# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 4 Simulations
### Evaluation of Entropy-Driven Exploration with Real Data
For the entropy driven approach, Table 1 shows a detailed summary of the exchanged data. For exploration with the SOF paradigm, we choose the ADMM parameter \(\rho\) according to [9] Section 3.4. There, the authors evaluate the primal and dual residuals and automatically choose \(\rho\) such that the difference between residuals is not too large, which is called residual balancing. However, for SOE, the residual balancing leads to an unstable solution of \(\rho\), which might be related to the high number of features \(N\) and the comparably smaller number of measurements \(M\). Therefore, we keep \(\rho=1.0\) as a constant for SOE, which has been evaluated by cross-validations (this empirical evaluation has to be done for each dataset, because the parameter \(\rho\) is highly data dependent) For both approaches, we set the parameter \(\delta=\lambda^{-2}\sqrt{2\log(M)}\). The estimations after 250 measurements of a single run for SOE are shown in the Figure 4a-c and for SOF in Figure 4d-f. The colored circles represent the measurement locations of the individual agents. The feature positions of the active features, i.e., the features with corresponding non-zero weight, are shown with black crosses. We observe that the entropy-driven methods tend to measure the border of the environments (cf. Figure 4a,d). This is a numerical effect of the evaluation of the entropy and has also been seen in [28]. The random walk has a uniformly distributed measurement location; see Figure 4b,e. It achieves a good coverage of the whole area. Therefore, active features might not be observed, and hence, the error is decreasing slower, compared to the entropy-driven method. The meander approach has always the drawback that it has to be carefully designed to achieve a good coverage within reasonable time. The trajectory design should reflect the observed process. However, this process is unknown from the beginning, and therefore, assumptions about measurement positions have to be made beforehand. This most often results in a bad performance or to many unuseful measurement positions. In summary, the entropy-driven methods have a better coverage, compared to the benchmark methods. Now, we would like to discuss the number of active components in all approaches, i.e., the sparsity of the estimations. The number of active features is shown in Figure 5. The exploration strategy for SOF, cf. Figure 5b, leads to measuring most often relevant features. Therefore, features are only neglected in the model, when a sufficient number of measurements and features already exists. The benchmark methods lead to similar results, however. SOE (see Figure 5a) has more fluctuations in the number of active components, because its number of features \(N\) is predefined from the beginning of the exploration. Therefore, in the beginning, the parameter \(\delta=\lambda^{-2}\sqrt{2\log(M)}\) is suboptimal due to the bad ratio of \(M\) to \(N\). However, when the number of measurements increase, active components are better identified, as can be seen in Figure 5a at 180  measurements. Now, we compare the performance of the algorithm with respect to the NMSE of the estimated field. The metric is defined as:  
\[e_{\text{NMSE, SOE}}=\frac{\|\mathbf{\
abla}(\mathbf{X},\mathbf{X}_{N})\mathbf{w}-\mathbf{f}\|}{ \|\mathbf{f}\|},\hskip 28.452756 pte_{\text{NMSE, SOF}}=\frac{\left\|\sum\limits_{k=1}^{K}\mathbf{ \
abla}(\mathbf{X},\mathbf{X}_{N_{k}})\mathbf{w}_{k}-\mathbf{f}\right\|}{\|\mathbf{f}\|}, \tag{40}\]  
for SOE and SOF, respectively, where \(\mathbf{f}=[f(\mathbf{x}_{0}),\ldots f(\mathbf{x}_{|\mathcal{X}^{\prime}|})]^{T},\mathbf{x} \in\mathcal{X}^{\prime}\) are all sampled values of the observed true magnetic field. Figure 4: Estimation results for a single run of the estimation algorithm with the SOF and SOE splitting approaches. The measurement positions of the individual agents are marked with colored circles. The active features are marked by with black crosses. (**a–c**) show the estimations of an SOE approach, and (**d–e**) show the estimations of an SOF approach.