Consider data of the general form \(\boldsymbol{P}(1:T):=(P^1(1), \dots, P^m(1), \dots, P^1(T), \dots, P^m(T))\) for an \(m\)-variate time series of length \(T\). We use the notation \(\boldsymbol{P}(t):=(P^1(t), \dots, P^m(t))\). To consider the general case, further assume that the mixture distributions, the initial mixture probabilities and transition probabilities can all depend on covariates \(\boldsymbol{z}(t)\).
The joint likelihood of observations \(\boldsymbol{P}(1:T)\) and latent states \(\boldsymbol{S}(1:T)\) given model parameters \(\boldsymbol{\theta}\) and covariates \(\boldsymbol{z}(1:T):=(\boldsymbol{z}(1), \dots, \boldsymbol{z}(T))\) can be written as \[ \mathbb{P}(\boldsymbol{P}(1:T), \boldsymbol{S}(1:T)|\boldsymbol{\theta}, \boldsymbol{z}(1:T))=\pi_{i}(\boldsymbol{z_{1}})\boldsymbol{b}_{S(1)}(\boldsymbol{P}(1)|\boldsymbol{z}(1))\prod_{t=1}^{T-1}\gamma_{i,j}(\boldsymbol{z}(t))\boldsymbol{b}_{S(t)}(\boldsymbol{P}(t+1)|\boldsymbol{z}(t+1)), \] where
Back to top