# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 4 Numerical Simulation  
This section demonstrates the mixed algorithm proposed above. The simulation experiment used 100 Monte Carlo simulations. Consider two linear dynamic systems called constant velocity (CV) and coordinated turn (CT), respectively, in which the state contains \(x=[p_{x},\dot{p}_{x},p_{y},\dot{p}_{y}]^{T}\):  
\[F_{k,1} = \begin{bmatrix}1&T&0&0\\ 0&1&0&0\\ 0&0&1&T\\ 0&0&0&1\end{bmatrix} \] \[F_{k,2} = \begin{bmatrix}1&\frac{\sin\omega T}{\omega}&0&-\frac{1-\cos \omega T}{\omega}\\ 0&\cos\omega T&0&-\sin\omega T\\ 0&\frac{1-\cos\omega T}{1}&1&\frac{\sin\omega T}{\omega}\\ 0&\sin\omega T&0&\cos\omega T\end{bmatrix} =F_{k,3}\]  
\[G_{k} = \begin{bmatrix}T^{2}/2&T&0&0\\ 0&0&T^{2}/2&T\end{bmatrix} ^{T}\] \[Q_{k} =G_{k}\Delta{G_{k}}^{T}\]  
where the sampling time is T = 1 s, \(\Delta=diag([\omega_{x}^{2}\quad\omega_{y}^{2}])\), and \(\omega_{x}^{w}=\omega_{y}^{2}\) = 0.1, \(\omega=\pi/40\) when i = 2 and \(\omega=-\pi/40\) when i = 3. The mode of maneuvering targets shifts between i = 1, 2, 3. For this simulation, the target starts with a CV motion and then two CT motions in the opposite direction respectively. The observation model is  
\[H_{k}^{n}= \begin{bmatrix}1&0&0&0\\ 0&0&1&0\end{bmatrix} \]  
The measurement noise variance is \(R=diag([15^{2}\quad 15^{2}])\) The sensor network comprises 15 nodes, and Figure 2 illustrates their network topology. The initial state and covariance are  
\[x_{0}=[2600\quad 20\quad 3800\quad 10]^{T}\]  
Figure 2: Topology of sensor network. The process noise and measurement noise distributions with outliers is  
\[\omega_{k}\sim\left\{ \begin{array}{l}N(0,Q)\;1-p_{0}\\ N(0,100Q)\;p_{0}\end{array} \right.\]  
\[v_{k}^{n}\sim\left\{ \begin{array}{l}N(0,R)\;1-p_{0}\\ N(0,100R)\;p_{0}\end{array} \right.\]  
where \(p_{0}\) is the probability of measuring outliers. The remainder of this section is devoted to presenting the results of the simulation experiments. Three methods are compared: (1) Distributed Consensus Kalman Filter, named DCKF; (2) Distributed Consensus Student-t Filter, named DCSTF; (3) Distributed Consensus Kalman Filter with IMM, named DCKFIMM; and (4) the proposed algorithm, named DDMDFIMM. The consensus iteration count for all of the above consensus filtering algorithms is 6, while the proposed algorithm just diffuses once. Target maneuvers are simulated using a Markov chain model as follows:  
\[ \begin{cases}F_{k}=F_{k,2}\;or\;F_{k,3}\;if\;rand>0.8\;;F_{k-1}=F_{k-1,1}\\ F_{k}=F_{k,1}\;or\;F_{k,3}\;if\;rand>0.8\;;F_{k-1}=F_{k-1,2}\\ F_{k}=F_{k,1}\;or\;F_{k,2}\;if\;rand>0.8\;;F_{k-1}=F_{k-1,3}\end{cases} \]  
At each time step, the target's motion model remains unchanged with a probability of 0.8. With a probability of 0.2, the model transitions to one of the two other models, each with a probability of 0.1. This results in a model transition probability matrix with dominant diagonal elements, reflecting the persistence of motion in real-world maneuvers. To evaluate performance in a Gaussian noise environment, the outlier occurrence probability \(p_{0}\) was set to zero. Figure 3 illustrates the actual and estimated trajectories from a single simulation. Figures 4 and 5 present the root mean squared error (RMSE) for position and velocity estimates generated by four algorithms. The DDMDFIMM algorithm demonstrates performance superior to other algorithms, while the DCKFIMM algorithm shows a considerably higher error. The DCSTF algorithm exhibits intermediate performance, while the DCKF shows considerably higher error. These findings indicate that, without outliers in maneuvering, Student-\(t\) models are more effective than Gaussian models, probably because Student-\(t\) treat dynamic measurement as outliers in the variance of motion mode. However, the DCKFIMM and DDMDFIMM algorithm, employing an IMM, achieves superior estimation accuracy; moreover, the DDMDFIMM employs an adaptive mixed model incorporating both Gaussian and Student-t distributions, showing better performance. We assigned an outlier occurrence probability of \(p_{0}=0.2\).