3.2.1. Global Moran's *I*

We use the Global Moran's *I* to study the overall spatial characteristics of haze, indicating whether the haze is related to space, as in (3).

$$I = \frac{n\sum\_{i=1}^{n}\sum\_{j=1}^{n} w\_{ij}(p\_i - \overline{p})\left(p\_j - \overline{p}\right)}{\sum\_{i=1}^{n}\sum\_{j=1}^{n} w\_{ij}\cdot\sum\_{i=1}^{n} \left(p\_i - \overline{p}\right)^2} \tag{3}$$

This paper cuts the satellite image into nine sub-regions of the same size, so *n* = 9 in the Equation. *pi*, *pj* denote the average levels for Block *i* and Block *j* in one of the seasons. *p* denotes the average haze level for all blocks in a given season. *wij* represents the weight between Block *i* and Block *j*. Since the image is equally divided, the areas on the diagonal are not considered to be adjacent. Then its spatial adjacency graph is as shown in Figure 3 below:

**Figure 3.** Spatial adjacency diagram.

From the adjacency relationship in the above figure, we can obtain its spatial adjacency matrix, the weight matrix in this section. We set each side to be 1, and the weight between two blocks is 1, as in Equation (4), and the weight matrix is as shown in Equation (5).

$$w\_{ij} = \begin{cases} 1 & \text{if Block } i \text{ is adjacent to Block } j \\ 0 & & \text{otherwise} \end{cases} \tag{4}$$

$$W = \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ \end{bmatrix} \tag{5}$$

In this section, the essential reason for using the Moran's *I* is to analyze and indirectly reflect the correlation between the two spatially adjacent areas of the haze concentration level. In order to facilitate the analysis and use of the Moran's *I*, we use (6)–(8) to simplify (3).

$$S\_0 = \sum\_{i=1}^{n} \sum\_{j=1}^{n} w\_{ij} \tag{6}$$

$$z\_i = (p\_i - \overline{p}) \tag{7}$$

$$z^T = [z\_1, z\_2, \dots, z\_n] \tag{8}$$

The simplified global Moran's *I* is given by (9).

$$I = \frac{n}{S\_0} \frac{\sum\_{i=1}^{n} \sum\_{j=1}^{n} w\_{ij} z\_i z\_j}{\sum\_{i=1}^{n} z\_i^2} = \frac{n}{S\_0} \frac{z^T W z}{z^T z} \tag{9}$$

The range of the global Moran's *I* is between [−1, 1]. If *I* > 0, the haze has a positive correlation with the space and, the closer the value is to 1, the stronger the correlation, and there is a strong positive correlation between haze and space. Conversely, if *I* < 0, the haze is negatively correlated with space, and the closer the value is to −1, the stronger the negative correlation.

The analysis methods of spatial autocorrelation generally consist of the following three types: 1. Local Indicators of Spatial Association (LISA); 2. G statistics; 3. Moran scatter plot; the first is the method of local spatial analysis, which will be used in the following experiment. In the global analysis, the Moran scatter diagram is adopted as the analysis method, and its four quadrants respectively represent the spatial relationship between the four blocks and their neighboring blocks, and the corresponding relationships are as follows:

As shown in Figure 4, the Moran scatter plot consists of four quadrants representing four different spatial association types. The relationship of the first quadrant is high–high, indicating that the haze concentration level of the area and the surrounding area are both high, and the spatial difference is slight. The spatial relationship is a positive correlation. The relationship of the second quadrant is low–high, indicating that the haze level of the area and the surrounding area differs significantly. The level of the area is low, and the haze level around it is high. The spatial relationship is now negatively correlated. The relationship of the third quadrant is low–low, indicating the haze concentration in this area is low, and the spatial relationship is positive. The relationship between the fourth quadrant is high–low, indicating that the haze level in the area is high. However, the haze concentration level in the surrounding area is low. The spatial relationship is negatively correlated. Quadrant 1 and quadrant 3 reveal positive local spatial autocorrelation, and quadrant 2 and quadrant 4 reveal negative local spatial autocorrelation.

**Figure 4.** Moran scatter image limit relationship.
