## 4 Experiments
### 2D Function Simulation  
For the external variation conditions, we also carried out relevant experimental verification. Figure 6a shows a two-dimensional thermodynamic conductivity diagram, the bottom of which has a fixed temperature value, where the external temperature varies with time, and different thermal conductivity coefficients in the X and Y directions are used. We imitate the basic coefficient of grain storage, where the length was 24 m and the height was 7 m. The thermal conductivity referred to the actual thermal conductivity of wheat in the actual storage. The initial condition was set as uniform temperature, and we conducted 360 simulations with days as the unit of time. The external conditions were fitted into the annual weather changes using a second-order function. Sampling points are set using a grid sampling method with six points in the X direction and five points in the Y direction. In addition, we randomly selected 90% of the data at all sampling points for each time period. We resampled the time-series on a seven-day basis, in order to obtain 50 time-series with sampling intervals of weeks. Taking the first 40 as training, we predicted the temperature of the 45 th. We added noise to the sampled data, using is a random distribution with a standard deviation of 0.5. The parameters selected for fitting were as follows:  
\[k(x_{a},x_{b})=\parallel x_{a},x_{b}\parallel_{2}^{2}log\parallel x_{a},x_{b} \parallel_{2},\qquad\mathbf{f}(x)=[\;1,x,y]^{T} \tag{47}\]  
We chose \(\mathbf{R}_{t}=0.5^{2}\mathbf{I}\) and \(a=40\). Figure 6b was obtained using our model. Figure 6c shows radial basis interpolation [36], where the radial basis function is:  
\[f(x_{a},x_{b})=\parallel x_{a},x_{b}\parallel_{2}^{2}(log\parallel x_{a},x_{ b}\parallel_{2}-1) \tag{48}\]  
As can be seen from Figure 6, the changes of external conditions were described more accurately using the proposed model. In particular, at \(t=23\), a small bull's-eye region appears obviously in the radial basis function fit, mainly caused by the random error of sampling. In our model, there was obvious consistency between the before and after, and the timing prediction with five steps was also described more accurately. The grain storage process is mainly affected by external weather conditions, which mainly present periodic changes. Therefore, we used the dynamic linear regression method to strengthen the multi-step prediction and obtain relatively more realistic results. At the same time, the spontaneous heat conduction inside the grain pile forms the distribution pattern of hot and cold zone, for which the Universal Kriging method can provide a more accurate description. Therefore, the prediction model can provide more accurate description and prediction for the grain storage process. In contrast, this linear regression model  
Figure 6: 2D sequence temperature simulation (times 1, 12, 23, 34 and 45 are shown): (**a**) Temperature simulation; (**b**) Fit by FUKSS; and (**c**) Fit by radial basis interpolation. conducts multi-step prediction mainly for spatio-temporal data with periodic changes in the boundary conditions. To further verify the reliability of our model, we used signal data to carry out fitting, in which changes in the temporal characteristics are more common. First of all, we used a sinc function to generate a time signal, and added spherical information on this basis, such that the simulation data contained spherical coordinate information and sinc function time information. As shown in Figure 7a, the height of the sinc function increased over time. We intercepted data from 25 equal time intervals, each of which was sampled on an equally spaced grid. Moreover, we added white noise to it, where the standard deviation was 0.1 times the maximum signal value of the whole process. The first 20 intervals were used for training and the last 5 were used for prediction. To estimate the original function from the noisy data, we applied the Kriging filter with data covariance function and the drift functions (according to Equation (47)).