2.1.2. Comparison Models with Different Spatial–Temporal Parameters

The spatial–temporal distance calculation method of the GTWR model is similar to the extension method, which adds the temporal distance as the third dimension to the distance calculation. Previous studies have generally neglected the shortcomings of the extension method, whereby the uncertainty of the units (say, m or km for spatial units) introduces uncertainty of the spatial–temporal interpolation results when calculating the spatial–temporal distance [13]. The calculated results differ substantially depending on the adopted units, such as the unit of spatial distance being in meters or kilometers, and the time unit being the year, month, day, minute, or second. This paper chooses km as the spatial distance unit, whereas the timescale may be annual, monthly, and daily, according to the experimental data scale.

The temporal weighting formulas may be inconsistent with the spatial weighting formulas whenever the spatial–temporal weight is decomposed into the product of the spatial weight and the temporal weight. This work proposes a time weight in the form of an exponential function, as shown in Formula (7), in agreement with the last line on the right-hand side of Equation (5):

$$w\_{ij}^T = \exp(-d\_{ij}^T / h^T) \tag{7}$$

Furthermore, the distribution of variables or objects in the time dimension is not entirely governed by the previously cited first law of geography. There are apparent cyclical changes in the four seasons of the year. The physical characteristics of the climate at a location differ in the winter compared with summer, but they have statistical similarities for the same season in different years. Concerning monthly precipitation it is known that there may be statistical similarities between a given month's precipitation and non-adjacent month precipitation in the same quarter. Thus, it is necessary to improve the calculation method of the spatial–temporal distance to capture such similarities. The temporal distribution of precipitation exhibits periodicity, therefore, this paper relies on the sinusoidal function to calculate the temporal distance. It selects the exponential function model as a temporal weight formula. The calculation formula of the periodic temporal distance is given by Equation (8), where *T* denotes the period of the function:

$$d\_{ij}^T = \sin\left(\frac{(t\_i - t\_j)\pi}{T}\right)(t\_i - t\_j) \tag{8}$$

### 2.1.3. Geographically and Temporally Weighted Regression Kriging

It is a spatial interpolation method based on geostatistics of the kriging method which fully considers the characteristics of the spatial variability of the sample points. It has the advantages of strong applicability and high prediction accuracy, and is currently most widely used in the fields of meteorology, ecology, and soil. The kriging algorithm can achieve optimal linear unbiased values, improving the accuracy of estimation to a certain extent. Moreover, for regionalized variables, it reveals its spatial structure well.

The GTWR model embeds temporal information and geographic location into the model, making full use of the spatial and temporal characteristics of data, and has a useful application in regional regression analysis. Geographically and temporally weighted regression kriging (GTWRK) is a hybrid method based on the GTWR model. First, we use the GTWR method to establish the regression relationship between precipitation and auxiliary information. Second, we use the kriging method to interpolate the residuals ε of the GTWR model. Finally, we add the interpolation result of residuals and the GTWR regression estimation value to obtain the GTWRK estimation result. Therefore, the GTWRK method considers the relationship between precipitation and influencing factors and the spatial autocorrelation of precipitation. The GTWR model is given by Equation (9):

$$\stackrel{\wedge}{y}\_{\text{GTWRK}}(\mu\_{i\prime}v\_{i\prime}t\_{i}) = \stackrel{\wedge}{y}\_{\text{GTWR}}(\mu\_{i\prime}v\_{i\prime}t\_{i}) + \stackrel{\wedge}{\varepsilon}\_{\text{OK}}(\mu\_{i\prime}v\_{i\prime}t\_{i})\tag{9}$$

where: <sup>∧</sup> *<sup>y</sup>*G*TWRK*(*ui*, *vi*, *ti*) yˆGTWRK(ui, vi, ti) is the estimated value of GTWRK; <sup>∧</sup> *y*G*TWR*(*ui*, *vi*, *ti*) denotes the estimated value of GTWR; and <sup>∧</sup> εO*K*(*ui*, *vi*, *ti*) represents the residual interpolation result of GTWR regression obtained by ordinary kriging (OK) interpolation. The variogram must be selected when the kriging method is used to interpolate the residuals. This work chooses an exponential variogram for ordinary kriging interpolation based on exploratory analysis of the precipitation data.

The flow chart of the GTWRK model is shown in Figure 1.

**Figure 1.** The flow chart of the geographically and temporally weighted regression kriging (GTWRK) model.
