## 1 Introduction  
Vegetation is a comprehensive indicator of the ecological environment [1; 2], and having suitable hydrothermal conditions is the key element for vegetation growth. Yet, the synergistic effect of multiple climatic factors on vegetation remains unclear. Spotvgt NDVI [3], Avhrr NDVI [4], Modis NDVI [5], and other datasets have been used by many scholars to carry out research on the characteristics of vegetation change and the vegetation response to natural factors such as P [6], T [7; 8], surface temperature, evaporation [9], soil moisture [10], land water storage [11], and other factors on different time scales, such as yearly [12; 13], seasonal [14; 15], and monthly [16]. Research methods include the standard regression coefficient [17], wavelet analysis [18], Mann-Kendall trend test [19], geographically weighted regression [20], BRT model [21], and ANOVA [22]. The study areas covered the whole world [23; 24; 25], Eurasia [7; 26; 27; 28], eastern Africa [29; 30], Australia [31; 32; 33], China [34; 35], and other areas. The results show that vegetation has been expanding in most parts of the world since the 1980s [11], and the net primary productivity has increased by about 13%. However, differences between the northern and southern hemispheres are significant. The vegetation in the northern hemisphere has been growing significantly due to increases in T, while that in the southern hemisphere has been declining due to a P limitation [23]. The vegetation in Eurasia has expanded as a whole, and there is a pronounced positive correlation with P in arid and semi-arid regions [26; 27]. Vegetation change in eastern Europe and western Siberia is closely related to the sea surface temperature, soil moisture, and other factors [9; 28]. In central Asia, vegetation and T complement each other, while in southern Asia, vegetation is mainly controlled by P during drought periods and T during rainy periods [36]. Vegetation in eastern Africa has shrunk, and water, altitude, and El Nino--Southern Oscillation are all important influencing factors [29; 30]. Except for in April and October, vegetation in Australia has decreased overall and is positively/negatively correlated with P/T, and this correlation has a lag period ranged from 0 to 3 months [32; 33]. The overall vegetation in China has increased [35]. For example, in northern China, the growing season NDVI increased at a rate of 0.0005 yr\({}^{-1}\) from 1982 to 2011 [37], and the average yearly NDVI increased by 0.001 yr\({}^{-1}\) between 1998 and 2015 in the north of Xinjiang Uygul Zizhiqu [38]. Generally, vegetation has positive correlations with P, Tave, surface temperature, average water vapor pressure, and relative humidity [10]. Some scholars have also proposed that the relationship between P and T has different impacts on different types of vegetation; for example, the warm-dry relationship forced the distribution boundary of _Pinus tabuliformis_ to shrink northward in the semi-arid region of northeast China and also caused the degeneration of meadows in Qinghai province [39; 40]. There have been many studies on the correlations between the NDVI and individual climate factors, such as P or T [6; 7; 8], but only a few studies have been conducted on the multivariate correlation between the NDVI and multiple climate factors, and the similarities and differences in the multivariate correlation between the NDVI and multiple climate factors are still unclear. It is also difficult to measure and quantify the synergistic effects of multiple climate factors for the NDVI on the basis of a correlation analysis, let alone revealing trends in such synergies. In summary, in order to ensure the research results were representative and comprehensive, northern China was chosen as the study area.