## 3 Simulations
### Case Study 1: Fault of Single Sensor  
In this case study, the sensor of ACE for the solar wind speed is faulted at 48 h from the beginning of the time development. Figure 10 shows the time development for the test case. The credibility of the solar wind speed observed by ACE, STEREO-A, and STEREO-B show rapidly decrease to less than 0.4 at the beginning of the simulation. Because the sensors of the solar wind speed also detected its rapid increase. The high-energy electron flux also decreases to small values. This change of the credibility corresponds to rapid increases of the high-energy electron flux and the solar wind speed observed by STEREO-B because the flux level grows to around 4.0 of the alarm level [31]. During this change, the virtual sensors for interpolation is not appeared. Further, Figure 10 shows the same change of the credibility for the sensors of the solar wind speed. The sensor for the solar wind speed of ACE keeps the very small credibility because the sensor is faulted at 48 h from the beginning. Figure 11 shows network snapshots for the test case shown in Figure 9. The sensor of ACE is identified as abnormal by other sensors associated in Figure 11a. The signs of all arcs to ACE show minus which means the observed value of ACE deviated from the predefined profile. The sensors of STEREO-A, STEREO-B and GOES-10 also show abnormal status. However, the credibility of STEREO-A, STEREO-B, and GOES-10 grows to 1.0 again shortly. The credibility of the solar wind speed of ACE keeps to zero to 72 h from the beginning of the time development. The sensor fault is identified if the credibility keeps the threshold for 24 h. In this test case, the sensors which are connected to the solar wind speed ACE, identify it as fault after 24 h later the failure occurs. After detecting sensor faults, the solar wind speed of ACE is replaced by the virtual sensor and its credibility is returned to 1 again. The credibility of the sensors keeps at 1.0 after the virtual sensor for the solar wind speed of ACE appears. All sensors are normal at 80 h from the beginning of the time development in Figure 11b. The time, 80 h from the beginning, corresponds to the recovery of the sensor fault of ACE. The sensors of the high-energy electron flux and its flux 24 h ahead, the credibilities of both sensors show different between observed data and prediction in Figure 10. The possible reason of this difference is due to replacement of the virtual sensor instead of the actual sensor. The sensor value of the virtual sensor is interpolated by the associated sensors. The interpolated sensor value is averaged values of the estimated values from the neighbor sensors, and the averaged value is interpolated value, not the observed value. In this test case, the solar wind speed of ACE gradually increases at 90 h from  
Figure 9: Time development of the test case for five days from 1 May 2008 at 00:00:00 UTC. The satellite names are labeled in horizontal axes. The plots in the first row and left one in the second row show the time development of solar wind speed. The right plot in the second row shows the time development of the high-energy electron flux. The solar wind plots for ACE and STEREO-B also show the abnormal data for the test (dashed line). the beginning. The virtual sensor trained with the data set is not able to follow this change of the solar wind speed. However, the simulation shows the different rate between observed and predicted data is 0.20. The prediction and interpolation of the virtual sensors are successful in this test case. Figure 11: Network snapshot of the dynamic relational network for the test data shown in Figure 9. Figure (**a,b**) show status of the nodes at 48 and 80 steps respectively. The circles and squares indicate the real and virtual sensor nodes respectively. The white (red) colored sensor nodes is normal (abnormal). The credibility for normal status is larger than 0.5, otherwise abnormal. The rectangles indicate region of the satellite.