The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

中文翻译：

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具有随机束缚的单粒子扩散的隐马尔可夫模型

粒子扩散运动的统计数据通常作为粒子与环境相互作用的实验代理。然而，从观察到的轨迹推断物理特性具有挑战性。受最近一项实验的启发，我们在这里分析了粒子在表面短暂束缚的情况下进行二维布朗运动的问题。我们将问题建模为隐马尔可夫模型，其中观察物理位置并隐藏束缚状态。我们开发了一种交替最大化算法来推断粒子的隐藏状态并估计系统的物理参数。我们方法的关键是鞍点近似，其中涉及找到最可能的隐藏状态序列并从中估计物理参数。大量的数值测试表明，我们的算法能够可靠地找到模型参数，并且对初始猜测不敏感。我们讨论物理参数的不同状态以及算法在这些状态下的性能。我们还提供算法的免费软件实现。