## 5 Case Study
### Road Segment Distribution Sampling Modelling and Clustering  
From the analysis in Figure 1, it can be seen that the overall level of the road network is relatively good. It is necessary to further filter out the poor road sections from the road network. The K-means algorithm can be used to cluster the level distribution of road section pavement performance. First, for each route, a continuous function model of the cumulative distribution of pavement performance indexes is established. Then, set the range of independent variables, and the range and number of independent variables for each route are the same. Here, the value range of the independent variable is set to [0.1, 100], and the number of independent variables is  
values is calculated, and the cumulative distribution values of each road segment are, respectively, formed into high-dimensional vectors with the same dimensions. For example, the cumulative distribution vector of pavement properties for \(R1\) is \(s_{R1}=[\left(p_{R1}\right)_{0.1},\left(p_{R1}\right)_{0.2},\ldots,\left(p_{R1} \right)_{100}]\). A road network with n routes contains 15 high-dimensional cumulative distribution vectors, \(s_{R1},s_{R2},\ldots,s_{R14},s_{R15}\). The cumulative distribution vector of the pavement performance of the above 15 road sections is used as the observation vector, and K-means clustering is performed to classify the road sections with similar road levels into one category. Set the value of K as an integer value from 1 to 15, calculate the sum of squares of errors after each classification, and draw the relationship between the sum of squares of errors and the value of K in Figure 2. As can be seen from the figure, K = 3 is an ideal value. The initial K value is set to 3, and clustering is performed according to the steps in \"Overall evaluation of road network pavement performance\". The clustering results are shown in Table 2 and Figure 3.  
Figure 1: Overall evaluation of the road network after weighting: (**a**) Frequency plot; (**b**) Cumulative distribution plot. Figure 2: The relationship between SSE and the number of classes. Analyzing Figure 3, the 15 routes are divided into three categories according to the level of pavement performance. Class III includes only one route, R11, and is the class with the worst road performance level. Class II includes four road sections. The road surface level of these road sections is slightly better than that of R11, but there is no road section with a PCI of 100, and the cumulative distribution reaches one in the 80-90 range. Class I contains many road sections and belongs to a category with better road performance. However, the composition of pavement performance levels of different road sections is also quite different. In some road sections, the pavement performance level of each road section exists and is balanced from 80 to 100; in some road sections, about 40% of the road sections have a PCI value of 100. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Highway Code** & **K = 3** & **K = 5** & **K = 8** \\ \hline R13 & & & 1 \\ \hline R2 & & & \\ \hline R3 & & & \\ \hline R12 & & A & \\ \hline R6 & Class I & & 2 \\ \hline R14 & & & \\ \hline R5 & & B & 3 \\ \hline R7 & & C & 4 \\ \hline R10 & & & 5 \\ \hline R8 & & & 5 \\ \hline R4 & & & 6 \\ \hline R15 & & & 7 \\ \hline R9 & & & 7 \\ \hline R11 & Class III & E & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of clustering results. Figure 3: Clustering diagram obtained by using K-means algorithm based on road performance distribution of road sections (K = 3). Therefore, K = 3 has already reduced the sum of squared errors to a certain level, and the intra-class similarity is high. However, in practical applications, further clustering is required to further identify road segments with more similar performance decays. By selecting a number of different K values and observing the clustering results, it is found that K = 5 and K = 8 can better describe the pavement performance decay characteristics of similar routes.