## 4 Adaptive Sample-Size Unscented Particle Filter with Partitioned Sampling  
### Partitioned Sampling  
In this section, we concentrate on the combination of partitioned sampling and high-maneuvering target tracking algorithm in three-dimensional motion space. In simple terms, partitioned sampling divides the system state space into several independent subspaces, and for each one, to apply the transition function and to perform a weighted resampling operation. The particles are independently sampled in each subspace with their respective best strategies, then we can further predict the optimal sub-states. Figure 2: Changes of sample size with different initial sample size. (**a**) The initial sample size is 100. (**b**) The initial sample size is 500. (**c**) The initial sample size is 1000. The final output is the weighted sum of all predicted sub-states. It is assumed that the state space \(\chi\) can be partitioned as \(\chi=\chi^{1}\times\cdots\times\chi^{M}\), while \(g(\cdot):\chi\rightarrow\mathbb{R}\) is a weighting function used to weight the sub-states in weighted resampling, and the particles are propagated by the proposal function \(f_{k}(\cdot):\chi\rightarrow\chi\) which can be decomposed as follows [41],  
\[f_{k}(X_{k-1},n_{k}^{X})=f_{k}^{M}\circ f_{k}^{M-1}\circ\cdots\circ f_{k}^{2} \circ f_{k}^{1}(X_{k-1}), \tag{26}\]  
where \(X_{k}\) is the state vector at time \(k\), \(n_{k}^{X}\) variable correlated with state X and time, \(\circ\) is the usual function composition operator. By definition, \(f_{k}^{i}\) can propagate particles over subspace \(\chi^{1}\times\cdots\times\chi^{M}\) and modify the sub-states of particles defined on \(\chi^{i}\) in practice. Exploiting the features of weighted resampling, partitioned sampling achieves the same result by substituting the \(f_{k}\) propagation by a sequence of applications of the \(f_{k}^{i}\) followed by weighted resampling, as shown in Figure 3. In this figure, operations \"\(\ast f_{k}^{i\omega}\) refer to propagations of particles using proposition function \(f_{k}^{i}\) as defined above, and operations \"\(\sim g_{k}^{i}\)\" refer to weighted resampling w.r.t. the importance function \(g_{k}^{i}\). It is noted that in the Cartesian coordinate system, the sub-states of the target in each direction of motion space are independent of each other, and are not affected by maneuvering motion in other directions, i.e., the motion space is three-dimensionally orthogonal. Based on the idea of dynamic decomposition and synthesis, the overall motion state of the target can be expressed as a superposition of motions in all directions. Figure 4 shows the structural diagram of three-dimensional space partitioned sampling. By using partitioned sampling, the three-dimensional motion space could be decomposed into three independent subspaces. Suppose that particles are sampled in each one-dimensional subspace, \(N_{x}\), \(N_{y}\) and \(N_{z}\) indicate the particles number of each direction respectively, then it is easily find the total number of particles is \(N=N_{x}+N_{y}+N_{z}\), and the number of particle species is \(N_{x}\times N_{y}\times N_{z}\). If particles  
Figure 4: Partitioned sampling structural diagram in three-dimensional motion space. Figure 3: Partitioned sampling condensation diagram. are sampled directly in three-dimensional space, the number of particle species is \(N_{x}+N_{y}+N_{z}\), with the same sample size as the former, and obviously, the spatial coverage of particles is lower than the former. Next, the algorithm complexity is compared in both cases. According to the analysis in Section 2 of this paper, the state vector dimension is 12, and the measurement vector dimension is 3. It is assumed that the state one-step prediction time in filtering is \(T_{f}\), and the measurement one-step prediction time is Th. Using partitioned sampling, the state space is decomposed into three independent subspaces, the state one-time prediction time is reduced to \(T_{f}/9\), and the measurement one-step prediction time is reduced to \(T_{h}/3\).