We propose a sparse vector autoregressive (VAR) hidden semi-Markov model (HSMM) for modeling temporal and contemporaneous (e.g. spatial) dependencies in multivariate nonstationary time series. The HSMM's generic state distribution is embedded in a special transition matrix structure, facilitating efficient likelihood evaluations and arbitrary approximation accuracy. To promote sparsity of the VAR coefficients, we deploy an $l_1$-ball projection prior, which combines differentiability with a positive probability of obtaining exact zeros, achieving variable selection within each switching state. This also facilitates posterior estimation via Hamiltonian Monte Carlo (HMC). We further place non-local priors on the parameters of the HSMM dwell distribution improving the ability of Bayesian model selection to distinguish whether the data is better supported by the simpler hidden Markov model (HMM), or the more flexible HSMM. Our proposed methodology is illustrated via an application to human gesture phase segmentation based on sensor data, where we successfully identify and characterize the periods of rest and active gesturing, as well as the dynamical patterns involved in the gesture movements associated with each of these states.

翻译：我们提出一种稀疏向量自回归（VAR）隐半马尔可夫模型（HSMM），用于对多变量非平稳时间序列中的时间依赖性和同期（例如空间）依赖性进行建模。该HSMM的通用状态分布被嵌入一种特殊的转移矩阵结构中，从而在实现高效似然评估的同时达到任意近似精度。为促进VAR系数的稀疏性，我们采用基于$l_1$球投影的先验分布，该先验在保持可微性的同时能以正概率获得精确零系数，从而在每个切换状态下实现变量选择。这一设计亦有助于通过哈密顿蒙特卡洛（HMC）进行后验估计。我们进一步在HSMM驻留分布参数上施加非局部先验，可提升贝叶斯模型选择区分"数据更支持简单隐马尔可夫模型（HMM）还是更灵活HSMM"的能力。通过基于传感器数据的人类手势相位分割应用，我们成功识别并刻画了静止期与主动手势期，以及各状态所涉及手势运动的动态模式。