2.4.2. Grey Relational Analysis

Grey relation refers to the uncertain correlation between things and system factors, or factors and the main behavior. GRA is based on the geometrical approximation of the sequence of behavioral factors and the main behavioral sequence, and it is used to analyze and determine the degree of influence between factors or the contribution measure of factors to the main behavior [45,46]. Not restricted by the type of samples and the distribution of probability distribution, GRA is a new method to study uncertain issues with limited data and information [30].

The reference factor sequence can be expressed as:

$$\mathbb{X}\_0 = \left[ \mathbb{X}\_0(1), \,\mathbb{X}\_0(2), \,\dots, \,\, \mathbb{X}\_0(\mathfrak{n}) \right] \tag{2}$$

The comparative factor sequence can be expressed as:

$$\mathbb{X}\_{\mathbf{i}} = [\mathbb{X}\_{\mathbf{i}}(1), \mathbb{X}\_{\mathbf{i}}(2), \dots, \mathbb{X}\_{\mathbf{i}}(\mathbf{n})], \mathbf{i} = 1, \ 2, \dots, \ m. \tag{3}$$

Considering the relevant factor sequence, the point relational coefficient is defined:

$$\mathbf{r}(\mathbf{x}\_0(\mathbf{k}), \mathbf{x}\_i(\mathbf{k})) = \frac{\mathbf{x}(\min) + \xi \mathbf{x}(\max)}{\Delta\_{0i}(\mathbf{k}) + \xi \mathbf{x}(\max)},\tag{4}$$

And

$$\Delta\_{0i}(\mathbf{k}) = |\mathbf{x}\_0(\mathbf{k}) - \mathbf{x}\_i(\mathbf{k})|,\tag{5}$$

$$\alpha(\text{min}) = \min\_{\mathbf{i}} \min\_{\mathbf{k}} \Delta\_{\mathbf{0}\mathbf{i}}(\mathbf{k}),\tag{6}$$

$$\text{ax}(\text{max}) = \max\_{\text{i}} \max\_{\text{k}} \Delta\_{\text{0i}}(\text{k}),\tag{7}$$

the Grey Relational Grade (GRG) of γ(X0, Xi) between Xi(i = 1, 2, ··· , m) and X0:

$$\gamma(\mathbf{x}\_{0\prime}, \mathbf{x}\_{i}) = \frac{1}{\mathbf{n}} \sum\_{\mathbf{k}=1}^{n} \mathbf{r}(\mathbf{x}\_{0}(\mathbf{k}), \mathbf{x}\_{i}(\mathbf{k})),\tag{8}$$

where γ(x0(k), xi(k)) is the relational coefficient between xi and x0 when it meets the condition of comparative factor k and the grey resolution coefficient ξ. The GRG is a measure of the influence of the comparative factor sequence on the reference factor sequence. When the value is closer to 1, the effect of the comparative factor sequence on the reference factor sequence becomes more significant. The value of the GRG can be used as an indicator that reflects the influence of comparative factor sequence on reference factor sequence [47,48].

In this study, EVI in the growing season of 2000–2015 was used as the reference factor sequence (X0), and the comparative factor sequence (Xi) was made up of three climatic factors of air temperature, relative humidity, and precipitation during 2000–2015.
