## 6 Conclusions  
In this paper, the spatial compactness and eco-efficiency of 16 cities in Shandong Province are studied in depth, and the inverse S-type circle decreasing law of the distribution of urban factor density and the developable space of urban centers are revealed through circle gradient analysis and other methods. Meanwhile, this paper measures the eco-efficiency of cities in Shandong Province based on the super-efficient SBM model with non-expected outputs, and explores its evolution trend and response law. Compared with previous studies, the highlights of this paper are the following:  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & **Quantile 0.2** & **Quantile 0.3** & **Quantile 0.4** & **Quantile 0.5** & **Quantile 0.6** & **Quantile 0.7** & **Quantile 0.8** \\ \hline \multirow{4}{*}{ \begin{tabular}{c} Jiaodong \\ Economic \\ Circle \\ \end{tabular}  } & const & \(-0.881\) (0.551) & \(-1.678\) (0.106) & \(-2.384\) & \(-2.798\) (0.015 \({}^{\ast\ast}\)) & \(-2.06\) (0.088 \({}^{\ast\ast}\)) & \(-2.157\) (0.041 \({}^{\ast\ast}\)) & \(-1.812\) (0.263) \\  & \multirow{2}{*}{ \begin{tabular}{c} In\_Per\_GDP \\ Economic \\ Circle \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} In\_Per\_GDP \\ En\_Str \\ Edu\_Per \\ Res\_Per \\ \end{tabular}  } & 0.664 (0.279) & 1.092 (0.035 \({}^{\ast\ast}\)) & 1.346 (0.055 \({}^{\ast\ast}\)) & 1.12 (0.024 \({}^{\ast\ast}\)) & 1.275 (0.014 \({}^{\ast\ast}\)) & 1.235 (0.008 \({}^{\ast\ast\ast}\)) & 1.438 (0.022 \({}^{\ast\ast}\)) \\  & \multirow{2}{*}{ \begin{tabular}{c} In\_Str \\ Edu\_Per \\ Res\_Per \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} In\_Str \\ 13.693 (0.778) \\ Res\_Per \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} In\_Str \\ 13.693 (0.778) \\ Res\_Per \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} In\_Str \\ 13.694 (0.320) \\ 0.331 \\ \end{tabular}  } & \multirow{2}{*}{
\begin{tabular}{c} In\_Str \\ 0.344 (0.000 \({}^{\ast\ast}\)) \\ 0.000 \({}^{\ast\ast}\) \\ 0.000 \({}^{\ast\ast}\) \\ 0.000 \({}^{\ast\ast}\) \\ 0.000 \({}^{\ast\ast}\) \\ 0.000 \({}^{\ast\ast}\) \\ 0.000 \({}^{\ast\ast\ast}\) \\ 0. First, this paper conducts a comprehensive study on both spatial structure and eco-efficiency of cities in Shandong Province, analyzing the relationship between the two and their influencing factors in depth. This comprehensive research method is rare in previous studies. Secondly, this paper adopts a variety of methods, such as Gini coefficient, quantile regression, and linear regression, to explore the evolutionary trend of urban compactness and eco-efficiency, and to analyze the situation that different influencing factors play different roles in the size and direction of each economic circle. This multi-angle and multi-dimensional research method makes the findings of this paper more comprehensive and accurate. And the following conclusions are drawn:  
1. During the sample examination period, the density distribution of factors in cities in Shandong Province generally conforms to the inverse S-type circle decreasing law. The density distribution curves of the cities in Shandong Province show the trend of inverse S-type decreasing from the city center to the outside. Specifically, the slopes of the curves in Qingdao and Jinan are larger, showing a fast decreasing and irregular trend, respectively. The curve of Laiwu city does not change much, probably due to the size of the city form and the overly centralized distribution of industries. The slopes of the density distribution curves of Heze city and Zaozhuang city have gradually increased in recent years, indicating that the urban centers are developing rapidly while the non-central areas are developing relatively slowly. The curve for Rizhao city, on the other hand, shows a slow decline followed by a rapid decline, indicating that urban resources are mainly concentrated in the center and the areas around the center, while the development of peripheral areas of the city is relatively lagging behind.