## 5 Discussion  
The FUKSS model presented in this paper is a step-by-step method which can be used to reconstruct the time-series of spatial functions from scattered data. As our model is based on the universal Kriging state-space model, the calculation process is based on Kriging's spatial statistics and the derivation is similar to that of a Kalman filter. In terms of space, we do not need to have strict distribution requirements for the sampled data. Using the statistical characteristics of Kriging, partial missing or new data have little impact on the estimate. In terms of time-series prediction, this model has the advantages of small memory requirement (constant extra space storage required \(O(N^{2})\)), acceptable computational speed (constant computational complexity \(O(n_{i}^{q})\)), and being suitable for dynamic problems. The keys to deducing the recursive filter is to use linear unbiased estimation, establishing the optimality of the recursive formulation. This method uses not only historical data but also the spatial relationship between the historical data, which optimizes the results. To further solve the problem that it is impossible to make long-term predictions, we introduce the dynamic linear regression model to carry out secondary processing on the estimated value of each step, such that the spatial information can still be retained, to some extent, in the long-term prediction. However, there are still some limitations to our model. As FUKSS is based on the universal Kriging state-space model, our algorithm can be simplified to a similar method of universal Kriging interpolation and Kalman filtering in special cases. If the sensor accuracy is not taken into account--namely, if the measurement data is unbiased--we can simplify it, to a certain extent, to universal Kriging interpolation fitting in this time period. If we do not consider the drift term in Equation (1), we can simplify it to space-time Kalman filtering [39]. Therefore, the parameters of universal Kriging and Kalman filtering are used simultaneously in our model, such that the model has the advantages and disadvantages of both. Our algorithm has a lot of prior variables; for example, we have to specify \(\mathbf{K}\), \(\mathbf{k}(x)\), \(\mathbf{\alpha}\), \(\mathbf{R}_{t}\), and \(\mathbf{F}\). The selection method for these parameters (except for \(a\)) is similar to that of universal Kriging, which has been described in detail in the literature [31]. In general, we recommend using a polynomial drift function for \(\mathbf{F}\), as discussed in the literature [40]. For \(\mathbf{K}\) and \(\mathbf{k}(x)\), we cannot calculate them statistically as with universal Kriging, or fit it in any other way, as they would take a lot of computational effort to determine. To simplify the calculation, we chose IRF, which are a more generalized covariance function [15; 16]. Figure 11: Prediction against sampled points for all real points. If enough data points are available, the selection of the covariance function is not critical. Finally, the noise covariance is mainly determined by the parameters of the sensor which has an effect similar to the smoothing parameters of the smooth spline. Compared with KKF, the advantage of our method is that it considers the development law of dynamic system through the state space model, so it does not need to spend a lot of prior calculation to obtain the initial parameters. However, in the calculation process of KKF, the parameters are estimated from all historical data, which is more universal than our algorithm. From the results, our model focuses more on temporal space, while KKF focuses more on time. To estimate past values of the function, our method is similar to estimation by Kalman smoothing, which can obtain a relatively good result. As the trend term \(F^{T}\mathbf{m}_{t}\) in our model is determined by the current sampling data \(\mathbf{Y}_{t}\), our model is more unsuitable for time prediction than KKF.