepsilon_{j}+S_{kg}\cdot b_{g}+S_{kk} \cdot b_{k}\end{array} \right.