Calculate continuous time transition probabilities

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

Arguments

Q

infinitesimal generator matrix of the continuous-time Markov chain of dimension c(N,N)

timediff

time differences between observations of length n-1 when based on n 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(N,N,n-1)

See also

Other transition probability matrix functions: generator(), 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)