How Poisson Hidden Markov Models Enable Count‑Based Time‑Series Regression

The article introduces a Poisson Hidden Markov Model (Poisson HMM) that combines two stochastic processes—a Poisson process for count data and a discrete k‑state Markov chain—to model integer‑valued time‑series such as daily click counts.

Poisson component

Observed values y_t are expressed as the sum of a predicted mean μ̂_t and a residual ε_t, where ε_t follows a zero‑mean normal distribution N(0,σ²). The count variable y_t is assumed to follow a Poisson distribution with mean μ_t, allowing each time step its own mean.

The regression matrix X (size n × (m+1)) includes an intercept column, and the fitted coefficient vector β̂ (size (m+1) × 1) yields the exponential linear predictor:

The dot product x_t·β̂ is exponentiated to guarantee a non‑negative mean, satisfying the key requirement for integer count modeling.

Hidden Markov component

A k‑state discrete Markov chain governs the hidden state j ∈ {1,…,k}. The Poisson mean is indexed by the hidden state, giving state‑specific means μ̂_t_j:

The transition matrix P (elements p_ij) and the state probability vector π_t evolve the hidden state over time. The probability of being in state j at time t is π_tj.

Because the true state is unknown, the model predicts a single mean μ̂_t by taking the expectation over all possible states:

Training and parameter estimation

Training the Poisson HMM involves estimating the coefficient matrix β̂_s and the transition matrix P. Maximum likelihood estimation (MLE) or the Expectation‑Maximization (EM) algorithm maximizes the joint likelihood of the observed series y:

The log‑likelihood is differentiated with respect to each transition probability p_ij and each state‑specific coefficient β̂_q_j. Setting the derivatives to zero yields a system of equations solved by optimization methods such as Newton‑Raphson, Nelder‑Mead, or Powell’s algorithm.

During optimization, each row of P must sum to 1 and each element stay within [0,1]. To enforce these constraints, a proxy matrix Q (size k × k) is introduced, allowing unconstrained optimization of q_ij and then normalizing to obtain valid probabilities:

References

Cameron, A., & Trivedi, P. (2013). Regression Analysis of Count Data (2nd ed.). Cambridge University Press.

James D. Hamilton, Time Series Analysis , Princeton University Press, 2020.