## 3 Prediction-Driven Measurement Sub-Partitioning
The lower and upper threshold of target number is calculated as \(K_{L}=J_{k-1}\) and \(K_{U}=J_{k-1}+J_{\beta,k}\times J_{k-1}+J_{\gamma,k}\), respectively. For each K value, there is an individual partition containing K subsets. Here, \(J_{\beta,k}\) and \(J_{\gamma,k}\) are approximated by 1 and \(J_{k-1}\), respectively. Figure 2: Sub-partitioning vs. prediction-driven measurement sub-partitioning. (**a**) The measurements are partitioned by sub-partitioning; (**b**) The measurements are partitioned by prediction-driven measurement sub-partitioning. Step 2. Judge whether the underestimation problem exists in the subsets of the first partition by using \(\frac{N_{x,k}}{\lambda}-J_{k-1}\). If \(\frac{N_{x,k}}{\lambda}-J_{k-1}\geq 1\), it means that the number of targets has been underestimated at the previous time \(k-1\), where \(\lambda\) is the expected number of observed measurements from each target and equal to the Poisson rate. Step 3. For each K value, the initial clustering centers \(C^{I_{k-1}}=\left\{c_{1},\cdots,c_{I_{k-1}}\right\}\) are obtained by the predicted positions of survival targets, and the initial partition \(P_{1}=\left\{W^{(i)}\right\}_{i=1}^{I_{k-1}}\) is gained by \(C^{I_{k-1}}\), where \(W^{(i)}\) is the \(i\)th subset. For each subset \(W^{(i)}\in P_{1}\), \(\frac{|W^{(i)}|}{\lambda}\geq 2\) is used to find the underestimated subset and corresponding predicted state \(x_{k|k-1}^{(i)}\), where \(\left|W^{(i)}\right|\) is the number of observed measurements in \(i\)th subset. Round \(\frac{|W^{(i)}|}{\lambda}\) up to the nearest whole number \(n_{i}\), and let \(n_{i}\) as the number of the alternative points of the predicted state \(x_{k|k-1}^{(i)}\). In this case, the predicted state \(x_{k|k-1}^{(i)}\) needs to be replaced by \(n_{i}\) alternative points \(P^{n_{i}}=\left\{p_{j}\right\}_{j=1}^{n_{i}}\). This strategy may help us to obtain the alternative initial clustering centers \(C_{n_{i}}^{i}=\left\{c_{j}^{i}\right\}_{j=1}^{n_{i}}\), where \(c_{j}^{i}\) is determined according to \(p_{j}\). Step 4. Calculate the distance between each measurement and the predicted state \(x_{k|k-1}^{(i)}\), and sort them from smallest to largest. Afterward, the one in the middle is selected as the first alternative initial clustering center \(c_{1}^{i}\). The distance between \(c_{1}^{i}\) and \(x_{k|k-1}^{(i)}\) is denoted by \(TH\). Let \(c_{1}^{i}\) be the first alternative point \(p_{1}\). The other \(n_{i}-1\) alternative points \(P^{n_{i}-1}=\left\{p_{2},p_{3},\cdots,p_{n_{i}}\right\}\) can be considered as the vertexes of a regular polygon, and their locations are calculated by Equation (3). Then, all alternative points \(P^{n_{i}}=\left\{p_{1},P^{n_{i}-1}\right\}\) are obtained to replace the corresponding predicted state \(x_{k|k-1}^{(i)}\). Step 5. Compute K-means clustering centers \(C_{n_{i}-1}^{i}=\left\{c_{j}^{i}\right\}_{j=2}^{n_{i}}\) with alternative points \(P^{n_{i}-1}\). Then, the alternative initial clustering center set \(C_{n_{i}}^{i}=\left\{c_{1}^{i},C_{n_{i}-1}^{i}\right\}\) is obtained to replace the corresponding clustering center gained by \(x_{k|k-1}^{(i)}\) in \(C^{I_{k-1}}\). Step 6. Repeat Steps 3-5 until finishing selecting the alternative initial clustering centers \(C^{under}=\left\{C_{n_{i}}^{i}\right\}_{i=1}^{N}\) in all the underestimated subsets, where \(N\) is the number of the underestimated subsets. These centers are included into \(C^{I_{k-1}}\). Other \(K-J_{k-1}\) measurements are selected as \(C^{K-I_{k-1}}=\left\{c_{I_{k-1}+1},c_{I_{k-1}+2},\cdots,c_{K}\right\}\), which satisfy the selection probability \(\frac{D{(x_{k}^{(i)})}^{2}}{\sum\limits_{z_{k}\in\mathcal{Z}_{k}}D{(z_{k})}^{2}}\), where \(D(x)\) is the shortest distance from a measurement \(z_{k}^{(i)}\) to the closest initial center. Step 7. \(J_{k-1}\) is changed to be \(J_{k-1}^{*}=J_{k-1}-N+\sum\limits_{i=1}^{N}n_{i}\), where \(N\) is the number of the underestimated subsets and \(n_{i}\) is the expected target number of the \(i\)th subset.