## 2 Extension of Dynamic Relational Network for Spatiotemporal Interpolation
### Modeling of Relations between Two Sensors  
This paper involves statistical properties of the observation data using the SVM [14]. The proposed method involves not only profiling of the observed data [3] but also spatiotemporal interpolation of the missing data. The SVM is a superior method in machine learning for classifying input data into two categories: \(+1\) and \(-1\). The sensor nodes appropriately evaluate the sensor status based on the relation between two sensors, so that the SVM classifies the input data nonlinearly into two categories. Further, the SVM also can be used for regression of the data. Therefore, this paper selects the SVM as a profiling method for estimating relations and interpolating data between two sensors.  
The profiles correspond to relations between two sensors. The profiles are described by parameters of the SVM. The profiles are created by training of the specified data set. The training input data consists of a quadruple and a desired output value (Figure 2). The quadruple consists of four values: averaged values for the past one day and their standard deviations. The averaged values and their standard deviations are calculated from a day in the past. In other words, twenty-four points of observations are used for making a data of the training data set. The desired output value can be binary values: Normal (0) and abnormal (1). The desired output value is determined by classifying conditions of the high-energy electron fluxes. The high-energy electron flux is also categorised into binary values: Quiet (\(-1\)) and alert (1) levels. The training data set for an abnormal condition, solar-wind speed and high-energy electron flux, are chosen from the data set corresponding to the condition of an alert  
Figure 2: Training data generation process for two sensors. The training data of the sensors are created from the observed data. The training data consists of averaged values, their standard deviations, and the desired output.  
level flux. In this paper, the level of the high-energy electron fluxes are quantified by a threshold. The condition of the high-energy electron fluxes are identified as normal where the flux level is less than the threshold. The threshold of classifying condition on high-energy electron flux is 3.0. This criterion is used in the space weather forecasting [28].  
Two kinds of profiles are created in order to evaluate and interpolate relations among sensors (Figures 3 and 4). For the former one, the profiles are used for evaluating relations between two sensors. For the latter one, the profiles are used for interpolating missing data between two sensors. The training process of profiles is different. The parameters of the SVM are repeatedly adjusted to make desired outputs by training input data.  
For interpolation of the missing data, the sensor nodes try to complement the faulted sensors. The faulted sensors are replaced with virtual sensors by actions of neighbor nodes. Eventually, the missing data in observations are interpolated by neighbor nodes. The missing data of observations are finalized by averaging interpolated values. The evaluations of the relations are diagnosed between two sensors in the dynamic relational network.  
Figure 4 shows the generation process for profiles aiming to interpolate missing data. The quadruple data for the test between two sensors is unavailable if either one of two sensors is faulted. Therefore, a tuple of the data composed of only available sensor is used for interpolation of missing data. The input data for interpolation By interpolation of missing data, the test process for the sensors is also applicable without modifying the process for quadruple data.