# A Formation Mechanism of Spatial Distribution Pattern of Industrial Clusters under Flow Space
## 3 Materials and Methods
### Formation Mechanism Method of Spatial Distribution Pattern of Industrial Clusters under Flow Space
#### 3.2.3 Spatial Interaction Regression Model  
Previous studies have mostly used multivariate linear or nonlinear regression models that do not consider the influence of geographic location or geographically weighted regressions, which means they fully comply with the first law of geography--distance decay effect [50; 51]. However, relevant basic research shows that the influencing factors of industrial agglomeration include the urban geographic factors within the region and the influencing factors outside the region. Since the transportation accessibility network, human resource flow trend network, and technology flow trend network are the main influencing factors of inter-city interaction, and their values express the intensity of inter-regional influence, we construct a three-dimensional vector based on the above three types of interaction influencing factors and represent the intensity of inter-regional influence using the modal length of the vector:  
\[w_{ij}=pow\left(\left(O_{ji}\right)^{2}+\left(POP_{ji}\right)^{2}+\left(TEC_{ ji}\right)^{2},0.5\right) \tag{11}\]  
\[w_{i}=\left(w_{i1},w_{i2},w_{i3},\ldots,w_{in}\right) \tag{12}\]  
\[w_{ii}=1 if(i\in n) \tag{13}\]  
\[w=\left(w_{1},w_{2},w_{3},\ldots,w_{n-1},w_{n}\right) \tag{14}\]  
where \(n\) denotes the number of cities, \(w_{ij}\) denotes the combined influence weight of city \(j\) on city \(i\) in terms of traffic link strength, population inflow trend, and technology inflow trend, and \(w_{i}\) denotes the weight matrix of the influence of \(n\) cities on city \(i\), which is mappedto geographic space to denote the magnitude of the influence weight of city \(k\) on city \(i\). \(w\) denotes the weight coefficient matrix composed for each city \(i\). The intensity of industrial agglomeration is the dependent variable on the research of the formation mechanism of urban industrial agglomeration. Industrial agglomeration area refers to the scope of high density industrial agglomeration area and low density industrial agglomeration area. Therefore, for each city, this paper takes the area of industrial agglomeration area as the weight to solve the urban industrial agglomeration intensity:  
\[den_{i}=\sum_{1}^{s}\frac{Area_{ik}}{Sum\_Area_{i}}\times den_{ik} \tag{15}\]  
\[Sum\_Area_{i}=\sum_{1}^{s}Area_{ik} \tag{16}\]  
\[den_{ik}=\frac{High\_Density_{ik}+Medium\_Density_{ik}}{Area_{ik}}\]  
where \(den_{i}\) denotes the industrial agglomeration intensity of city \(i\), \(den_{ik}\) denotes the average agglomeration density of the \(k\)th industrial agglomeration in city \(i\), \(Area_{ik}\) denotes the area of the \(k\)th industrial agglomeration in city \(i\), \(sum\_area_{i}\) denotes the total area of industrial agglomeration in city \(i\), \(s\) denotes the number of industrial agglomerations in city \(i\). \(High\_Density_{ik}\) denotes the product of the area of high-density aggregation and the value of high-density aggregation of industrial agglomeration area \(k\), and \(Medium\_Density_{ik}\) denotes the product of the area of medium-density aggregation and the value of medium-density aggregation of industrial agglomeration area \(k\). Since the traditional geographically weighted regression model adopts the weight coefficient that follows the laws of geography, it is difficult to consider the \"distance failure\" in the physical space. Therefore, in this paper, we take the average intensity of urban industrial agglomeration \(den_{i}\) as the dependent variable \(Y\), the level of economic foundation, the level of human resources and the ability of innovative technology as the dependent variable \(X\) and take the three network values as the spatial influence weight. Finally, we further construct a weighted regression model considering the intensity of flow space:  
\[Y=(Y_{1},Y_{2},Y_{3},\ldots,Y_{i},\ldots,Y_{n-1},Y_{  
Figure 4: Results of nuclear density analysis in the Yangtze River Delta.