# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 1 Introduction  
Filtering is a commonly used method for estimating the state from noisy measurements, and it has been widely applied in fields such as target tracking and multi-sensor fusion [1]. The Kalman filter (KF) and its variants, such as the extended Kalman filter (EKF) [2] and unscented Kalman filter (UKF) [3], can provide reliable results. In the domain of target tracking, the Kalman filter continues to hold a prominent position, drawing the attention of numerous researchers. However, the dynamic and multifaceted characteristics of real-world environments have given rise to novel challenges, including non-Gaussian noise distributions, multi-sensor information fusion requirements, and outliers stemming from maneuvering targets. Conventional Kalman filters and their related extensions are insufficient to address the challenges posed by these emerging issues. With the development of sensor networks, researchers have begun to consider distributed multi-sensor scenarios [4, 5, 6, 7]. Several distributed frameworks, such as consensus [4, 5] and diffusion architectures [6, 7], have been developed. Both of the above methods are commonly used distributed filtering information fusion algorithms, which have advantages such as scalability and real-time performance. Among these, consensus algorithms offer the benefits of estimate consistency and tolerance for loss of some communication nodes. However, they exhibit sensitivity to local estimation errors, potentiallyconverging to suboptimal solutions and communication burden of repeated iterations. In contrast, diffusion-based fusion offers lower communication requirements compared to consensus algorithms and is applicable in a wider range of scenarios. Moreover, diffusion algorithms demonstrate greater robustness and reduced sensitivity to local errors, making them suitable for use with maneuvering targets with heavy-tailed noise. Model uncertainty is a common challenge in filtering or multiple-sensor fusion [8], arising from a lack of prior knowledge about the system or disturbances [9]. This can lead to the statistical characteristics of model noise deviating from a Gaussian distribution, often exhibiting heavy-tailed behavior [10]. Since standard KF assumes Gaussian noise, they are ill-suited for environments with model uncertainty. As a result, various robust methods have been developed. Monte Carlo methods [11], for instance, approximate arbitrary distributions through sampling techniques, but they are computationally expensive. Some researchers have adopted optimization criteria distinct from those used in Kalman filtering. In [12], a filter based on the maximum entropy Kerman criterion is proposed, which can achieve relatively robust results. During observation, sensors can be susceptible to abrupt interference, human influence, or intrinsic saturation limits, leading to measurements frequently contaminated by heavy-tailed noise. Some researchers have directly chosen distributions with heavy-tailed characteristics to model uncertainties [13; 14]. In [15], an algorithm based on the Student's \(t\)-distribution is proposed, where the degrees of freedom of the distribution are adaptively adjusted to obtain reliable estimation results. Similarly, in [16], an algorithm based on the Laplace distribution is introduced, which, compared to the Student's \(t\)-distribution, requires fewer estimation parameters. Considering that non-Gaussian noise is not prevalent in practical environments, these algorithms, while yielding relatively stable results, tend to be overly conservative in estimation. The Multiple Model (MM) algorithms have garnered significant attention for addressing estimation problems involving model uncertainty [17; 18]. Process models are typically represented by Markov jump systems, where multiple models are governed by a Markov chain [7]. A widely applied approach is the classical IMM algorithm [18], which effectively balances estimation accuracy with computational complexity. Inspired by multi-model approaches, [PERSON].