tpm_ct: Calculate continuous time transition probabilities

In LaMa: Fast Numerical Maximum Likelihood Estimation for Latent Markov Models

Calculate continuous time transition probabilities

Description

A continuous-time Markov chain is described by an infinitesimal generator matrix Q. When observing data at time points t_1, \dots, t_n the transition probabilites between t_i and t_{i+1} are caluclated as

\Gamma(\Delta t_i) = \exp(Q \Delta t_i),

where \exp() is the matrix exponential. The mapping \Gamma(\Delta t) is also called the Markov semigroup. This function calculates all transition matrices based on a given generator and time differences.

Usage

tpm_ct(Q, timediff, rates = NULL, ad = NULL, report = TRUE)

tpm_cont(Q, timediff, rates = NULL, ad = NULL, report = TRUE)

Arguments

Q	infinitesimal generator matrix of the continuous-time Markov chain of dimension c(nStates, nStates)
timediff	time differences between observations of length nObs-1 when based on nObs observations
rates	optional vector of state-dependent rates for MM(M)PP fitting. For the MM(M)PP likelihood, the matrices needed in the forward algorithm are \exp((Q - \Lambda) \Delta t), where \Lambda is a diagonal matrix with the state-dependent rates on the diagonal.
ad	optional logical, indicating whether automatic differentiation with RTMB should be used. By default, the function determines this itself.
report	logical, indicating whether Q should be reported from the fitted model. Defaults to TRUE, but only works if ad = TRUE.

Value

array of continuous-time transition matrices of dimension c(nStates, nStates, nObs-1)

See Also

Other transition probability matrix functions: generator(), generator_g(), tpm(), tpm_emb(), tpm_emb_g(), tpm_g(), tpm_g2(), tpm_p()

Examples

# building a Q matrix for a 3-state cont.-time Markov chain
Q = generator(rep(-2, 6))

# draw random time differences
timediff = rexp(100, 10)

# compute all transition matrices
Gamma = tpm_cont(Q, timediff)

LaMa documentation built on July 25, 2026, 1:08 a.m.