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），用于对多元非平稳时间序列中的时间依赖性和同期（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，从而实现了高效的似然计算与任意近似精度。为促进VAR系数的稀疏性，我们采用$l_1$-球投影先验，该先验结合了可微性与以正概率获得精确零值的特性，从而在每个切换状态下实现变量选择。这一设计还有助于通过哈密顿蒙特卡洛方法进行后验估计。进一步地，我们在HSMM驻留分布的参数上设置非局部先验，增强了贝叶斯模型选择区分更简单的隐马尔可夫模型（HMM）与更灵活的HSMM何者更优地支持数据的能力。通过基于传感器数据的人体手势阶段分割应用，我们验证了所提方法的有效性，成功识别并刻画了静止期与活跃手势期，以及这些状态所对应手势运动中涉及的动态模式。