## 4 Experiments
### Experiments on Thermodynamic Model  
There have been many studies on thermal spreading models in the field of spatio-temporal analysis and prediction [35]. In the aspect of spatio-temporal correlation, they have very obvious characteristic. At the same time, their complexity is manageable. A 1D simple thermodynamic model is:  
\[a(\frac{\partial^{2}T}{\partial x^{2}})=\frac{\partial T}{\partial t}, \tag{45}\]  
where \(T\) is the temperature field, including time \(t\) and position \(x\), and \(a\) is the thermal conductivity coefficient.Using the finite difference method, it can be discretized as  
\[T_{t+1}^{i}=T_{t}^{i}+a\Delta t\frac{(T_{t}^{i-1}-2T_{t}^{i}+T_{t}^{i+1})}{ \Delta x^{2}},\forall i\in 1,\ldots,n. \tag{46}\]  
The time of thermodynamic simulation was 200 time steps, and 41 discrete points were selected for calculation to construct the data set. The initial temperature distribution was \(25-40*(x-0.5)^{2}\), which is a second-order distribution, and the boundary conditions were fixed. The result is shown in Figure 3a. White noise with a standard deviation of \(\sigma=0.5\) was also added to the simulated data, the results are shown in Figure 3b. We evenly chose 10 points in the interval at which to observe the functions. The sampling model is shown in Figure 3c. We applied the FUKSS with model parameters \(k(x_{a},x_{b})=\parallel x_{a}-x_{b}\parallel_{2}^{3}\), \(\mathbf{f}(x)=\left[ \begin{array}{cc}1,x\end{array} \right]^{T}\), \(\mathbf{R}_{t}=0.5^{2}\mathbf{I}\), and \(\alpha=5\). In the actual work process, we often cannot determine the trend term order of the distribution, and generally choose a low possible order. Therefore, we chose first-order \(\mathbf{f}(x)\) in this experiment, and proved that this choice also achieved good results. For the parameter \(\mathbf{R}_{t}\), we chose the observation error as a Gaussian distribution with standard deviation of 0 and variance of 0.5\({}^{2}\). The selection of other parameters is detailed in Section 3.2 and the discussion. The resulting function estimates are depicted in Figure 3f. The same sampled data were used in linear and Kriging. Note, that we adopted the same covariance parameter as FUKSS, as the fitting of the covariance function could not accurately describe the actual situation due to the small amount of sampling data for Kriging interpolation. Compared with the experiment, we found that the three methods could fit the actual situation well, but the proposed method had a smaller error. At the same time, it can be seen, from Figure 3f, that the Kalman filtering method could effectively eliminate the observation noise. We further compared different sampled data at different times, and the results are shown in Figure 4. Different sampling methods had an obvious influence on the spatial interpolation but, through iteration, our model obtained better results, compared with the other spatial interpolation models. At the same time, comparing Figures 3f,i and 4f,i, under the condition of the same number of observable sensors, multiple sampling methods can improve the fitting accuracy of the model. In Figure 5, we took out 40 time-steps equally from 200 time-steps, in order to simplify the calculation process. The first 30 time-steps were trained and the last 10 time-steps were used as prediction test. We applied the KUKF Prediction algorithm with cubic exponential smoothing model parameters \(\zeta=0.8\), \(\tau=0.3\), \(\gamma=0.2\), and \(L=0\).