# A Study on the Normalized Delineation of Airspace Sectors Based on Flight Conflict Dynamics
## 4 Sector-Delineation Method Based on Mean Shift Clustering Algorithm
### Integrated Network Indicator Classification  
Firstly cluster analysis of aircraft nodes is carried out and the distribution of regional centers of aircraft nodes is obtained by clustering with an EM algorithm. The EM algorithm [22] is an iterative algorithm for the maximum likelihood estimation of the probability model parameters with hidden variables. It is a kind of algorithm in unsupervised learning. It has the advantages of stable and accurate calculation results and self-convergence.  
The first step of the EM algorithm is to compute the expectation (E) and then maximize (M) the parameter found at step E. The parameter estimates found on step M are used in the next step of computation at step E. The process keeps on alternating until it converges  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Serial Number**} & \multicolumn{6}{c}{**Evaluation Index**} & \multirow{2}{*}{**Expert Judgement**} \\ \cline{2-2} \cline{5-8}  & \(k_{\text{sum}}\) & \(\overline{k}\) & \(\overline{s}\) & \(\overline{C}\) & \(\boldsymbol{ND}\) & \(\boldsymbol{NE}\) & \(\boldsymbol{CNM}\) \\ \hline Network 1 & 56 & 0.747 & 0.2590 & 0.0738 & 0.0107 & 0.0117 & 9.06 & Busy \\ Network 2 & 66 & 0.857 & 0.2909 & 0.0784 & 0.0113 & 0.0150 & 10.6 & Busy \\ Network 3 & 11 & 0.154 & 0.0351 & 0.0073 & 0.0062 & 0.0054 & 1.8 & Idle \\... &... &... &... &... &... &... &... &... \\ Network 98 & 42 & 0.792 & 0.2950 & 0.1224 & 0.0152 & 0.0160 & 6.85 & Busy \\ Network 99 & 70 & 0.959 & 0.3243 & 0.1646 & 0.0133 & 0.0197 & 11.3 & Busy \\ Network 100 & 46 & 0.7301 & 0.2606 & 0.1429 & 0.0118 & 0.0127 & 7.6 & Busy \\ \hline \hline \end{tabular}
\end{table}
Table 2: Sample set.  
Figure 4: Multi-factor interaction matrix for evaluation indicators.  
Step E: In spatial partitioning, EM searches for node clusters, by determining the parameters of the probability density function suitable for a given node set, and assigns nodes to different clusters; each cluster is modeled by a probability distribution. Each node approximately obeys the normal distribution. Therefore, assume that the joint distribution of latent variables and special variables \(N\big{(}a_{i};\mu_{j},\sigma_{j}\big{)}\). Calculate the sample weights \(r_{ij}\) of the kth (k is the number of clusters) normal distribution of the aircraft node sample \(a_{i}\):  
\[r_{ij}=p(j|a_{i})=\big{[}x_{j}N\big{(}a_{i};\mu_{j},\sigma_{j}\big{)}\big{]} \big{/}\sum\limits_{k=1}^{m}\big{(}a_{i};\mu_{k},\sigma_{j}\big{)} \tag{24}\]  
Step M: Update clustering parameters:  
\[\mu_{j}=\frac{\sum\limits_{i=1}^{n}r_{ij}a_{i}}{n_{j}} \tag{25}\]  
\[\sigma_{j}^{2}=\frac{\sum\limits_{i=1}^{n}r_{ij}\big{(}a_{i}-\mu_{j}\big{)} \big{(}a_{i}-\mu_{j}\big{)}^{\text{T}}}{n_{j}} \tag{26}\]  
\(n_{j}=\sum\limits_{i=1}^{n}a_{ij}\), \(a_{j}=\frac{n_{j}}{n}\). Equations (25) and (26) are repeated until the difference of K sample weights is less than the given threshold, and the final estimate is obtained. The EM algorithm can be implemented by the normal distribution model to obtain the probability that each node belongs to each cluster, continuously update the estimated parameters, and then gradually modify the center value of each cluster. Finally, the classical maximum likelihood algorithm is used to obtain the regional node center.