Example 1: two states
Setting parameters for simulation
We start by setting parameters to simulate data. In this example, state 1 has a smaller rate and the state dwell time in state one follows and \(Exp(0.5)\) distribution, i.e. it exhibits longer dwell times than state 2 with rate 1.
Simulating data
We simulate the continuous-time Markov chain by drawing the exponentially distributed state dwell-times. Within a stay, we can assume whatever structure we like for the observation times, as these are not explicitly modeled. Here we choose to generate them by a Poisson process with rate \(\lambda=1\), but this choice is arbitrary. For more details on Poisson point processes, see the MM(M)PP vignette.
set.seed(123)
k = 200 # number of state switches
trans_times = s = rep(NA, k) # time points where the chain transitions
s[1] = sample(1:2, 1) # initial distribuion c(0.5, 0.5)
# exponentially distributed waiting times
trans_times[1] = rexp(1, -Q[s[1],s[1]])
n_arrivals = rpois(1, trans_times[1])
obs_times = sort(runif(n_arrivals, 0, trans_times[1]))
x = rnorm(n_arrivals, mu[s[1]], sigma[s[1]])
for(t in 2:k){
s[t] = c(1,2)[-s[t-1]] # for 2-states, always a state swith when transitioning
# exponentially distributed waiting times
trans_times[t] = trans_times[t-1] + rexp(1, -Q[s[t], s[t]])
n_arrivals = rpois(1, trans_times[t]-trans_times[t-1])
obs_times = c(obs_times,
sort(runif(n_arrivals, trans_times[t-1], trans_times[t])))
x = c(x, rnorm(n_arrivals, mu[s[t]], sigma[s[t]]))
}Let’s visualise the simulated continuous-time HMM:
color = c("orange", "deepskyblue")
n = length(obs_times)
plot(obs_times[1:50], x[1:50], pch = 16, bty = "n", xlab = "observation times",
ylab = "x", ylim = c(-5,25))
segments(x0 = c(0,trans_times[1:48]), x1 = trans_times[1:49],
y0 = rep(-5,50), y1 = rep(-5,50), col = color[s[1:49]], lwd = 4)
legend("topright", lwd = 2, col = color,
legend = c("state 1", "state 2"), box.lwd = 0)
Writing the negative log-likelihood function
The likelhood of a continuous-time HMM for observations \(x_{t_1}, \dots, x_{t_T}\) at irregular time
points \(t_1, \dots, t_T\) has the
exact same structure as the regular HMM likelihood: \[
L(\theta) = \delta^{(1)} \Gamma(t_1, t_2) P(x_{t_2}) \Gamma(t_2, t_3)
P(x_{t_3}) \dots \Gamma(t_{T-1}, t_T) P(x_{t_T}) 1^t,
\] where \(\delta^{(1)}\), \(P\) and \(1^t\) are as usual and \(\Gamma(t_k, t_{k+1})\) is computed as
explained above. Thus we can fit such models using the standard
implementation of the general forward algorithm forward_g()
with time-varying transition probability matrices. We can use the
generator() function to compute the infinitesimal generator
matrix from an unconstrained parameter vector and
tpm_cont() to compute all matrix exponentials.
nll = function(par, timediff, x, N){
mu = par[1:N]
sigma = exp(par[N+1:N])
Q = generator(par[2*N+1:(N*(N-1))]) # generator matrix
Pi = stationary_cont(Q) # stationary dist of CT Markov chain
Qube = tpm_cont(Q, timediff) # this computes exp(Q*timediff)
allprobs = matrix(1, nrow = length(x), ncol = N)
ind = which(!is.na(x))
for(j in 1:N){
allprobs[ind,j] = dnorm(x[ind], mu[j], sigma[j])
}
-forward_g(Pi, Qube, allprobs)
}