## 3 Prediction-Driven Measurement Sub-Partitioning
More importantly, the problem of underestimation in the target number is considered and solved explicitly. \(\frac{N_{z,k}}{\lambda}-J_{k-1}\geq 1\) is presented to judge whether there is the underestimation of the target number, and \(\frac{|W^{(i)}|}{\lambda}\geq 2\) is used to find the underestimated subset and corresponding predicted state if the condition is satisfied (i.e., the number of targets have been underestimated). In this way, \(n_{i}\) alternative points of the predicted state can be found by introducing the Geometry of a regular polygon, which serves to determine the alternative initial centers of the underestimated subset. The whole procedures of PMS are detailed in Algorithm 1. ### Update

As the accurate measurement partitioning results can be obtained by the PMS, the subsets of each partition is used to update each predicted Gaussian Mixture components. By using the measurement pseudo-likelihood function \(L_{Z_{k}}(x)\), the update equation is given by

\[D_{k|k}(x)=L_{Z_{k}}(x)D_{k|k-1}(x) \tag{10}\]

\[L_{Z_{k}}(x)=\underbrace{1-(1-e^{-\gamma(x)})p_{D}(x)}_{\text{No detections of the target}}+\] \[\underbrace{e^{-\gamma(x)}p_{D}(x)\times\sum_{P\angle Z_{k}} \omega_{P}\sum_{W\in P}\frac{\gamma(x)}{d_{W}}\cdot\prod_{z_{k}\in W}\frac{ \phi_{z_{k}}(x)}{\lambda_{k}\epsilon_{k}(z_{k})}}_{\text{At least one detection of the target}} \tag{11}\]

where \(D_{k|k}(x)\) is the posterior intensity, \(D_{k|k-1}(x)\) is the predicted intensity, \(p_{D}(x)\) is the detection probability of target, \(\gamma(x)\) is the expected number of measurements from the target, \(\phi_{z_{k}}(x)\) is the Gaussian likelihood function of the generate measurement for a single target, \(\lambda_{k}\) is the mean of clutter measurements, and \(c_{k}(z_{k})\) is the spatial distribution of the clutter over the surveillance volume. Note that the non-negative coefficients \(\omega_{P}\) and \(d_{W}\) are defined for \(P\) and \(W\), respectively. \(P\angle Z_{k}\) denotes that there are \(P\) partitions, each of which is able to divides the measurement set \(Z_{k}\) into several non-empty subsets, and \(W\in P\) denotes that \(W\) is a subset in the partition \(P\). \(L_{Z_{k}}(x)\) can be separated into two parts according to the situation of measurement detection. Thus, the above update equation can be evolved to a more clear form containing two aspects (i.e., the update in the case with no detection of targets and the update in the case with detection of targets), which are as follows,

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]0where \(D_{k|k}^{ND}(x)\) serves to the case of non-detected targets and \(D_{k|k}^{D}(x,W)\) serves to the case of detected targets. In these cases, \(\omega_{k|k}^{(j)}\) is used to denote the updated weight of the Gaussian component, \(m_{k|k}^{(j)}\) is the updated mean of the Gaussian component, and \(P_{k|k}^{(j)}\) is the updated covariance of the Gaussian component. The update for no detection case is given as follows,

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]1

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]2

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]3

where \(p_{D}(j)\) is the detection probability of target and \(\gamma(j)\) is the expected number of measurements from the target. The update for detected targets is

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]4

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]5

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]6

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]7

where the coefficient \(\Gamma^{(j)}\) and \(\Phi_{W}^{(j)}\) are

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]8

\(\delta_{|W|,1}\) is the Kronecker delta and the coefficient \(\phi_{W}^{(j)}\) is

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]9

\[z_{k}^{(i)}=H_{k}x_{k-1}^{(i)}+e_{k} \tag{5}\]0

\[z_{k}^{(i)}=H_{k}x_{k-1}^{(i)}+e_{k} \tag{5}\]1

Here, the standard Kalman filter is used to obtain the mean \(m_{k|k}^{(j)}\) and covariance \(P_{k|k}^{(j)}\) of the Gaussian components.