We propose and investigate a hidden Markov model (HMM) for the analysis of dependent, aggregated, superimposed two-state signal recordings. A major motivation for this work is that often these signals cannot be observed individually but only their superposition. Among others, such models are in high demand for the understanding of cross-talk between ion channels, where each single channel cannot be measured separately. As an essential building block, we introduce a parameterized vector norm dependent Markov chain model and characterize it in terms of permutation invariance as well as conditional independence. This building block leads to a hidden Markov chain sum process which can be used for analyzing the dependence structure of superimposed two-state signal observations within an HMM. Notably, the model parameters of the vector norm dependent Markov chain are uniquely determined by the parameters of the sum process and are therefore identifiable. We provide algorithms to estimate the parameters, discuss model selection and apply our methodology to real-world ion channel data from the heart muscle, where we show competitive gating.

翻译：我们提出并研究了一种隐马尔可夫模型，用于分析依赖性的、聚合的、叠加的双态信号记录。该工作的主要动机在于，这些信号通常无法单独观测，只能观测到其叠加结果。这类模型在理解离子通道间的串扰方面需求尤为迫切，因为每个单独通道无法被独立测量。作为核心构建模块，我们引入了一种参数化的基于向量范数的马尔可夫链模型，并从置换不变性和条件独立性角度对其进行了刻画。该构建模块进而形成了一种隐马尔可夫链求和过程，可用于在隐马尔可夫模型框架内分析叠加双态信号观测的依赖结构。值得注意的是，基于向量范数的马尔可夫链的模型参数由求和过程的参数唯一确定，因此是可辨识的。我们提供了参数估计算法，讨论了模型选择问题，并将该方法应用于来自心肌的真实离子通道数据，在其中展示了竞争性门控现象。