# Variance of Linear MMSE estimator from three measurements.

I am here presenting a question on calculating the variance of an Estimator. This is a problem in localization of vehicle using ranging from Vehicular Ad hoc Network and Change in location from Inertial Navigation System: So I have three measurements as follows
\begin{equation*} \Delta \hat{x}_{i,i-1} = \Delta x_{i,i-1} + \epsilon_{x_{INS}} = \Delta x_{i,i-1} + \mathcal{N}(0,{\sigma}_{x_{INS}}^2) \end{equation*} \begin{equation*} \Delta \hat{y}_{i,i-1} = \Delta y_{i,i-1} + \epsilon_{y_{INS}} = \Delta y_{i,i-1} + \mathcal{N}(0,{\sigma}_{y_{INS}}^2) \end{equation*} $(\Delta \hat{y}_{i,i-1}, \Delta \hat{x}_{i,i-1})$ are the estimated change in position, which is shown as sum of actual position and an error(a normal distribution). This is data from an Inertial navigation system \begin{equation*} \hat{r}_i = r_i + \epsilon_r = r_i + \mathcal{N}(0,{\sigma_r}^2) \end{equation*} $\hat{r}_i$ is the estimated range measurement, which is represented as sum of actual range and an error in ranging(a normal distribution again). This is data from ranging between vehicle and a Road side unit $RSU_j$ that is located at $(h_j,k_j)$

After collecting the above three data for $i=1,2,\cdots,m$ number of time instances, the Minimum Mean Square Error (MMSE) estimator for $m^th$ instance location of vehicle: $$\hat{\vec{x}} = (A^T A)^{-1} A^T \vec{b}$$

Where $A$ is of size $(m-1 \times 2)$\begin{equation*} A = \begin{bmatrix} -2\sum_{a = 1}^{m-1} \Delta \hat{x}_{a+1,a} & -2\sum_{a = 1}^{m-1} \Delta \hat{y}_{a+1,a} \\ \vdots & \vdots\\ -2\sum_{a = i}^{m-1} \Delta \hat{x}_{a+1,a} & -2\sum_{a = i}^{m-1} \Delta \hat{y}_{a+1,a} \\ \vdots & \vdots\\ -2\Delta \hat{x}_{m,m-1} & -2 \Delta \hat{y}_{m,m-1} \end{bmatrix} \end{equation*} $\vec{x}$ represents the most current two dimensional location of the vehicle: \begin{equation*} \vec{x} = \begin{bmatrix} x_m \\ y_m \end{bmatrix} \end{equation*} $\vec{b}$ is of size $(m-1 \times 1)$ as below:

\begin{eqnarray*} \begin{bmatrix} {\hat{r}_1}^2 - {\hat{r}_m}^2 - 2 h_j \sum_{a=1}^{m-1} \Delta \hat{x}_{a+1,a} - 2 k_j \sum_{a=1}^{m-1} \Delta \hat{y}_{a+1,a} -{\sum_{a=1}^{m-1}\Delta \hat{x}_{a+1,a} }^2 - {\sum_{a=1}^{m-1}\Delta \hat{y}_{a+1,a} }^2 \\ \vdots\\ {\hat{r}_i}^2 - {\hat{r}_m}^2 - 2 h_j \sum_{a=i}^{m-1} \Delta \hat{x}_{a+1,a} - 2 k_j \sum_{a=i}^{m-1} \Delta \hat{y}_{a+1,a} -{\sum_{a=i}^{m-1}\Delta \hat{x}_{a+1,a} }^2 - {\sum_{a=i}^{m-1}\Delta \hat{y}_{a+1,a} }^2 \\ \vdots\\ \hat{r}_{m-1}^2 - {\hat{r}_m}^2 - 2 h_j \Delta \hat{x}_{m,m-1} - 2 k_j \Delta \hat{y}_{m,m-1} -{\Delta \hat{x}_{m,m-1}}^2 - {\Delta \hat{y}_{m,m-1} }^2\\ \end{bmatrix} \end{eqnarray*}

My question:'' How does one determine the variance of $\hat{\vec{x}}$ ". My idea of finding this variance is to understand how error in three types of input data $\epsilon_{x_{INS}},\epsilon_{y_{INS}},\epsilon_r$ is effecting the error in $\hat{\vec{x}}$ estimate. Looking forward to all thoughts from the community.

• what have you worked on so far ? any thoughts of how to tackle this problem ? – Ahmad Bazzi Aug 27 '18 at 12:46
• I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $\Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables? – prithvi shenoy Aug 27 '18 at 15:08
• Correct me if I am wrong with calculating the variance of $A$ \begin{equation*} \sigma_A = \begin{bmatrix} 4\times(m-1) \times \sigma_{x_{INS}}^2 & 4\times(m-1) \times \sigma_{y_{INS}}^2 \\ \vdots & \vdots\\ 4\times(m-i) \times \sigma_{x_{INS}}^2 & 4\times(m-i) \times \sigma_{y_{INS}}^2\\ \vdots & \vdots\\ 4\times \sigma_{x_{INS}}^2 & 4\times\sigma_{y_{INS}}^2 \end{bmatrix} \end{equation*} I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance. – prithvi shenoy Aug 27 '18 at 15:28
• An approximation to the covariance is as below: $$P_1 = [A^T A]^{-1} 2 \hat{r}_1^2 \sigma_{\epsilon_{r}}^2$$ Assuming that RSU is far away from vehicle, which means $$\hat{r}_1 \approx \hat{r}_2 \cdots \approx \hat{r}_m$$ and $$\hat{r}_i$$ is large enough that the INS terms can be ignored in matrix $$\vec{b}$$ – prithvi shenoy Sep 19 '18 at 11:16