## 2 Extension of Dynamic Relational Network for Spatiotemporal Interpolation
### Tests of Relations in Dynamic Relational Network  
In the test phase, the diagnosis of the relation between two sensors is executed with two sensor values shown in Figure 5. Similar to the training phase, the input data for the test is comprised using  
Figure 4: Training process of a profile between two sensors for spatiotemporal interpolation of missing data. The training data consisting of tuple data set are input to an SVM. The SVM learns desired outputs (interpolated value) from the training data set. This profile is used for interpolation. On the other hand, the profile shown in Figure 3 is used for evaluating relations between two sensors. Figure 3: Training process of a profile between two sensors. The training data set are input to an SVM. The SVM learns desired outputs from the training data set. the values of the two sensors that are observed values are averaged a day in the past. The test data for the diagnosis is input to the SVM, which represents the relation between two sensors. The SVM outputs the result whether the test data satisfies the predefined conditions of the space environment. The SVM outputs 1 where the input data for the test is contained in the profile at the quiet level of the high-energy electron flux, otherwise \(-1\). Before the diagnosis, however, the quadruple data for tests is not always available. The missing data is interpolated by two sensors associated by using the SVM if the observation data is lacking. The interpolated data is calculated by using the SVM with the data observed by the single sensor. In Figure 6, the missing data of Sensor A is interpolated by the value of Sensor B using the SVM with the predefined profile. In this case, the missing data of Sensor A is necessary for interpolation since Sensor A is faulty. The tests of relations among sensors are executed after interpolating missing data. Figure 7 shows the replacement process of the faulted sensor in the dynamic relational network. In the proposed model, the sensors are diagnosed from other sensors connected. The neighbor sensors would identify the faulted sensors as abnormal by the distributed manner. The faulted sensor is disjointed from the dynamic relational network where the sensor status identified as abnormal for 24 h  
Figure 5: Test of observed data recorded by two sensors. The test data are input to an SVM based on the profile. The SVM classifies the condition of the space environment as binary values 1 (normal) or \(-1\) (abnormal). Figure 6: Interpolation process of missing data. The interpolation of the missing data is calculated by the data last one day of the opposite side. The interpolated value is calculated by the SVM with the input data and the predefined profile. continuously. In Figure 7a, the solar-wind sensor of ACE is diagnosed as abnormal from other three nodes. In Figure 7b on the other hand, the solar wind sensor of ACE is replaced a circle node to rectangle one. The rectangle node indicates the virtual sensor behaving as virtual sensor which is created from information of neighbor nodes. The virtual sensor is also diagnosed from other sensor nodes. In Figure 8, the virtual sensors are constrained to diagnose from the neighbor nodes. The interpolated value of the virtual sensor is not reliable since it is the result that are accumulated sensor values by the neighbor nodes. In the proposed model, the diagnosis between virtual sensors for interpolation is not allowed. The virtual sensors \(N_{1}\) and \(N_{2}\) (\(N_{3}\)) are not allowed to diagnosis each other. Because the virtual sensors simultaneously need to get the data from each other at that time. The sensor value of one sensor depends on the data of another one. Therefore, the sensor value of the virtual sensor interpolated by another virtual one is not determined simultaneously. However, the virtual sensors for prediction and interpolation are allowed to diagnosis each other. Figure 8: An illustrative example of a constraint on virtual sensors in the dynamic relational network.