# Hidden semi-Markov models

#### Jan-Ole Koslik

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

So-called hidden semi-Markov models (HSMMs) are a flexible generalisation of HMMs to a semi-Markovian state process which is motivated by the fact that for homogeneous HMMs, the time spent in a hidden state, also called the state dwell time or sojourn time is necessarily geometrically distributed as a consequence of the Markov assumption. HSMMs are designed to mitigate this often unrealistic assumption by allowing for arbitrary distributions on the positive integers to be estimated for the state dwell time. Inference in such models becomes more involved, but Langrock and Zucchini (2011) showed that HSMMs can be estimated conveniently via approximating them by HMMs with an extended state space. Each state of the HSMMs is represented by a state aggregate of several states and the transition probabilities within each aggregate are designed carefully to represent the chosen dwell-time distribution. For more details see Langrock and Zucchini (2011) or Zucchini, MacDonald, and Langrock (2016). Due to this approximate inference procedure, such models can again be fitted by numerically maximising the (approximate) likelihood which can be evaluated using the forward algorithm.

## Simulation example

We begin by considering homogeneous HSMMs. In such models, each state has an associated state dwell-time distribution. The transition probability matrix of a regular HMM is replaced by these distributions and the conditional transition probabilities given the state is left.

### Setting parameters

Here we choose the simplest case of dwell times that are shifted Poisson distributed. We have to specify the Poisson mean for each state, the conditional transition probability matrix called \(\Omega\) and the parameters of the state-dependent process.

    # loading the package
    library(LaMa)

    lambda = c(7, 4, 4)
    omega = matrix(c(0, 0.7, 0.3,
                     0.5, 0, 0.5,
                     0.7, 0.3, 0), nrow = 3, byrow = TRUE)
    mu = c(10, 40, 100)
    sigma = c(5, 20, 50)

    color = c("orange", "deepskyblue", "seagreen2")
    curve(dnorm(x, mu[1], sigma[1]), lwd = 2, col = color[1], bty = "n",
          xlab = "x", ylab = "density", xlim = c(0, 150), n = 300)
    curve(dnorm(x, mu[2], sigma[2]), lwd = 2, col = color[2], add = T)
    curve(dnorm(x, mu[3], sigma[3]), lwd = 2, col = color[3], add = T)

### Simulating data

We simulate data by drawing dwell times from the dwell-time distribution of the current state and then draw the next state using the conditional transition probabilities. The state-dependent process is drawn conditional on the current state.

    set.seed(123)

    k = 50 # number of stays
    s = rep(NA, k)
    s[1] = sample(1:3, 1) # uniform initial distribution
    staylength = rpois(1, lambda[s[1]]) + 1 # drawing dwell time from shifted Poisson
    C = rep(s[1], staylength)
    x = rnorm(staylength, mu[s[1]], sigma[s[1]])

    for(t in 2:k){
      # conditionally drawing state
      s[t] = sample(c(1:3)[-s[t-1]], 1, prob = omega[s[t-1], -s[t-1]])
      staylength = rpois(1, lambda[s[t]]) + 1 # drawing dwell time from shifted Poisson

      C = c(C, rep(s[t], staylength))
      x = c(x, rnorm(staylength, mu[s[t]], sigma[s[t]]))
    }

    plot(x, pch = 20, col = color[C], bty = "n")
    legend("topright", col = color, pch = 20,
           legend = paste("state", 1:3), box.lwd = 0)

### Writing the negative log-likelihood function

We now write the negative log-likelihood function for an approximating HMM. As a semi-Markov chain is specified in terms of state-specific dwell-time distributions and conditional transition probabilities given that the current state is left, we have to compute both (here called `dm` and `omega`). For the latter, we can use the function `tpm_emb()` that constructs a transition probability matrix via the inverse multinomial logit link (softmax), where the diagonal entries are forced to equal zero.

The transition probability matrix of the approximating HMM can then be computed by the function `tpm_hsmm()` where the exact procedure is detailed by Langrock and Zucchini (2011). We need the extra argument `agsizes` to specify the aggregate sizes that should be used to approximate the dwell-time distributions. These should be chosen such that most of the support of the state-specific dwell-time distributions is covered.

    nll = function(par, x, N, agsizes){
      mu = par[1:N]
      sigma = exp(par[N+1:N])
      lambda = exp(par[2*N+1:N])
      omega = if(N==2) tpm_emb() else tpm_emb(par[3*N+1:(N*(N-2))])
      dm = list() # list of dwell-time distributions
      for(j in 1:N) dm[[j]] = dpois(1:agsizes[j]-1, lambda[j]) # shifted Poisson
      Gamma = tpm_hsmm(omega, dm, sparse = FALSE)
      delta = stationary(Gamma)
      allprobs = matrix(1, length(x), N)
      ind = which(!is.na(x))
      for(j in 1:N){
        allprobs[ind,j] = dnorm(x[ind], mu[j], sigma[j])
      }
      -forward_s(delta, Gamma, allprobs, agsizes)
    }

### Fitting an HSMM (as an approximating HMM) to the data

    # initial values
    par = c(10, 40, 100, log(c(5, 20, 50)), # state-dependent
                   log(c(7,4,4)), # dwell time means
                   rep(0, 3)) # omega

    agsizes = qpois(0.95, lambda)+1

    system.time(
      mod <- nlm(nll, par, x = x, N = 3, agsizes = agsizes, stepmax = 2)
    )
    #>    user  system elapsed
    #>   0.470   0.006   0.476

Fitting HSMMs is rather slow (even using C++) as we translate the additional model complexity into a higher computational overhead (31 states here).

