tpm_cont function

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 [REMOVE_ME]$\Gamma(\Delta t_i) = \exp(Q \Delta t_i), [REMOVE_ME_2]$

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.

tpm_cont(Q, timediff, 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
• 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.

Returns

array of continuous-time transition matrices of dimension c(N,N,n-1)

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

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.

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)

See Also

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