## 2 Data and Sensor System
### Modeling: Identifying Relations among Sensors
Figure 1: Dynamic relational network for space weather forecast using multiple sensors. (**a**) Four-sensor model (**b**) Eight-sensor model. \(V_{i}\) and \(B_{i}\) (\(i=1\), 2, 3) represent sensor nodes of solar wind speed and north-south component of interplanetary magnetic field. \(E\) and \(E_{24}\) indicate, respectively, a present high-energy electron flux and its flux 24 h ahead. Rectangles represent regions of satellites equipped with sensors for observation. Solid line arcs represent evaluations from a source node to a target node. Dashed line arcs indicate evaluations involving a prediction from a source node to a target node. The source node evaluates the target as abnormal, if the relation between two sensors deviates from a predefined condition. For instance, if the relationship between solar wind speed and high-energy electron flux deviates from the predefined condition, then the source node evaluates the target node as abnormal. We use \(R_{i}\) to denote the credibility of sensor node \(i\). The credibility is a time variable, so the value will be changed by mutual evaluation according to time development. We denote the relationship between two sensors as \(T_{ij}\). The value of the parameter \(T_{ij}\) is determined by the diagnosis from the source node \(i\) to the target node \(j\). The source node \(i\) evaluates the target node \(j\) based on their relation. The relation is expressed as an arc in the network to be constructed using SVM. The dynamics of the dynamic relational network are described as follows [22]:  
\[\frac{d{r_{i}}(t)}{dt}=\sum_{j}T^{+}_{ji}R_{j}(t)-r_{i}(t) \tag{1}\]  
where:  
\[R_{i}(t)=\frac{1}{1+\exp(-r_{i}(t))}. \tag{2}\]  
\[T^{+}_{ij}= \begin{cases}T_{ij}+T_{ji}-1&\text{if there are arcs between $i$ and $j$}\\ \\ \\ 0&\text{if there is no arc between $i$ and $j$}\end{cases}  \tag{3}\]  
\(r_{i}(t)\)\(\in\)( \(-\infty\), \(+\infty\)) is a time variable which represents an accumulation of a diagnosis result. Equation (1) describes the dynamics of the credibility for a sensor node \(i\). Equation (2) defines a mapping function of \(r_{i}(t)\) to \(R_{i}\) (\(t\)) \(\in\) [0,1]. The time variable \(r_{i}(t)\) is translated into the credibility \(R_{i}\) through Equation (2). The relation between two sensors is estimated using an SVM. The parameters of the SVM represent the relation between two sensors as a profile. In the previous model [13], the relations were estimated by Vector Autoregressive Models (VAR models). The space environment data, the interplanetary magnetic field in particular, changes rapidly in a short duration, thereby requiring statistical parameters for some duration to detect the change. The VAR models are not sufficient for prediction because they do not follow chaotic changes of the observed values. The north-south component of IMF changes irregularly between south and north directions. For our purpose, the VAR models could not predict the values from time-series data. We could not incorporate the statistical parameters in VAR models, but they can be involved in the SVM. The SVM is an outstanding method in machine learning for classifying input data into two categories: \(+1\) and \(-1\). The status of normal and fault for the sensors correspond to two categories \(+1\) and \(-1\). The SVM evaluates the relation between two sensors appropriately so that the SVM classifies the input data nonlinearly into two categories. Therefore, the SVM is selected as a profiling method for estimating relations between two sensors. For applying the SVM to space weather forecasting, the training and test process are necessary for consideration. The input parameters and desired output are determined for training process. For the test process, the input data is passed to the SVM and its outputs the class normal/fault. The appropriate selection of the input parameters for the SVM is crucial in order to bring out its classification performance.