2.3.3. Hurst Index

The Hurst index is an effective method to describe the information dependence of long time series [48]. For the NDVI time series, NDVI (τ), = 1, 2, 3, 4, ... , n. For any positive integer ≥ 1, the mean series of the time series is defined as follows:

$$\overline{\text{NDVI}}\_{\left(\tau\right)} = \frac{1}{\tau} \sum\_{t=1}^{\tau} \text{NDVI}\_{\left(\tau\right)} \qquad \tau = 1, 2, \cdots, n \tag{3}$$

The Hurst index is calculated as follows:

$$\frac{\mathcal{R}\_{\left(\tau\right)}}{\mathcal{S}\_{\left(\tau\right)}} = \left(c\tau\right)^{H} \tag{4}$$

The relevant values involved in the calculation of the Hurst index are calculated as follows:

(1) The cumulative deviation is as follows:

$$X\_{(t,\tau)} = \sum\_{t=1}^{t} (\text{NDVI}\_{(t)} - \overline{\text{NDVI}\_{(\tau)}}), 1 \preccurlyeq t \preccurlyeq \tau \tag{5}$$

(2) The range sequence is as follows:

$$R\_{(\tau)} = \max\_{1 \le t \le \tau} X\_{(t,\tau)} - \min\_{1 \le t \le \tau} X\_{(t,\tau)}, \tau = 1, 2, \dots, n \tag{6}$$

(3) The standard difference order is as follows:

$$S\_{\left(\tau\right)} = \left[\frac{1}{\tau} \sum\_{t=1}^{\tau} (\text{NDVI}\_{\left(t\right)} - \text{NDVI}\_{\tau})^2\right]^{\frac{1}{2}}, \tau = 1, 2, \dots, n \tag{7}$$

The Hurst index (value) may reflect the persistent nature of the NDVI time series. In the Hurst exponent (value), when 0 < H < 0.5, this indicates that the change is continuing to decline, meaning that the future change trend is opposite to the past change trend. When = 0.5, this indicates that the NDVI time series is a random series and there is no long-term correlation. When 0.5 < H < 1, the time series are characterized by long-term dependence and persistence, meaning that the future change is consistent with the past trend. In other words, areas that have tended to increase in past years are likely to increase in years to come, and vice versa. The closer it is to 1, the stronger the persistence.

#### 2.3.4. Analysis of Correlation

To explore the response of vegetation dynamic changes in Miaoling to climate change factors, we conducted a partial correlation analysis between NDVI and climate factors (Table 3) to reveal the main driving forces controlling the interannual changes of NDVI from 2000 to 2020.

$$R\_{12,3} = \frac{r\_{12} - r\_{13}r\_{23}}{\sqrt{(1 - r\_{13}^2)(1 - r\_{23}^2)}}\tag{8}$$

where *R*12,3, *R*13,2, *R*23,1 are the correlation coefficients among the variables; *R*12,3 is the partial correlation coefficient between r1 and r2 after fixing the variable r3. The value range of the partial correlation coefficient ranges from −1 to 1. When *R*12,3 > 0, the correlation is positive, meaning that both factors correlate in the same direction. When *R*12,3 < 0, the correlation is negative. The higher the partial correlation coefficient, the stronger the correlation between the two elements at the pixel.

**Table 3.** Statistics of the partial correlation coefficient and significance reveal the intricate relationship between climate factors and NDVI.


Note: X1, precipitation seasonality (CV); X2, VPD; X3, precipitation of wettest quarter; X4, MAP; X5, soil moisture; X6, MAT.

#### 2.3.5. Geographic Detector

Geographic detection is a new spatial statistical method to detect spatial differentiation and reveal driving factors [49]. It contains the following four detectors: factor detection, interaction detection, risk detection, and ecological detection. The first two parts apply here. The model is shown below:

(1) Factor detector

A factor detector could determine the effect of detecting the spatial heterogeneity of vegetation change. The spatial heterogeneity of X to Y could be expressed as q × 100%, and the greater the number, the greater the influence of the detection factors on vegetation change [50], which is as follows:

$$\mathbf{q} = 1 - \frac{\sum\_{\mathbf{h}=1}^{\mathcal{L}} \mathbf{N}\_{\mathbf{h}} \sigma\_{\mathbf{h}}^2}{N \sigma^2} \tag{9}$$

where h is the vegetation change or detection factor hierarchy; N is the number of class h or total region units; and Y is the change in class h or total region Y value.

(2) Interaction detector

The interaction detector is appropriate to identify the impact of the combination of the detection drivers *Xa* and *Xb* on the heterogeneity of the spatial variation of vegetation. The five interaction results are as follows: [50].

Miaoling's vegetation change trend from 2000 to 2020 was regarded as the dependent variable Y. The selected driving factors mainly include the key factors selected by partial correlation analysis as the detection factor X, as shown in Table 4.

**Table 4.** The q value of each driving factors.


Note: X1, precipitation seasonality (CV); X2, VPD; X3, precipitation of wettest quarter; X4, MAP; X5, soil moisture; X6, MAT; X7, elevation; X8, slope; X9, aspect; X10, karst; X11, non-karst; X12, land use change; X13, NLI.
