2.3.2. Hot Spot Analysis

Hot Spot Analysis is often used to identify potential spatial agglomeration characteristics of PM2.5 pollution, and PM2.5 levels are divided into cold spots, insignificant points, and hot spots. The Getis-Ord *Gi*\* of ArcGIS was used to calculate the *Gi*\* of each city in the study area. The principle formulae are as follows [18]:

$$G\_i^\* = \frac{\frac{\sum\_{j=1}^n w\_{ij} \mathbf{x}\_j - \overline{\mathbf{x}} \sum\_{j=1}^n w\_{ij}}{\mathcal{S}\sqrt{\frac{\left[n \sum\_{j=1}^n w\_{ij}^2 - \left(\sum\_{j=1}^n w\_{ij}\right)^2\right]}{n-1}}} \tag{5}$$

$$S = \sqrt{\frac{\sum\_{j=1}^{n} \mathbf{x}\_j^2}{n} - \left(\overline{\mathbf{x}}\right)^2} \tag{6}$$

where *xj* is the annual PM2.5 concentration of city *j*; *ωij* is the spatial weight between city *i* and city *j*, and *n* = 56 represents the number of cities in the study area.

#### 2.3.3. Spatial Lag Model

Socioeconomic variables, such as GDP, population size, and traffic, greatly affect local PM2.5 concentrations. In this study, the Spatial Lag Model (SLM) was used to determine the influence of different socio-economic factors on PM2.5 concentration, which could be explained by Formula (7):

$$\mathcal{Y} = \rho \mathcal{W} \mathcal{Y} + \mathcal{X}\mathcal{\beta} + \varepsilon, \ \varepsilon \sim \mathcal{N}\left[0, \sigma^2 I\right] \tag{7}$$

where *Y* indicates the PM2.5 concentration; *X* expresses the independent variables, including all introduced socioeconomic factors; *ρ* is the spatial effect coefficient, and its value ranges from 0 to 1. The spatial matrix is represented by *W*, which indicates whether two spatial elements have a common boundary; *β* represents the regression coefficient of explanatory variables; and *ε* is the error term.
