# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming
## 3 Results
### Evaluation of Performance under Different \(\eta_{b}\) and \(\eta_{m}\) Conditions  
Based on the realistically simulated EEG data, the estimation process was performed for all the possible combinations of three candidate positions from our pool (up to 551,300 in our case). Hence, the indices in (7), (13) and (15) were computed for \(\theta_{\text{\tiny{model}}}=[\mathbf{r}_{c1},\mathbf{r}_{c2},\mathbf{r}_{c3}]^{T}\), with \(\{\mathbf{r}_{c1},\mathbf{r}_{c2},\mathbf{r}_{c3}\}\in\text{ROI}\). The estimated parameter \(\widehat{\mathbf{\theta}}_{k}\) for MAI corresponded to the value of \(\theta_{\text{\tiny{model}}}\) for which (7) was maximum, while the estimated parameter for our proposed indices where those according to (16). Next, we used the following metrics to evaluate the performance of all the estimations:  
* the individual bias of the estimates, given by \(b_{l,k}=||\mathbf{r}_{l}-\hat{\mathbf{r}}_{l,k}||_{2}\);
* their sum-of-squares: \(\text{SS}_{k}=\sum_{l=1}^{3}b_{l,k}^{2}\);
* the maximum bias: \(b_{\text{\tiny{MAI}},k}=\max_{l}b_{l,k}\). The results are finally presented for different combinations of \(\eta_{b}\) and \(\eta_{m}\) as an average over the \(K=100\) trials. Hence, we compared the average sum-of-squares (denoted as \(\overline{\text{SS}}\)), and the standard deviation of the maximum bias (denoted as \(\sigma_{b_{\text{\tiny{MAI}}}}\)). The first provides us with an overall view of estimation errors as it adds up the squared-bias in the estimation of the three sources, while the second give us the worst case scenario, i.e., the largest variability in the estimation of each of the sources. These results are shown in Figures 6 and 7, respectively. Note that both \(\overline{\text{SS}}\) and \(\sigma_{b_{\text{\tiny{MAI}}}}\) are dimensionless as we normalized them against the mean node distance of tessellated model of the brain cortex, as the minimum spatial resolution that we could achieve in our simulations was constrained by the size of the triangles in our mesh. Additionally, note that in those figures we have included (only for comparison purposes) the result of a source localization based only on minimizing the power of \(Q_{0}\) as a function of \(\mathbf{\theta}\), which is denoted as \(P(W_{0})_{\text{\tiny{min}}}\). The results in Figure 6 show that \(\overline{\mathrm{SS}}\approx 10\) in the case of \(\iota_{\mathrm{{}_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}}}\) for all \((\eta_{m},\eta_{b})\) pairs we evaluated. Furthermore, we noticed in our experiments that the most common case was that of two dipoles correctly estimated while the third was off the true value of \(\mathbf{r}_{l}\), hence accounting for most of the bias. Under that consideration, the average error in the estimation is as large as 3.16 times the mean-node distance of our mesh modeling the brain for \(\iota_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}\). As expected, \(\iota_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}\) keeps a lower error only in high \(\eta_{b}\) (Figure 6a,b), but the spatially correlated biological noise rapidly degrades its performance as \(\eta_{b}\) gets lower. The standard deviation of the worst-case scenario is shown in Figure 7. There, \(\sigma_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}\approx 0.2\) times the mean-node distance for \(\iota_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}\). In the case of \(\iota_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}\), this value is only lower than the one of our proposed index for the best conditions \((\eta_{m},\eta_{b})=\) (10 dB, 10 dB). For all other conditions, \(\sigma_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}\) increases up to 0.525 times the mean-node distance. Clearly, \(\iota_{\mathrm{{}_{\mathrm{{}_{\mathrm{{}_{\mathrm{{max}}}}}}}}}\) provides a biased solution with low variability for all noise conditions, regardless of time and spatial correlation between the sources.