## 3 Materials and Methods
### Dynamic Linear Autoregressive Model Specification  
The prediction ability discussed in this paper is mainly for multi-step prediction ability. In this experiment, the time interval between the time-series data is equal, and the current data are used for estimation and revised in our calculations. Therefore, this method is not suitable for prediction, due to the lack of a specific model to exclude the prediction of future functions. Compared with spatial interpolation, our model can eliminate the observation error, to a certain extent, with an operation similar to that of a Kalman filter. To solve this problem, we introduce the dynamic linear autoregressive model to solve the prediction problem. During prediction, the predicted value is closer to the current value, as illustrated in the subsequent tests. A simpler alternative model, based on the dynamic linear model, is the Holt-Winters model. This model has three main components: A smoothing sequence (\(\mathbf{S}_{t}\)), a trend sequence (\(\mathbf{B}_{t}\)), and a periodic smoothing sequence (\(\mathbf{C}_{t}\)). We can combine these components together with several methods. To make the model more general, we use an additive model, which integrates the components together to obtain the full model. \[\mathbf{Y}_{t+1}=\mathbf{S}_{t}+\mathbf{B}_{t}+\mathbf{C}_{t-L+1}, \tag{40}\]  
where \(L\) is the length of periodic sequence. Inside the model, we use the cubic exponential smoothing method to yield the components. \(\zeta\), \(\tau\), and \(\gamma\) are the parameters used, which have values between 0 and 1. The equations are given below. \[\widehat{Y}_{t}(\mathbf{X})=\widehat{\varepsilon}_{t-1}+(\mathbf{G}_{t}+\mathbf{F}^{T} \mathbf{M}_{t})(\mathbf{Y}_{t}-\mathbf{W}_{t}\widehat{\varepsilon}_{t-1}), \tag{41}\]  
\[\mathbf{S}_{t}=\zeta(\widehat{Y}_{t}(\mathbf{X})-\mathbf{C}_{t-L})+(1-\zeta)(\mathbf{S}_{t-1}+ \mathbf{B}_{t-1}), \tag{42}\]  
\[\mathbf{B}_{t}=\tau(\mathbf{S}_{t}-\mathbf{S}_{t-1})+(1-\tau)\mathbf{B}_{t-1}, \tag{43}\]  
\[\mathbf{C}_{t}=\gamma(\widehat{Y}_{t}(\mathbf{X})-\mathbf{S}_{t})+(1-\gamma)\mathbf{C}_{t-L}. \tag{44}\]  
The initial data have little impact on the whole model, so we choose \(\mathbf{S}_{0}=\widehat{Y}_{0}(\mathbf{X})\), \(\mathbf{B}_{0}=\widehat{Y}_{1}(\mathbf{X})-\widehat{Y}_{0}(\mathbf{X})\), and \(\widehat{Y}_{0}(\mathbf{X})=\mathbf{0}\). We introduce exponential smoothing prediction to provide additional quasi-observable values for multi-step prediction. The detailed steps are shown in Algorithm 2. Exponential smoothing prediction is introduced to enhance FUKSS multi-step prediction. In the prediction process, we first use our proposed model to re-estimate the observed values, in order to improve the accuracy of exponential smoothing prediction, which can realize data cleaning (e.g., denoising) to a certain extent. For the prediction stage, we use the updated parameters of the iteration to re-estimate the quasi-observed values, which makes the actual predicted value more in line with the spatial distribution law set by the model. However, this prediction error also increases with the increase of the prediction step size. At the same time, due to the limitation of the dynamic linear model, it can predict and describe the time-series with trend and periodic change more accurately. ```
Initialize: \(t=1\), \(\mathbf{P}_{0}=\alpha\mathbf{K}\), \(\mathbf{c}_{0}=\mathbf{0}(N*1)\).