Identifiable Hierarchical Factor Disentanglement in Dual Non-IID Time Series

Many nonstationary time series contain two coupled latent scales: a persistent sequence-level environment and recurrent mechanism states within each sequence. Flat switching representations conflate these sources of variation, obscuring the hierarchy, state cardinalities, and cross-environment state correspondence. We formalize this setting as dual non-IID time series and study when the persistent–recurrent hierarchy is identifiable from unlabeled sequences. Under identifiable within-sequence switching and translated invariant transition signatures, we recover the environment partition, both unknown cardinalities, and a shared indexing of recurrent states across environments. With sufficient transition variation, dynamic coordinates are identifiable up to a component permutation and componentwise transformations within contexts. Cross-context anchoring yields a global chart, and unique minimal target sufficiency further identifies the target-relevant subset. We also characterize representative recovery objects and ambiguities at the corresponding identification boundaries. These recovery stages motivate Hierarchical Dual Non-IID Component Estimation (HiDICE), a hierarchical variational state-space model with persistent–recurrent inference and additive signature sharing. Controlled experiments reproduce the predicted hierarchy recovery and stagewise behavior at the identification boundaries. Among evaluated label-free methods, HiDICE performs best under shortcut reversal on HPR-MNIST. It also achieves the lowest mean leave-one-subject-out heart rate error on PPG-DaLiA.