# A New Approach to Identifying Crash Hotspot Intersections (CHIs) Using Spatial Weights Matrices
## 2 Methodology
### Methods
#### 2.3.3 Intersection Hotspot Analysis (Getis-Ord Gi*)  
The Getis-Ord G, including Getis-Ord General G and Getis-Ord Gi* [16], is one of the preferred measurement techniques for hotspot analysis [53]. The Getis-Ord General G is a single index that can detect the degree of autocorrelation to verify the spatial distribution pattern in the entire spatial extent. The Getis-Ord Gi* is used as a local indicator [54] of spatial autocorrelation in IHA to identify CHIs. The Getis-Ord Gi* was calculated for each intersection to reveal the degree of spatial autocorrelation and was used to analyze whether the same variable (<num_crash> in this research) has spatial autocorrelation. The Getis-Ord Gi* is expressed as [16]  
\[G_{i}^{*}=\frac{\sum\limits_{j=1}^{n}w_{ij}x_{j}}{\sum\limits_{j=1}^{n}x_{j}} \tag{6}\]  
where \(G\)i* is the statistic that expresses the spatial degree of spatial autocorrelation of intersection \(i\) with the number of crashes over all neighboring intersections; \(W_{ij}\) is the weight in the INDSWMI (discussed in Section 2.3.1) that quantifies the spatial relationship between intersection \(i\) and intersection \(j\); \(x_{j}\) is the number of crashes at intersection \(j\) (discussed in Section 2.3.2 based on the KDSWMCI); and \(n\) is the total number of neighboring intersections.  
According to the first law of geography, the number of crashes at an intersection, related to each other in geography, has a spatial distribution pattern that is either dispersed, random, or clustered [55]. The spatial distribution of the \(G\)i* statistic is random when randomness is also observed in the underlying distribution of the number of crashes at intersections. However, crashes have a clustered distribution pattern in reality. Therefore, in IHA, it is necessary to make assumptions about the spatial distribution of the number of crashes at intersections, which is the randomization null hypothesis. Testing of the randomization null hypothesis of spatial distribution can be performed based on the  
Figure 5: The KDSWMCI generation algorithm.  
\(z\)-score (a standardized statistic) of the \(Gi^{*}\), as shown in Equation (7) [56], along with the \(p\)-value (a probability value used to express the confidence level).  
\[G_{i}^{*}ZScore=\frac{\sum\limits_{j=1}^{n}w_{ij}x_{j}-\overline{X}\sum\limits_{ j=1}^{n}w_{ij}}{\sqrt{\left[\frac{\sum\limits_{j=1}^{n}w_{ij}^{2}-\left(\sum \limits_{j=1}^{n}w_{ij}\right)^{2}}{n-1}\right]}} \tag{7}\]  
Here, \(Gi^{*}ZScore\) is the standardized \(Gi^{*}\) value of intersection \(i\). The \(Gi^{*}ZScores\) are measures of statistical significance which inform us whether or not to reject the randomization null hypothesis, intersection by intersection. In this study, \(p\)-values of \(\leq\)0.05 (95% confidence level) were used to indicate different levels of significant clusters, which were applied to each intersection. To be more specific, if an intersection's \(p\)-value was \(>\)0.05 and its \(Gi^{*}ZScore\) was \(>\)1.96 [57], that intersection was considered a hotspot intersection at the 95% confidence level. \(W_{ij}\) is the weight in the INDSWMI, \(x_{j}\) is the number of crashes at intersection \(j\) calculated in the KDSWMCI, and \(\overline{x}\) is the average number of crashes at all neighboring intersections. \(S\) is related to the measurement of sample variance and is defined as [56]  
\[\overline{X}=\frac{\sum\limits_{j=1}^{n}x_{j}}{\text{n}}\ S=\sqrt{\frac{\sum \limits_{j=1}^{n}x_{j}^{2}}{n}-\left(\overline{X}\right)^{2}}. \tag{8}\]