## 1 Introduction  
On an ecosystem level, water-use efficiency (WUE) relates carbon assimilation to the water consumption of vegetation [1, 2, 3] and it is usually calculated as the ratio of gross primary productivity (GPP) to evapotranspiration (ET) [4, 5, 6]. As an important parameter that connects the carbon and water cycles of a terrestrial ecosystem [7, 8, 9], WUE not only reflects the interaction between the carbon and water cycles [10, 11, 12], but is also a key variable indicating the responses of an ecosystem to global climate change [13, 14, 15]. The accurate quantization of spatiotemporal variations in WUE and revealing its drivers are conducive to understanding the mechanism of an ecosystem response to environmental change [16, 17, 18]. It has important significance for regional vegetation growth management, and the reasonable allocation and protection of water resources in an ecosystem [19]. The Inner Mongolia (IM) in China is located in a semi-arid and arid agropastoral ecotone and it is one of the most sensitive regions to climate change with a vulnerable ecological environment [20,21]. Since the 1970s, the central and local governments have implemented a series of ecological restoration projects, which have improved the local vegetation coverage to different extents [22, 23, 24]. These projects have also changed the local hydrological circulation, thus resulting in WUE variation [25, 26, 27]. Large-scale af-forestation in semi-arid and arid regions could cause excessive water consumption, which may further influence the carbon and water cycles of an ecosystem [28, 29, 30, 31]. Therefore, a quantitative assessment of the spatiotemporal variation of WUE and its cause in IM is vital to understanding the interaction between a local ecosystem and environmental change and to formulate water-conservation protection schemes. Spatiotemporal variations of WUE are influenced by biotic and abiotic factors [1,5,32]. The WUE of different vegetation types responds differently to climate change and hydrological conditions [11,12,15]. Previous studies have mainly focused on the variation of WUE and its relationship with air temperature, precipitation and drought. For example, based on eddy flux measurement data, [PERSON] et al. [3] found a good correlation between WUE and leaf-area index (LAI) in IM, and the WUE of meadows could be predicted by LAI. [PERSON] et al. [33] assessed the responses of WUE to drought and found that the WUE in the middle of IM often increased upon the occurrence of drought events. [PERSON] et al. [34] found a rising trend of WUE in the Beijing-Tianjin Sandstorm Source Region from 1982 to 2015, during which WUE was positively correlated with temperature and precipitation. However, such correlations only came from a simple correlation or partial correlation analysis, in which it was assumed that sample variables met the joint normal distribution, and that the analysis results could easily be influenced by sample size and extreme values of samples [35, 36, 37]. Grey relational analysis can solve the poor reliability of results caused by a small sample size, missing information and atypical distribution, and can measure the relation strength among different variables [38, 39, 40]. In addition, the contributions of climate and vegetation to the variation of WUE remain unclear. The multiple-regression residual analysis and path analysis have been widely used to quantify the contributions of independent variables to dependent variables [5,41, 42, 43]. However, complicated multicollinearity is a common issue in multiple linear regression [44, 45], which can be solved by ridge regression [27,46]. In this study, the Global Land Surface Satellite (GLASS) products were used to study the spatial pattern of the WUE in IM from 2001 to 2018.