3.2.2. Local Moran's *I*

Like the global Moran's *I*, the local Moran's *I* focuses on a specific block to describe the similarity between Block *i* and its adjacent areas, as shown in (10).

$$I\_i = \frac{n(p\_i - \overline{p})\sum\_{j=1}^n w\_{i,j}(p\_j - \overline{p})}{\sum\_{i=1}^n (p\_i - \overline{p})^2} = \frac{nz\_i \sum\_{j=1}^n w\_{i,j} z\_j}{z^T z} \tag{10}$$

The standardized statistics for local Moran's *I* monitoring are specified in the following form:

$$Z(I\_l) = \frac{I\_l - E(I\_l)}{\sqrt{Var(I\_l)}}\tag{11}$$

This formula can be used to study the spatial heterogeneity of each region, and can also study and analyze the relative spatial relationship and its changes, where *E*(*Ii*) represents the mathematical expectation of the Local Moran's *I* of the *i-th* Block under the condition of no spatial autocorrelation, and its formula is expressed as:

$$E(I) = -\frac{1}{n-1} \tag{12}$$

*Var*(*Ii*) represents the standard deviation of the Local Moran's *I* in the region. Since the LISA method is relatively intuitive, in the local analysis the Local Moran's *I* of LISA is used for spatial analysis.

By comparing the sign of *Z*(*Ii*) and the value of the local correlation coefficient *Ii*, the spatial units whose local correlation index reaches a certain threshold can be divided into four types of spatial autocorrelation relationships, as shown in the following Table 3.

**Table 3.** Corresponding values of time–space relationship.


Among these, the local Moran's *I*, whose significance level reaches a certain threshold, indicates a positive correlation in the spatial relationship. If it is significantly negative, it indicates a negative correlation between the two research areas in the space–time relationship. Combined with the standardized measurement *Z*, the time–space relationship can be analyzed. The high–high type indicates that the haze density level of area *i* and its neighboring blocks are relatively high. This area is a point where haze occurs frequently. The low–low type indicates that the haze concentration levels of the research block *i* and the surrounding adjacent blocks are relatively low. It indicates an area with lighter haze pollution. The other two types, low–high and high–low respectively, indicate that the high-pollution area surrounds the polluted area, and the high-pollution area is surrounded by the low-pollution area, showing a negative correlation. Compared with the spatial analysis obtained by general actual monitoring sites, the research in this section has the advantage that the research areas are distributed in equal blocks in time and space, and the distances are equal. Therefore, the accuracy of the weight matrix is higher.
