## 1 Introduction  
According to the Fifth Assessment Report of the IPCC, the global average surface temperature increased by 0.85 \({}^{\circ}\)C from 1880 to 2010 [1]. The temperature rise has led to significant changes in regional rainfall amounts and frequencies, further resulting in the occurrence of numerous extreme weather events [2; 3; 4]. As a vital component of the terrestrial ecosystems, vegetation plays a crucial role in the carbon-water cycle and is the most direct reflection of the natural landscapes [5]. The existing research indicates that vegetation covers 20% of the Earth's surface, and the growth of the vegetation is highly sensitive to regional climate changes [6]. Therefore, studying the response of the regional climate change to long-term changes in vegetation, based on remote-sensing monitoring, has become a hot topic in research, and research related to this is of significant importance for understanding the mechanisms of the vegetation ecosystems [7; 8; 9]. Climate, either directly or indirectly, regulates heat, moisture, and nutrients, impacting the growth and coverage of the vegetation [10; 11]. When the climate changes, the structural and physiological changes in vegetation affect canopy conductance, latent heat flux, and surface albedo. Subsequently, the exchange of the energy between the land and the atmosphere further influences vegetation dynamics [12; 13; 14]. [PERSON] et al. (1993) summarized the analysis methods for exploring the impact of the climate factors on vegetation changes, which mainly include deductive, correlation, and biophysical methods [15]. Relevant scholars have carried out extensive research on the response of vegetation to climate at regional and global scales. [PERSON] et al. compiled approximately 5600 global data points using regression models and estimated the global vegetation net primary productivity using climate and land-cover data. The results indicate that, compared to temperature, precipitation has a stronger global correlation with vegetation net primary productivity, and precipitation explains larger annual productivity variations in grassland- or shrubland-dominated systems than in forest-dominated systems [16]. [PERSON] et al. extracted Normalized Difference Vegetation Index (NDVI) values and corresponding environmental variables from 23 different forest sites in India. They employed the partial least-squares multiple-regression method to emphasize the control of climate on the vitality of the local evergreen forest vegetation. Their findings suggest that indices combining temperature and precipitation have a more significant positive impact on local vegetation compared to bioclinatic indices that rely solely on precipitation parameters [17]. [PERSON] et al. 's study on vegetation in Switzerland found that NDVI had a low correlation with precipitation, while temperature was the main driving factor [18]. Vegetation and ecosystems often require time to adapt to new climate conditions or environmental changes [19]. For instance, when climate change leads to rising temperatures or altered precipitation patterns, vegetation may need some time to adapt to these new conditions. Therefore, the response of the vegetation to climate is not always immediate; the life cycles of plants, their reproductive periods, and ecological processes within populations can all contribute to lagged effects [20; 21]. [PERSON] et al. considered the lagged and cumulative effects of the vegetation on meteorological factors and found that, compared to temperature, precipitation and solar radiation exhibit more significant temporal cumulative effects throughout the vegetation growing season. Additionally, these effects differ during the growth and senescence stages of the vegetation in the high-latitude regions of the Northern Hemisphere [22]. [PERSON] et al. found that the monthly variation in GPP in the Arctic North is mainly determined by the spatial distribution of the meteorological factors and shows different time lags for different meteorological factors [23]. [PERSON] et al.