**6.1 Two-sensor network**

We compare the Monte Carlo Gauss-Seidel iterative algorithm with the centralized algorithm and the iterative algorithm based on the Riemann sum approximation in [31] by using the receiver operating characteristics (ROC) curves.

In this case, we know *<sup>μ</sup>*<sup>0</sup> <sup>¼</sup> ½ � 0, 0 *<sup>T</sup>*, *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> ½ �Þ 1, 1 *<sup>T</sup>* and <sup>Σ</sup><sup>0</sup> <sup>¼</sup> ½ � <sup>0</sup>*:*6, 0; 0, 0*:*<sup>6</sup> , <sup>Σ</sup><sup>1</sup> <sup>¼</sup> ½ � 1, 0*:*4; 0*:*4, 1 . Some discrete values of *a* and *b* are used to plot ROC curves. We refer to the optimal importance sampling density *g y*ð Þ∝∣*PH*<sup>0</sup> 0 ð Þ*<sup>y</sup> L y* ^ð Þ<sup>∣</sup> in Section 2.2 and <sup>∣</sup>*L y* ^ð Þ<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>*ap y*ð Þ� <sup>j</sup>*H*<sup>1</sup> *bp y*ð Þ <sup>j</sup>*H*<sup>0</sup> <sup>∣</sup>. The form is similar to the mixture-Gaussian distribution. Therefore, the importance sampling density *g y*ð Þ is chosen to be the mixture-Gaussian distribution. The effects of choosing different *g y*ð Þ in terms of the performance of the Monte Carlo method were shown in [21] *via* numerical examples. For Algorithm 1, we take *N* ¼ 200 samples from the density *g y*ð Þ. For the Riemann sum approximation iterative algorithm in [31], we take a discretized step-size *Δ* ¼ 0*:*09, *yi* ∈½ � �8, 10 , i.e., *N*<sup>1</sup> ¼ *N*<sup>2</sup> ¼ *N* ¼ 200. The ROC curves for three important fusion rules: AND, OR, and XOR rules with *p* ¼ 0*:*05, 0*:*15, 0*:*3 are plotted in **Figure 2**. We compare the computational time of the two algorithms with *p* ¼ 0*:*15 in **Figure 3**. Note that the analytical solution is used for the AND rule and the OR rule. Since the XOR rule is not a *K*-out-of-*L* rule, we use Algorithm 1 to search for the sensor rules.

Some observations in **Figures 2** and **3** are presented as follows:

• Given the fusion rule, the two points 0, 0 ð Þ and 1, 1 ð Þ may not be the beginning or ending points of the ROC curves, which is different from the case in the ideal channel cases. In addition, the larger the probability of channel errors is, the

**Figure 2.** *Two-sensor ROC curves with the probability of channel errors p* ¼ 0*:*05, 0*:*15, 0*:*3*.*

farther away from 0, 0 ð Þ or 1, 1 ð Þ the ROC curves are. A possible reason is that the detection probability is not equal to 0 or 1, even when the false alarm probability is 0 or 1 in the presence of channel errors.


**Figure 3.** *Two-sensor computational time as N increases with the probability of channel errors p* ¼ 0*:*15*.*
