Hidden Markov Models and Shifted Compound Poisson-Gamma Renewal Process

Provides functions to define, simulate and fit hidden Markov models, hidden semi-Markov models, and their time-varying versions, to incompletely observed point process records. Such data arise when some events of an underlying point process are not recorded or are otherwise missing. The models treat the number of missing events between each pair of consecutively observed events of a point process record as a hidden process. The number of missing events may be fixed or modelled as a random variable whose distribution can vary over time. When the number of missing events is fixed, the observed event times can be modelled as a compound gamma renewal process. When it is random, the observed process is represented by a shifted compound Poisson-gamma renewal process, which may also depend on time, leading to a time-dependent shifted compound Poisson-gamma renewal process. These modelling frameworks are described in detail in Shahzadi, Wang, Parry and Bebbington (2023) and Shahzadi, Wang, Parry and Bebbington (2026) . The model parameters are estimated through direct nonlinear numerical optimization of the corresponding likelihood function, and each fit reports whether the change points of its time-dependent components are identified by the data. The package also provides methods for assessing model adequacy through residual analysis. In addition, it calculates and plots the most probable sequences of hidden states, allowing users to examine changes in the underlying missing-event process over time.