Continuous-time HMMs

Jan-Ole Koslik

Before diving into this vignette, we recommend reading the vignette Introduction to LaMa.

The regular HMM formulation needs a key assumption to be applicable, namely the data need to be observed at regular, equidistant time-points such that the transition probabilties can be interpreted meaningfully w.r.t. a specific time unit. If this is not the case, the model used should acocunt for this by building on a mathematical formulation in continuous time. The obvious choice here is to retain most of the HMM model formulation, but replace the unobserved discrete-time Markov chain with a continuous-time Markov chain. However, here it is important to note that the so-called snapshot property needs to be fulfilled, i.e. the observed process at time $t$ can only depend on the state at that time instant and not on the interval since the previous observation. For more details see Glennie et al. (2023).

A continuous-time Markov chain is characterised by a so-called (infinitesimal) generator matrix $$Q = \begin{pmatrix} q_{11} & q_{12} & \cdots & q_{1N} \\ q_{21} & q_{22} & \cdots & q_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ q_{N1} & q_{N2} & \cdots & q_{NN} \\ \end{pmatrix},$$ where the diagonal entries are $q_{ii} = - \sum_{j \neq i} q_{ij}$, $q_{ij} \geq 0$ for $i \neq j$. This matrix can be interpreted as the derivative of the transition probability matrix and completely describes the dynamics of the state process. The time-spent in a state $i$ is exponentially distributed with rate $-q_{ii}$ and conditional on leaving the state, the probability to transition to a state $j \neq i$ is $\omega_{ij} = q_{ij} / -q_{ii}$. For a more detailed introduction see Dobrow (2016) (pp. 265 f.). For observation times $t_1$ and $t_2$, we can then obtain the transition probability matrix between these points via the identity $$\Gamma(t_1, t_2) = \exp(Q (t_2 - t_1)),$$ where $\exp()$ is the matrix expoential. This follows from the so-called Kolmogorov forward equations, but for more details see Dobrow (2016).

Example 1: two states

Setting parameters for simulation

# loading the package
library(LaMa)
#> Loading required package: RTMB

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.

# generator matrix Q:
Q = matrix(c(-0.5, 0.5, 1, -1), 
           nrow = 2, byrow = TRUE)

# parameters for the state-dependent (normal) distributions
mu = c(5, 20)
sigma = c(2, 5)

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)
}

Fitting a continuous-time HMM to the data

par = c(mu = c(5, 15), # state-dependent means
        logsigma = c(log(3), log(5)), # state-dependent sds
        qs = c(log(1), log(0.5))) # off-diagonals of Q

timediff = diff(obs_times)

system.time(
  mod <- nlm(nll, par, timediff = timediff, x = x, N = 2)
)
#>    user  system elapsed 
#>   0.259   0.237   0.255

Results

N = 2
# mu
round(mod$estimate[1:N],2)
#> [1]  5.06 20.24
# sigma
round(exp(mod$estimate[N+1:N]))
#> [1] 2 5
Q = generator(mod$estimate[2*N+1:(N*(N-1))])
round(Q,3)
#>        [,1]   [,2]
#> [1,] -0.479  0.479
#> [2,]  0.905 -0.905

Example 2: three states

Setting parameters for simulation

# generator matrix Q:
Q = matrix(c(-0.5,0.2,0.3,
             1,-2, 1,
             0.4, 0.6, -1), nrow = 3, byrow = TRUE)

# parameters for the state-dependent (normal) distributions
mu = c(5, 15, 30)
sigma = c(2, 3, 5)

Simulating data

The simulation is very similar but we now also have to draw which state to transition to, as explained in the beginning.

set.seed(123)

k = 200 # number of state switches
trans_times = s = rep(NA, k) # time points where the chain transitions
s[1] = sample(1:3, 1) # uniform initial distribuion
# 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){
  # off-diagonal elements of the s[t-1] row of Q divided by the diagonal element
  # give the probabilites of the next state
  s[t] = sample(c(1:3)[-s[t-1]], 1, prob = Q[s[t-1],-s[t-1]]/-Q[s[t-1],s[t-1]])
  # 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]]))
}

Fitting a 3-state continuous-time HMM to the data

par = c(mu = c(5, 10, 25), # state-dependent means
        logsigma = c(log(2), log(2), log(6)), # state-dependent sds
        qs = rep(0, 6)) # off-diagonals of Q

timediff = diff(obs_times)

system.time(
  mod2 <- nlm(nll, par, timediff = timediff, x = x, N = 3, stepmax = 10)
)
#>    user  system elapsed 
#>   2.003   2.745   1.584
# without restricting stepmax, we run into numerical problems

Results

N = 3
# mu
round(mod2$estimate[1:N],2)
#> [1]  4.90 15.45 29.10
# sigma
round(exp(mod2$estimate[N+1:N]),2)
#> [1] 1.80 2.58 5.06
Q = generator(mod2$estimate[2*N+1:(N*(N-1))])
round(Q, 3)
#>        [,1]   [,2]   [,3]
#> [1,] -0.888  0.565  0.323
#> [2,]  2.821 -3.469  0.647
#> [3,]  0.000  0.770 -0.770

References

Dobrow, Robert P. 2016. Introduction to Stochastic Processes with r. John Wiley & Sons.

Glennie, Richard, Timo Adam, Vianey Leos-Barajas, Théo Michelot, Theoni Photopoulou, and Brett T McClintock. 2023. “Hidden Markov Models: Pitfalls and Opportunities in Ecology.” Methods in Ecology and Evolution 14 (1): 43–56.