2.3.4. The GWR Model

The GWR model is an extension of the ordinary linear regression analysis method [39], which can effectively estimate the data with spatial autocorrelation and reflect the spatial heterogeneity of parameters in different regions. The multi-variate linear regression equation is given by:

$$y\_i = \beta\_0(\boldsymbol{u}\_{i\prime}\boldsymbol{v}\_i) + \sum\_{j=1}^n \beta\_j(\boldsymbol{u}\_{i\prime}\boldsymbol{v}\_i)\boldsymbol{x}\_{ij} + \varepsilon\_i \tag{11}$$

where *β*<sup>0</sup> represents the intercept; (*ui*, *vi*) represent the coordinates of the *i*-th sampling point; *β*<sup>j</sup> (*ui, vi*) the *j*-th regression parameter on the *i*-th sampling point, which has geographic significance; *xi*1, *xi*2, *xi*3,··· , *xin* are *n* regression variables at this point; and ε represents random error. Finally, the revised Akaike Information Criterion (*AIC*) was compared with the ordinary least squares (OLS) results. *AIC* is defined as:

$$AIC = -2InL(\mathfrak{e}\_{L'}y) + 2c \tag{12}$$

where *y* represents the sample set of the fitting value of the dependent variable SPEI, *L*(*êL*, *y*) is the likelihood function, *ê<sup>L</sup>* is the maximum likelihood estimate of *eL*, and c is the number of unknown parameters. The smaller the *AIC* is, the higher the fitting degree will be.

**Figure 2.** Factor grading (MAT(**a**), MP (**b**), MWS (**c**), MSD (**d**), DTR (**e**), DTC (**f**), Elevation (**g**), Aspect (**h**), Slope (**i**), AOPD (**j**), POS (**k**), LUCC (**l**) (CPL: CropLand; F: Frost; CRL: Crass Land; W: Water Area; COL: Construction Land; UL: Unused Land)).

#### **3. Results and Analysis**

*3.1. Spatiotemporal Variation Characteristics of SPEI*

The result of SPEI interpolation cross-validation shows a Pearson correlation coefficient of r = 0.85 and root-mean-square error RMSE = 1.15, indicating that the SPEI interpolation result has high accuracy. The statistical results of SPEI in the study area over the years show that (Table 3) the average annual SPEI of the growing season in Inner Mongolia from 2000 to 2018 is −0.03, representing a mild drought. The area in mild drought during the growing season reached 532,600 km2, accounting for 52.39% of the total study area. The area in mild drought was the largest in 2000, accounting for 99.60% of the total, followed by 2001 and 2017, and that in 2012 was the smallest, accounting for only 6.01% of the total. Among the various types of droughts, the average annual area of mild drought accounted

for about 74.34%, the highest proportion. The largest areas of moderate drought and severe drought occurred in 2000 and 2005, accounting for 73.10% and 36.17%, respectively.


**Table 3.** Change of drought area and proportion of various types of drought area in the study area from 2000 to 2018.

The study area has high elevation in the west and low in the east (Figure 1a), and high in the south and low in the north. The spatial distribution of SPEI shows an increasing pattern from west to east with a rate of change of 0.008/degree and an increase of 0.01/degree from south to north (Figure 3). SPEI is highly sensitive to elevation gradients. Areas with high SPEI were mainly distributed in the 40~52◦N area below 800 m in elevation, including Hinggan League, Hulunbuir City, Bairin Left Banner of Chifeng City, and other areas (Figure 1b); low SPEI appeared in areas with elevation between 1100 m~1400 m, in longitude between 105~115◦E, and latitude between 40~45◦N, mainly including Bayannur City, Baotou City, Ulanqab City, and West Ujimqin Banner of Xilingol League. In the Banner area, land covers are mainly grasslands, meadows, and deserts (Figure 1b).

**Figure 3.** Spatial distribution of multi-year mean values of SPEI during the growing season in the study area from 2000 to 2018. (**a**) Longitude and elevation statistics. (**b**) Latitude and elevation statistics.

Trend analysis results show that (Figure 4a) there are significant differences in SPEI changes between the east and west of the study area. SPEI decreased with time significantly at a rate of −0.40~−0.25·(10a−1) in the west including Alxa Left Banner in Alxa League, Dorbod Banner in Ulanqab City, Darhan Muminggan United Banner in Baotou City, and Wuhai City, while SPEI increased significantly with time at a rate of change of 0.25~0.75·(10a−1) in Hulunbuir City, Hinggan League, Tongliao City, and the eastern part of Xilingol League. On the whole, the area with elevated SPEI was about 819,190 km2, accounting for 79.70% of the study area. The land cover in the study area was relatively high in grassland, desert, and cultivated vegetation, reaching 43.68%, 11.94%, and 11.07%, respectively. The area of marsh was the smallest, accounting for only 3.69%. There are significant differences in the spatial distribution of vegetation (Figure 4b). The results of the SPEI variation trend in different land cover types showed that the area with a significantly higher SPEI (SPEI > STD) accounted for about 44.10% of the study area. The increasing trends of SPEI in swamp, coniferous forest, and broad-leaved forest were the most obvious, accounting for more than 85%. These land covers were located in a high-latitude, low-altitude forest area. The area has a large amount of precipitation, abundant water resource, and a low probability of drought. About 93.12% of the area where SPEI dropped significantly was located in the desert, accounting for about 61.53% of the total desert area. In the past 20 years, the mean annual precipitation in the desert areas of Inner Mongolia was less than 150 mm. Under the high-temperature and high-evaporation climatic conditions, water loss became severe and terrestrial carbon productivity was restricted, leading to an increased risk of drought [40].

**Figure 4.** (**a**) The distribution of SPEI trend and (**b**) area percent in different land cover type.
