## 2 Extension of Dynamic Relational Network for Spatiotemporal Interpolation
### Dynamics of Dynamic Relational Network Model  
For computation, three kinds of representative models had been proposed for a dynamic relational network [5]: Black and white model, skeptical model, and gray model. The basic model is black and white model. This paper uses the gray model for the simulations.  
In the dynamic relational network, the sensor status is represented by a credibility: normal (credibility = 1) or faulty (credibility = 0). This paper uses \(R_{i}\) to denote a credibility of a sensor node \(i\). The credibility is a time variable so that the value will be changed by mutual evaluation according to a time development. We denote the relationship between two sensors as \(T_{ij}\). The value of the parameter \(T_{ji}\) 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. An arc represents the relation between two sensors described as profiles using the SVM.  
For the black and white model, the dynamics of the dynamic relational network are described as follows:  
\[\frac{dr_{i}(t)}{dt}=\sum_{j}T_{ji}R_{j}+\sum_{j}T_{ij}R_{j}-\frac{1}{2}\sum_{ i}(T_{ij}+1) \tag{1}\]  
where  
\[R_{i}(t)=\frac{1}{1+exp(-r_{i}(t))} \tag{2}\]  
\(r_{i}(t)\in(-\infty,+\infty)\) 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 \(r_{i}\) is mapped into the credibility \(R_{i}\) through Equation (2). The convergence of the credibilities are theoretically guaranteed in the black and white model. The credibilities are converged to 0 or 1.  
In the gray model, the credibility can be an intermediate value between 0 and 1. The credibility of the gray model can reflect ambiguous states that do not determine exactly whether the status of the  
Figure 1: Dynamic relational network for space weather forecast using multiple sensors. The node \(V_{i}\) represents the sensor for the solar wind speed. The nodes \(E\) and \(E_{24}\) indicate present high-energy electron flux and flux twenty-four hours ahead. The rectangle represents an internal regions of GOES-10 which equips with couple sensors for observation. Solid line arcs represent a diagnosis flow from a source node to a target node. Dashed line arcs indicate a diagnosis flow involving a prediction from a source node to a target node.  
sensors is normal or abnormal. Space weather forecasting of high-energy electron flux also involves ambiguous situations by the change of unpredictable flux. With the gray model, the risk for changes of high-energy electron flux can be considered by the credibility.  
\[\frac{dr_{i}(t)}{dt}=\sum_{j}T_{ji}^{+}R_{j}(t)-r_{i}(t) \tag{3}\]  
where  
\[T_{ji}^{+} = \begin{cases}T_{ij}+T_{ji}-1&\text{(if there are arcs between i and j))}\\ 0&\text{(if there is no arc between i and j)}\end{cases} \tag{4}\]  
In this model, the time variable \(r_{i}(t)\) is also mapped into the credibility \(R_{i}\) by Equation (2).