Builds the transition probability matrix of an HSMM-approximating HMM

Hidden semi-Markov models (HSMMs) are a flexible extension of HMMs, where the state duration distribution is explicitly modelled. For direct numerical maximum likelhood estimation, HSMMs can be represented as HMMs on an enlarged state space (of size \(M\)) and with structured transition probabilities.

This function computes the transition matrix to approximate a given HSMM by an HMM with a larger state space.

Usage

tpm_hsmm(omega, dm, Fm = NULL, sparse = TRUE, eps = 1e-10)

Arguments

omega: embedded transition probability matrix of dimension c(N,N) as computed by tpm_emb.
dm: state dwell-time distributions arranged in a list of length(N). Each list element needs to be a vector of length N_i, where N_i is the state aggregate size.
Fm: optional list of length N containing the cumulative distribution functions of the dwell-time distributions.
sparse: logical, indicating whether the output should be a sparse matrix. Defaults to TRUE.
eps: rounding value: If an entry of the transition probabily matrix is smaller, than it is rounded to zero. Usually, this should not be changed.

Value

extended-state-space transition probability matrix of the approximating HMM

Examples

# building the t.p.m. of the embedded Markov chain
omega = matrix(c(0,1,1,0), nrow = 2, byrow = TRUE)
# defining state aggregate sizes
sizes = c(20, 30)
# defining state dwell-time distributions
lambda = c(5, 11)
dm = list(dpois(1:sizes[1]-1, lambda[1]), dpois(1:sizes[2]-1, lambda[2]))
# calculating extended-state-space t.p.m.
Gamma = tpm_hsmm(omega, dm)