## 4 Results
### Analysis of Urban Spatial Structure in Shandong Province
#### 4.1.3 Characterization of Urban Factor Agglomeration  
The compactness indicator K derived from the inverse S function can quantitatively describe the compactness of the distribution of urban elements [64], and the compactness of each city is shown in Table 2. Laiwu city has the highest average compactness, followed by Qingdao. The reason for the high degree of compactness in Laiwu city may be because the area of Laiwu city's jurisdiction is small, the type of industry is more homogenized, and the industrial distribution is highly concentrated. Qingdao has a high level of economic development, strong attraction for business concentration and investment, and a dense population. Taian has the lowest degree of compactness, indicating that the development of the city is more balanced and that the agglomeration of the city center is not prominent. Comparing the indicators of urban compactness in different years, it is found that most of the cities in Shandong Province have declined in compactness indicators, indicating that most of the urban development is still in the stage of development from single-center to multicenter cities. The compactness indicators of Jining city and Binzhou city have risen. Combined with the analysis in Figure 3, the degree of centralization of these cities is not strong, and there is still more space for development in the urban center area. In accordance with the Peninsula City Circle Plan for Shandong Province, the regional divisions of Qingdao, Weihai, Weifang, Rizhao, and Yantai (five cities) are divided into the Jiaodong Economic Circle; Linyi, Zaozhuang, Jining, and Heze (four cities) are divided into the Lunan Economic Circle; and the eight cities of Jinan, Zibo, Tai'an, Liaocheng, Dezhou, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Name** & **2014** & **2016** & **2018** & **2020** & **2021** &
\begin{tabular}{c} **Average K-Value (Outlier** \\ **Elimination)** \\ \end{tabular} \\ \hline Ji’nan & \(1.1324\) & \(0.4120\) & \(0.4927\) & \(0.2708\) & \(0.2573\) & \(0.35819\) \\ Qingdao & \(0.9213\) & \(1.2224\) & \(0.1062\) & \(0.7156\) & \(0.8471\) & \(0.647569\) \\ Zibo & \(0.3225\) & \(0.4886\) & \(0.3070\) & \(0.4089\) & \(0.4466\) & \(0.394707\) \\ Zazohuang & \(0.1483\) & \(0.2246\) & \(0.0842\) & \(-0.0364\) & \(0.0337\) & \(0.122689\) \\ Dongying & \(0.4752\) & \(0.3757\) & \(0.3123\) & \(0.3553\) & \(0.2797\) & \(0.359644\) \\ Yantai & \(0.3602\) & \(0.3522\) & \(-5.5807\) & \(0.4801\) & \(0.3631\) & \(0.38891\) \\ Weifang & \(-2.5648\) & \(-0.2496\) & \(-0.4179\) & \(-1.6654\) & \(2.14\) & \(2.14\) \\ Jining & \(-0.5847\) & \(0.3871\) & \(0.3929\) & \(0.1081\) & \(0.1469\) & \(0.258733\) \\ Tai’an & \(-0.0008\) & \(-0.0498\) & \(0.0400\) & \(-0.8550\) & \(-0.3556\) & \(0.04004\) \\ Weihai & \(-0.2182\) & \(-0.2702\) & \(0.3501\) & \(-0.0596\) & \(-12.7113\) & \(0.35006\) \\ Rizhao & \(0.4268\) & \(0.4238\) & \(0.4207\) & \(0.4101\) & \(0.4119\) & \(0.418652\) \\ Laiwu & \(-1.9315\) & \(3.5623\) & \(2.3553\) & & & \(2.958801\) \\ Linyi & \(0.1683\) & \(-0.2544\) & \(0.0335\) & \(-0.0030\) & \(0.0406\) & \(0.08081\) \\ Dezhou & \(0.1346\) & \(0.1050\) & \(-0.3865\) & \(-0.0669\) & \(-1.6749\) & \(0.119798\) \\ Liaocheng & \(0.0807\) & \(-0.0389\) & \(-0.0548\) & \(0.4892\) & \(-0.0401\) & \(0.284985\) \\ Binzhou & \(-0.0238\) & \(-0.4316\) & \(0.0222\) & \(0.0415\) & \(0.2533\) & \(0.105666\) \\ Heze & \(-0.0967\) & \(0.0592\) & \(-0.5746\) & \(-0.6452\) & \(-1.7884\) & \(0.059192\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Compactness indicator K for cities in Shandong Province. Binzhou, Dongying, and Laiwu (although Laiwu was not assigned to Jinan before 2019) are divided into the Provincial Capital Economic Circle. The Jiaodong Economic Circle has the highest degree of urban compactness, followed by the Provincial Capital Economic Circle, and the Lunan Economic Circle has the lowest degree, which is consistent with the reality of Jiaodong > Provincial Capital > Lunan in combination with the economic and social development of each economic circle.