# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 3 Distribution Diffusion Multi-Distribution Filtering of IMM
### The DDMDIM Filter  
This subsection presents a novel multi-distribution filter designed for single sensors operating under both heavy-tailed process and measurement noise. Existing approaches, which rely solely on Student-\(t\) or Gaussian filters, suffer from inherent limitations. Student-\(t\) filters can exhibit performance degradation or require frequent parameter tuning, while Gaussian filters can diverge in the presence of outliers. To leverage the advantages of both filter types, we propose the following two-hypothesis framework. In this paper, it is mostly important to fuse the above parallel filters so that a probability \(\mu_{k}^{h},\sum_{k\in\mathcal{H}}\mu_{k}^{h}=1,\mathcal{H}=\{0,1\}\) is proposed as the weight of different filter, where \(\mu_{k}^{0}\) is probability of Gaussian and \(\mu_{k}^{h}\) is one of Student-\(t\). According to the full probability theorem,  
\[\bar{p}(x_{k}|Z_{k})=\sum_{i\in\mathcal{N}}\kappa_{k}^{i}p^{i}(x _{k}|Z_{k})\propto\sum_{h\in\mathcal{H}}\mu_{k}^{h}p^{i}(x_{k}|Z_{k},\mathcal{ H}_{h}) \tag{19}\] \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]  
Given the distribution probability \(\mu_{k-1}^{r}\), according to [27], is associated with likelihood. Under the Gaussian distribution, the diffusion Bayesian filter can be represented as a information filter-like iteration with prior state estimate \(\hat{x}_{0}^{0}\) and covariance matrix \(\hat{P}_{0}^{0}\), iterates according to the following steps for k > 1:  
(1) Time update  
\[\hat{x}_{k|k-1}^{i,0}=F_{k}\hat{x}_{k-1}^{i,0} \tag{20}\]  
\[P_{k|k-1}^{i,0}=F_{k}P_{k-1}^{i,0}F_{k}^{T}+Q_{k-1}^{0} \tag{21}\]  
(2) Measurement update  
\[\hat{z}_{k}^{i,0}=z_{k}^{i,0}-H_{k}^{i}\hat{x}_{k|k-1}^{i,0} \tag{22}\]  
\[S_{k}^{i,0}=\sum_{j\in\mathcal{N}}\left(H_{k}^{j}\right)^{\top}(R_{k}^{j,0})^{ -1}H_{k}^{j} \tag{23}\]  
\[q_{k}^{i,0}=\sum_{j\in\mathcal{N}}\left(H_{k}^{j}\right)^{\top}(R_{k}^{j,0})^{ -1}\hat{z}_{k}^{i,0} \tag{24}\]  
\[\left(P_{k}^{i,0}\right)^{-1}\hat{x}_{k}^{i,0}=\left(P_{k-1}^{i,0}\right)^{-1} \hat{x}_{k|k-1}^{i,0}+\left(H_{k}^{i}\right)^{\top}(R_{k}^{i,0})^{-1}\hat{z}_{ k}^{i,0} \tag{25}\]\[{(P_{k}^{i,0})}^{-1}={(P_{k|k-1}^{i,0})}^{-1}+{S_{k}^{i,0}} \tag{26}\]  
Similarly, under the Student-\(t\) distribution hypothesis, and given the linearity of the system model, an information filter is implemented according to [28]. The recursive algorithm, initialized with prior state estimate \(\hat{x}_{0}^{1}\) and covariance matrix \(\hat{P}_{0}^{1}\), iterates according to the following steps for k > 1:  
1. Time update \[\hat{x}_{k|k-1}^{i,1}=F_{k}\hat{x}_{k-1}^{i,1}\] (27) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]0 (28)
2. Measurement update \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]1 (29) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]2 (30) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]3 (31) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]4 (32) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]5 (33) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]6 (34) \[=\mu_{k}^{0}\text{N}(x_{k};\hat{x}_{k}^{0},P_{k}^{0})+\mu_{k}^{1} \text{St}(x_{k};\hat{x}_{k}^{1},P_{k}^{1},\eta_{k})\]7 (35)  
The foregoing description details the concurrent processing of two parallel filters. The mixed posterior estimation state \(\hat{x}_{k}^{i}\) and the covariance \(P_{k}^{i}\) corresponding to the mixed posterior distribution can be found in our previous work [27].