# Analysis of Development Coordination Levels between Skywalk Systems and Urban Spatial Environments
## 3 Materials and Methods
### Experimental Design
#### 3.2.1 Experiment 1: Indicator System Construction  
Firstly, the two systems of aerial walking system and urban spatial environment are selected, and the main factors affecting the development of skywalk construction are screened from the natural environment, socio-economic environment, urban built environment, and skywalk construction based on the literature research, so as to construct the two indicator systems. Secondly, the entropy value method is used to determine the weight of each indicator, which will enable the calculation of the coupling degree between each urban aerial walking system and the urban spatial environment in the next step. The calculation formula is as follows:  
1. Information entropy value e Calculate the weight of the ith sample value under the jth indicator for that indicator. \[\mathrm{P_{ij}}=\frac{\mathrm{X_{ij}}}{\sum_{i=1}^{n}\mathrm{X_{ij}}}\] (1) where i is the ith row, and j is the jth column. Calculate the information entropy for each indicator (column). \[\mathrm{e_{j}}=-\mathrm{K}*\sum_{i=1}^{n}(\mathrm{P_{ij}}*\mathrm{In}(\mathrm{ P_{ij}}))\] (2) where n is the row. There are logarithms in Equation (3) to determine if there is any significance. Usually, k takes the following values: \[\mathrm{k}=\frac{1}{\mathrm{ln}(n)},\left(0\leqslant\mathrm{e_{j}}<1\right)\] (3)
2. Information utility value \[\mathrm{d_{j}}=1-\mathrm{e_{j}}\] (4) where \(\mathrm{d_{j}}\) is the coefficient of variation of the jth indicator. 3. Weighting coefficient value \[\mathrm{w_{j}}=\frac{\mathrm{d_{j}}}{\sum_{j=1}^{n}\mathrm{d_{j}}}\] (5) The weight values obtained will be used for subsequent composite score calculation. Figure 3: Experimental flowchart. 3.2.2 Experiment 2: Evaluating the Coordination Levels between Skywalk Systems and Urban Development  
In this study, the coupling degree model is used to analyze the development level and degree of coordination of skywalk systems in North America and Asia. First, the degree of coupling and coordination of the skywalk system is evaluated by calculating the coupling degree between the skywalk system and the urban spatial environment in each city, and analyzing the interactions and influences between the two [58]. Second, each city is classified into a different type according to the coupling results, so as to better understand the relationship between the skywalk system and urban development. Based on the literature, the coupling degree model of the two systems is constructed as follows:  
\[C=\sqrt[n]{\frac{\prod_{i=1}^{n}X_{i}}{\left(\frac{1}{n}\sum_{i=1}^{n}X_{i} \right)^{n}}} \tag{6}\]  
where n is the number of subsystems, and n = 2, and X represents the value of each subsystem, with a distribution interval of [0, 1]. The coupling degree C-value interval is [0, 1]. Only two subsystems are involved in this study, so n = 2 and the traditional model function can be simplified. \[C=\frac{2\sqrt{x}}{x+1}=\frac{2}{\sqrt{x}+\frac{1}{\sqrt{x}}},x=\frac{\min(X_ {1},X_{2})}{\max(X_{1},X_{2})},x\in(0,1] \tag{7}\]  
In the traditional model for calculating the degree of coupling, the uneven distribution of C-values can have a great impact on the evaluation results. In addition, the use of traditional models involves a variety of criteria for categorizing the degree of coupling, which can undermine the validity of the use of the C-value [59]. Therefore, referring to related studies [60], the original coupling degree calculation formula is amended as follows:  
\[C^{*}=\sqrt{\left[1-\frac{\sum_{i>j=1}^{n}\sqrt{\left(X_{i}-X_{j}\right)^{2}} }{\sum_{\lambda=1}^{n}\lambda}\right]}\times\sqrt[n]{\left(\prod_{i=1}^{n} \frac{X_{i}}{\max(X_{i})}\right)} \tag{8}\]  
where \(x_{i}\in[0,\,1]\) and \(C\in[0,\,1]\). When the subsystem is more discrete, the C-value is lower, and vice versa. Compared with the traditional model, the distribution of the C-value in [0, 1] is as decentralized as possible, making the difference in C-value more pronounced and suitable for studying the two systems in this study.