Time-series segmentation

Under the hidden Markov model, the time-series ${\displaystyle {\boldsymbol {y}}_{1:T}=({\boldsymbol {y}}_{1},...,{\boldsymbol {y}}_{T})}$ is assumed to have been generated as the system transitions among a set of discrete, hidden states ${\displaystyle z\in \{1,2,...,n\}}$. At each time ${\displaystyle t}$, a sample ${\displaystyle {\boldsymbol {y}}_{t}}$ is drawn from an observation (or emission) distribution indexed by the current hidden state, i.e., ${\displaystyle {\boldsymbol {y}}_{t}\sim P_{z_{t}}({\boldsymbol {y}}_{t})}$. The goal of the segmentation problem is to infer the hidden state at each time, as well as the parameters describing the emission distribution associated with each hidden state. Hidden state sequence and emission distribution parameters can be learned using the Baum-Welch algorithm, which is a variant of expectation maximization applied to HMMs. Typically in the segmentation problem self-transition probabilities among states are assumed to be high, such that the system remains in each state for nonnegligible time. More robust parameter-learning methods involve placing hierarchical Dirichlet process priors over the HMM transition matrix.[3]