# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 3 Distribution Diffusion Multi-Distribution Filtering of IMM
### The DDMDIM Filter
Use Equations (20)-(35). **Step 3 Calculate distribution probability**  
Update probability  
\[P_{k|k-1}^{i,0}=F_{k}P_{k-1}^{i,0}F_{k}^{T}+Q_{k-1}^{0} \tag{21}\]1  
where the likelihoods of the Gaussian and student-\(t\) filter \(\Lambda_{k,i}^{n,0}\) and \(\Lambda_{k,i}^{n,1}\) can be obtained using [27]. **Step 4 Fuse the mixed PDF**  
\[P_{k|k-1}^{i,0}=F_{k}P_{k-1}^{i,0}F_{k}^{T}+Q_{k-1}^{0} \tag{21}\]2  
\[P_{k|k-1}^{i,0}=F_{k}P_{k-1}^{i,0}F_{k}^{T}+Q_{k-1}^{0} \tag{21}\]3  
\[ \begin{split}& P_{mix,k}^{i,r}\\ &=\mu_{k}^{i,r,0}P_{k}^{i,r,0}+\mu_{k}^{i,r,1}\hat{P}_{k}^{i,r,1} \\ &+\sum_{s\in\mathcal{H}}\mu_{k}^{i,r,s}(\hat{x}_{k}^{i,r,s}-\hat {x}_{mix,k}^{i,r})(\hat{x}_{k}^{i,r,s}-\hat{x}_{mix,k}^{i,r})^{T}\end{split}  \tag{49}\]  
**Step 5 diffusion on fused PDF**  
Locally update  
\[P_{diff,k}^{i,r}=\sum_{i\in\mathcal{N}}\kappa_{k}^{i|j}(P_{k}^{i,r})^{-1} \tag{50}\]  
\[\hat{x}_{diff,k}^{i,r}=P_{diff,k}^{i,r}\sum_{i\in\mathcal{N}}\kappa_{k}^{i|j}(P _{k}^{i,r})^{-1}\hat{x}_{k}^{i,r} \tag{51}\]  
For each model \(r\in\mathcal{M}\), Steps 2-5 are processed concurrently and iteratively. **Step 6 Calculate model probability**  
Update model probability by the likelihoods, which can be expressed as follow for each model:  
\[\Lambda_{k}^{i,r}=\sum_{s\in\mathcal{H}}\mu_{k}^{i,r,s}\Lambda_{k}^{i,r,s} \tag{52}\]  
Then,  
\[\rho_{k}^{r}=\frac{1}{cc_{k}}\Lambda_{k}^{i,r}\cdot c_{k-1}^{r} \tag{53}\]  
\[cc_{k}=\sum_{r\in\mathcal{M}}\Lambda_{k}^{i,r}\cdot c_{k-1}^{r} \tag{54}\]  
The outputs of Steps 5 and 6 become the input for Step 1 in the next cycle. **Step 7 Output extraction**  
\[\hat{x}_{k}^{i}=\sum_{r\in\mathcal{M}}\rho_{k}^{r}\cdot\hat{x}_{k}^{i,r} \tag{55}\]  
The workflow of the proposed algorithm is depicted in Figure 1, with the corresponding pseudocode outlined in Algorithm 1. ```
Haven the initial values \(\hat{x}_{0}^{i,r,h}\), \(p_{0}^{i,r,h}\), \(\rho_{0}^{r}\) for \(r\in\mathcal{M}\) and \(\mu_{0}^{h}\) for \(h\in\mathcal{H}\) and the dof \(\eta\) for each iteration \(k\) at each node \(i\), starting the following iterationIterationIteration \(\hat{x}_{im,k-1}^{i,l}=\sum\limits_{r\in\mathcal{M}}\rho_{k-1}^{r|l}\cdot\hat{x }_{k-1}^{i,r}\)\(P_{im,k-1}^{i,l}\)\(=\sum\limits_{r\in\mathcal{M}}\rho_{k-1}^{r|l}\{P_{k-1}^{i,r}+[\hat{x}_{k-1}^{i,r}- \hat{x}_{im,k-1}^{i,r}]\cdot[\hat{x}_{k-1}^{i,r}-\hat{x}_{im,k-1}^{i,r}]^{T}\}\)For all\(\mathbf{r}\in\mathcal{M}\)doParallel filtering  Gaussian: \(\hat{x}_{0}^{i,r,0}\), \(P_{0}^{i,r,0}\) Student-\(t\): \(\hat{x}_{0}^{i,r,1}\), \(P_{0}^{i,r,1}\)Distribution probability \(\mu_{k}^{i,r,h}=\frac{\Lambda_{k}^{i,r}h_{k}^{i,r,h}}{\sum_{k\in\mathcal{M}} \Lambda_{k}^{i,r}h_{k-1}^{i,r,h}}\)Fusion \(\hat{x}_{mix,k}^{i,r,h}=\sum\limits_{s\in\mathcal{H}}\phi_{k}^{i,r,s}\hat{x}_ {k}^{i,r,s}\)\(\hat{P}_{k}^{i,r,1}=\frac{\eta_{k}^{i,r}}{\eta_{k}^{i,r}-2}P_{k}^{i,r,1}\)\(P_{mix,k}^{i,r,h}=\phi_{k}^{i,r,0}P_{k}^{i,r,0}+\phi_{k}^{i,r,1}\hat{P}_{k}^{i,r,1}\)\(+\sum\limits_{s\in\mathcal{H}}\phi_{k}^{i,r,s}\hat{x}_{k}^{i,r,s}-\hat{x}_{mix,k}^{i,r })^{T}\)Difusion \(P_{diff,k}^{i,r}=\sum\limits_{i\in\mathcal{N}}\kappa_{k}^{i|j}(P_{k}^{i,r})^{-1}\)\(\hat{x}_{diff,k}^{j,r}=P_{diff,k}^{i,r}\sum\limits_{i\in\mathcal{N}}\kappa_{k}^{i|j} {(P_{k}^{i,r})^{-1}\hat{x}_{k}^{i,r}}\)EndforUpdate model probability \(\rho_{k}^{r}=\frac{1}{\alpha_{k}}\Lambda_{k}^{i,r}\cdot c_{k-1}^{r}\)Extraction \(\hat{x}_{k}^{i}=\sum\limits_{r\in\mathcal{M}}\rho_{k}^{r}\cdot\hat{x}_{k}^{i,r}\)
```  
**Algorithm 1**Distributed diffusion multi-distribution of IMM filter. Figure 1: Architectural diagram of DMDDIMM filter.