# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 3 Distribution Diffusion Multi-Distribution Filtering of IMM
### The DDMDIM Filter
Note that because of the dof, the covariance of the Student-\(t\) distribution is different from the one of the Gaussian distribution when fusing different distributions. For Gaussian distribution, there are  
\[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]8  
For the t distribution, the variance is  
\[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]9  
The expression for the covariance of the mixed posterior distribution is thus derived. \[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]0  
Note that the increasing dof with each measurement update, leads to a more Gaussian noise distribution, causing the algorithm to converge to a Kalman filter. This necessitates the use of approximate methods. One of the simplest ways is to enforce \(\eta_{k}=\eta_{k-1}\) so that the dof will not increase in each iteration. But the actual posterior density is \(p(x_{k})=\text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k-1}+n_{z})\) instead of \(q(x_{k})=\text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k-1})\). In order to make \(q(x_{k})\) close to \(p(x_{k})\), a scaling factor \(c\) needs to be added. In this paper, we use the moment matching method to make the variances of \(q(x_{k})\) and \(p(x_{k})\) equal, thereby making their distributions close. This method has been described in detail in our previous research [27]. For the sake of convenience, it is not be discussed here while just describing as follows:  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]1  
where \(\eta>2\) and \(\tilde{\eta}=\eta+n_{z}>2\). In the diffusion algorithm, a single node in a network of nodes accomplishes information fusion similar to central filtering using interaction covariance in response to observations from neighboring nodes. After obtaining the mixed state and covariance of the multi-distribution nodes, the information from a single node is diffusion across the distributed network. It can be obtained by the mixed interaction covariance in distributed network, which can be described like in central filtering as follows:  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]2  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]3  
The combination of IMM and diffusion algorithms used in this paper involves performing diffusion first, followed by IMM. The advantage of this approach is that only one IMM calculation is required, rather than a separate calculation for each node. This greatly reduces the computational overhead when there are a large number of nodes and motion models. Having established the single-sensor algorithm and the multi-sensor diffusion strategy for distribution, we now present the extension of these results to an IMM. For following algorithm, \(i,j\in\mathcal{N}\) means node, \(r,l\in\mathcal{M}\) means model, and \(h,s\in\mathcal{H}\) means distribution, while the scripts with \(ina\) means interacting model data, mix means mixed distribution data, and \(dif\) means diffusion data. **Step 1 Input interacting**  
For the posteriori state \(x_{k-1}\) and posteriori error covariance matrix \(P_{k-1}\) at \(k-1\) instant, they can be inputs at k instant through a series of interaction calculations:  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]4 \[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]5  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]6 \[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]7 \[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]8  
where \(\rho_{k-1}^{r|l}\) is the transition probability from model r to l, which is related to the probability as follows:  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]9  
\[P_{k|k-1}^{i,0}=F_{k}P_{k-1}^{i,0}F_{k}^{T}+Q_{k-1}^{0} \tag{21}\]0  
**Step 2 Parallel filtering**  
The interaction data of each model of all nodes from the preceding step served as input for the subsequent filtering of two parallel filter of \(n\) node.