# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 2 Preliminaries and Problem Formulation
### Multi-Noise Distribution  
Realistic noise distributions are often subject to disturbances that can generate outliers, leading to measurement noise that deviates from a Gaussian distribution. Consequently, traditional state estimation methods based on Gaussian assumptions may perform poorly. While Gaussian noise assumptions are frequently employed, the presence of unknown perturbations can result in outliers, which are often characterized by heavy-tailed distributions. To address these deviations, robust statistical models are commonly utilized. Therefore, it is necessary to consider both Gaussian and robust distributions, and a model base of distributions can be described as \(\mathcal{H}\triangleq\{h|h=0,1\}\).  
Generally, the process and measurement noise of Gaussian \(h=0\) can be described as follows: \(p(w_{k})=\mathrm{N}(w_{k};0,Q_{k}^{0})\)\(p(v_{k})=\mathrm{N}(v_{k};0,R_{k}^{0})\) One of the robust statistical modelsabout noise distribution is Student-\(t\) distribution. Similar to Gaussian distribution, it can be described as an equation such that a random variable \(x\) obeys the following equation:  
\[St(x;m,P,\eta)=M(1+\frac{1}{\eta}(x-m)^{T}P^{-1}(x-m))^{-\frac{\eta+nx}{2}} \tag{3}\]  
where  
\[M=\frac{\Gamma(\frac{\eta+n_{x}}{x})}{\Gamma(\frac{1}{2})}\frac{1}{(\eta\pi) \frac{n_{x}}{2}}\frac{1}{\sqrt{\det(P)}}\]  
where \(m\) is the mean, \(\eta\) is the degrees of freedom (dof), \(P\) is a positive definite symmetric matrix, \(\Gamma\) denotes the Gamma function, and \(n_{x}\) is the dimension of \(x\). Meanwhile, the process and measurement noise of Student-\(t\)\(h=1\) can be described analogously as follows:  
\[p(w_{k})=\text{St}(w_{k};0,Q_{k}^{1},\eta_{k}) \tag{4}\]  
\[p(v_{k})=\text{St}(v_{k};0,R_{k}^{0},\eta_{k}) \tag{5}\]  
While the Student-\(t\) distribution is well-suited for modeling heavy-tailed noise due to its inherent long-tailed property, it may not accurately represent normal noise conditions. Given that Gaussian noise is more prevalent in typical applications, relying solely on the Student-\(t\) distribution, while enhancing robustness, can degrade overall estimation accuracy. To address the tracking problem of maneuvering targets, common methods include variational Bayes and IMM approaches, with IMM being the most extensively researched. The challenges in maneuvering target tracking are multifaceted, and this paper addresses three specific aspects. First, the target's maneuvering, potentially due to its own control issues, can lead to signal anomalies received by the sensor. As discussed earlier, the Student-\(t\) distribution can be used to mitigate this issue. Second, target maneuvering introduces uncertainty in position information, potentially exceeding the coverage of a single sensor and necessitating collaborative localization through a multi-sensor network. This paper employs a diffusion algorithm within a distributed sensor network to address this challenge. Third, accurate target state prediction requires an appropriate motion model; this paper utilizes the widely adopted IMM algorithm for this purpose.  
Therefore, the core problem addressed in this paper is how to effectively utilize the IMM algorithm to balance robustness and estimation accuracy for maneuvering target estimation in diffusion sensor networks operating in environments with potential heavy-tailed noise.