# Distributed Localization with Complemented RSS and AOA Measurements: Theory and Methods
## 2 RSS-AOA Distributed Localization Problem
### Performance Metrics  
In general, requirements and performance metrics depend on the application for which an estimator will be employed. In practice, some intrinsic uncertainties are always present in the system (such as measurement noise), which naturally lead to errors in the node's location estimate. The location error of the \(i\)-th node is defined as \(e_{i}=\|\mathbf{x}_{i}-\mathbf{\hat{x}}_{i}\|\), where \(\mathbf{\hat{x}}_{i}\) is the estimated location of the true node's location, \(\mathbf{x}_{i}\). One can use this error to build a variety of statistics that can be utilized as performance metrics. One of the most popular in the localization literature is the root mean square error (RMSE), given by  
\[\text{RMSE}=\sqrt{\mathbb{E}\left(e_{i}^{2}\right)}.\]  
However, in practical performance evaluation tests, the expectation is approximated by a set of Monte Carlo (\(M_{c}\)) trials. In addition, due to the consideration of \(M\) target nodes in the network, one can actually define the average RMSE (ARMSE) as  
\[\text{ARMSE}=\sqrt{\frac{1}{M_{c}}\frac{1}{M}\sum_{i=1}^{M_{c}}\sum_{j=1}^{M} \|\mathbf{x}_{ij}-\hat{\mathbf{x}}_{ij}\|^{2}}, \tag{5}\]  
where \(\hat{\mathbf{x}}_{ij}\) represents the estimate of the true \(j\)-th target location, \(\mathbf{x}_{ij}\), in the \(i\)-th Monte Carlo run. ARMSE  can be thought of as accuracy, since it is the rate of the statistical deviation of the location estimate from its real location.  
In order to get a notion of how good the performance of an estimator actually is, it is useful to compare it against some benchmark. Theoretically, the best achievable performance is often bench-marked by the Cramer-Rao lower bound (CRLB) [56] (Chapter 3). If one defines the co-variance matrix \(\mathbf{C}\) of an estimator, where the estimate is denoted by, for instance \(\hat{\mathbf{y}}\), with  
\[\mathbf{C}=\mathbb{E}\left(\left(\mathbf{y}-\hat{\mathbf{y}}\right)\left(\mathbf{y}-\hat{\mathbf{y }}\right)^{T}\right),\]then we know that the covariance is bounded by  
\[C\succeq J^{-1},\]  
where \(J\) represents the Fisher information matrix (FIM) and \(A\succeq B\) means that the matrix \(A-B\) is non-negative definite. The interested reader can find the derivation of the CRLB for the considered localization problem in the Appendix A.  
The reason why ARMSE is the most widely used performance metric in the localization literature is the fact that it is the best finite-sample approximation of the standard error, which is usually the most desired optimality criterion in terms of errors. Obviously, the ARMSE might not capture completely the accuracy of an estimator, and one may opt to employ other performance metrics. These include localization error outage, cumulative distribution function of the localization error, average Euclidean error, geometric average error, etc. [57; 58].  
It should be noted that performance evaluation is never completely fair nor impartial. To some extent, this is owed to the fact that elementary metrics are not able to capture a complete picture of an estimator, while those that are more complete are too complex and are often biased due to subjective interpretations (for instance, a metric can be in favor of an estimator that implicitly tries to optimize this same metric) [57]. To prevent such issues, in general, one should choose a metric that is the most relevant for the intended application.