## 3 Prediction-Driven Measurement Sub-Partitioning
To correct the unreasonable target prediction, the unreasonable target prediction should be replaced by alternative points, the number of which is the estimated correct target number. At last, based on geometry, the locations of alternative points can be determined by the spatial distribution of the target prediction and measurements. Through analyzing the distribution of the measurements and target prediction, the prediction \(x_{k|k-1}^{(i)}\) corresponding to \(i^{th}\) unreasonable subset \(W^{(i)}\) can be approximated as the geometrical center of a polygon composed of real clusters' centers. It can be seen in Figure 1, which represents the spatial distribution of four true clusters of measurements that are observed from four crossed extended targets. Target number underestimation occurs in this tracking scenario with tracks-crossed, i.e., these measurements from four targets are mistakenly put into one subset according to the clustering center calculated by the target prediction. From Figure 1, the target prediction can be treated as the geometrical center of the polygon composed of these true target centers. Thus, the alternative points can be roughly approximated by the vertices of polygons, as shown in Figure 1.  
Figure 1: The measurement distribution in unreasonable subset. (**a**) The geometrical distribution of the target prediction and the centers of 4 clusters. (**b**) The selection of alternative points in prediction-driven measurement sub-partitioning (PMS) algorithm. According to the above analysis of the spatial distribution of the measurements and \(x_{k|k-1}^{(i)}\), we consider replacing the predicted target state \(x_{k|k-1}^{(i)}\) with reasonable alternative points (e.g., the vertices of a regular polygon) based on the geometry thought. Thus, the position \((p_{j,x},p_{j,y})\) of alternative point can be calculated as follows,  
\[ \begin{split} p_{j,x}&=x_{k|k-1,x}^{(i)}+TH\times \cos(\theta-\frac{2\pi(j-1)}{n_{i}})\\ p_{j,y}&=x_{k|k-1,y}^{(i)}+TH\times\sin(\theta- \frac{2\pi(j-1)}{n_{i}})\end{split}  \tag{3}\]  
where \(\theta=\arctan\left(\frac{p_{1,y}-x_{k|k-1,x}^{(i)}}{p_{1,x}-x_{k|k-1,x}^{(i)}}\right)\) is the angle between the X-axis and the line passing through \(p_{1}\) and \(x_{k|k-1}^{(i)}\), \(TH\) is the distance between \(x_{k|k-1}^{(i)}\) and the first alternative point which is decided according to the distribution of the distances between \(x_{k|k-1}^{(i)}\) and the measurements, and \(j=2,\cdots,n_{i}\). As the number of polygon vertices, \(n_{i}\) is the expected target number of the underestimated subsets. The above strategy is more reliable and effective than the existing measurement partitioning algorithm (e.g., sub-partitioning algorithm) for repartitioning unreasonable subsets. Take Figure 2 as an example. There is a measurement set consists of \(N_{z,k}=31\) measurements (originated from three closely spaced targets). From Figure 2a, we can see that sub-partitioning cannot partition the measurements into three relatively accurate subsets, which unreasonably treats the distant measurement as an individual subset. Compared with sub-partitioning, our method successfully divides the set of densely distributed measurements into three subsets (as shown in Figure 2b), which has more accurate measurements partitioning results and fits well with the tracking of extended targets with tracks-crossed. After getting the clustering centers (including the alternative points of the unreasonable target predictions), a new partition can be obtained by K-means++ clustering, i.e., put the measurements closet to the above position into the same subset. Suppose that \(X_{S,k|k-1}=\left\{x_{k|k-1}^{(i)}\right\}_{i=1}^{k-1}\) is the predictions of existing targets, \(Z_{k}\) is the measurement set at time \(k\), \(N_{z,k}\) is the number of the observed measurements, and \(J_{k-1}\) is the estimated number of targets at time \(k-1\). The whole procedure of the proposed algorithm is detailed as follows. Step 1.