### Results

    N = 3
    (mu = mod$estimate[1:N])
    #> [1]  10.16569  39.06161 107.66034
    (sigma = exp(mod$estimate[N+1:N]))
    #> [1]  4.78882 19.35639 48.56115
    (lambda = exp(mod$estimate[2*N+1:N]))
    #> [1] 6.942983 4.595469 3.354765
    (omega = tpm_emb(mod$estimate[3*N+1:(N*(N-2))]))
    #>           S1        S2        S3
    #> S1 0.0000000 0.5541030 0.4458970
    #> S2 0.5040937 0.0000000 0.4959063
    #> S3 0.6654703 0.3345297 0.0000000

## Real-data application

We now want to briefly show the analysis of a real data set using hidden semi-Markov models. For this purpose we use the movement track of an Arctic muskox contained in the `R` package `PHSMM`. Originally these data were collected by Beumer et al. (2020) and have already been analysed by Pohle, Adam, and Beumer (2022).

    library(PHSMM)
    data = muskox[1:1000, ] # only using first 1000 observations for speed
    head(data)
    #>             date tday        x       y       step
    #> 88273 2013-10-12   15 513299.2 8264867  17.998874
    #> 88274 2013-10-12   16 513283.4 8264875   8.214733
    #> 88275 2013-10-12   17 513284.3 8264883   7.205098
    #> 88276 2013-10-12   18 513280.4 8264877  53.378332
    #> 88277 2013-10-12   19 513252.0 8264922 719.242687
    #> 88278 2013-10-12   20 513386.7 8265629  10.797127

As these data have already been preprocessed, we can immediately write the negative log-likelihood function. When modeling the dwell-time distribution of real processes, it is typically advisable to use a more flexible distribution than the shifted Poisson distribution, as the latter cannot account for overdispersion. Here, we will employ the shifted negative binomial distribution that yields the Poisson distribution as a special case for the dispersion parameter equal to zero. The state-dependent step lengths are modelled by gamma distributions, where we reparametrise the gamma distribution in terms of its mean and standard deviation as opposed to shape and scale for better interpretability using `dgamma2()`.

    nll_muskox = function(par, step, N, agsizes){
      # parameter transformation from working to natural
      mu = exp(par[1:N]) # step mean
      sigma = exp(par[N+1:N]) # step standard deviation
      mu_dwell = exp(par[2*N+1:N]) # dwell time mean
      phi = exp(par[3*N+1:N]) # dwell time dispersion
      omega = if(N==2) tpm_emb() else tpm_emb(par[4*N+1:(N*(N-2))])
      dm = list() # list of dwell-time distributions
      for(j in 1:N){
        dm[[j]] = dnbinom(1:agsizes[j]-1, mu=mu_dwell[j], size=1/phi[j])
      }
      Gamma = tpm_hsmm(omega, dm, sparse = FALSE)
      delta = stationary(Gamma)
      allprobs = matrix(1, length(step), N)
      ind = which(!is.na(step))
      for(j in 1:N) allprobs[ind,j] = dgamma2(step[ind], mu[j], sigma[j])
      -forward_s(delta, Gamma, allprobs, agsizes)
    }

### Fitting an HSMM (as an approximating HMM) to the muskox data

    # intial values
    par = c(log(c(4, 50, 300, 4, 50, 300)), # state-dependent mean and sd
                   log(c(3,3,5)), # dwell time means
                   log(c(0.01, 0.01, 0.01)), # dwell time dispersion
                   rep(0, 3)) # omega

    agsizes = c(11,11,14)

    system.time(
      mod_muskox <- nlm(nll_muskox, par, step = data$step, N = 3,
                        agsizes = agsizes, iterlim = 500)
    )
    #>    user  system elapsed
    #>   2.338   0.013   2.352

### Results

We retransform the parameters for interpretation

    par = mod_muskox$estimate; N = 3
    (mu = exp(par[1:N])) # step mean
    #> [1]   4.408371  55.522382 306.546349
    (sigma = exp(par[N+1:N])) # step standard deviation
    #> [1]   3.148482  50.329985 331.573580
    (mu_dwell = exp(par[2*N+1:N])) # dwell time mean
    #> [1] 2.545409 2.659182 5.540232
    (phi = exp(par[3*N+1:N])) # dwell time dispersion
    #> [1] 4.399426e-05 3.742076e-02 2.975531e-09
    (omega = tpm_emb(par[4*N+1:(N*(N-2))])) # embedded t.p.m.
    #>           S1        S2           S3
    #> S1 0.0000000 0.6865562 3.134438e-01
    #> S2 1.0000000 0.0000000 4.126327e-08
    #> S3 0.8201504 0.1798496 0.000000e+00

In this case the Poisson distribution would have been sufficiently flexible as all dispersion parameters were estimated very close to zero. We can easily visualise the estimated state-specific dwell-time distributions:

    oldpar = par(mfrow = c(1,3))
    for(j in 1:N){
      plot(1:agsizes[j], dnbinom(1:agsizes[j]-1, mu=mu_dwell[j], size = 1/phi[j]),
           type = "h", lwd = 2, col = color[j], xlab = "dwell time (hours)",
           ylab = "probabilities", main = paste("state",j), bty = "n", ylim = c(0,0.25))
    }

    par(oldpar)

## References

Beumer, Larissa T, Jennifer Pohle, Niels M Schmidt, Marianna Chimienti, Jean-Pierre Desforges, Lars H Hansen, Roland Langrock, Stine Højlund Pedersen, Mikkel Stelvig, and Floris M van Beest. 2020. “An Application of Upscaled Optimal Foraging Theory Using Hidden Markov Modelling: Year-Round Behavioural Variation in a Large Arctic Herbivore.” Movement Ecology 8: 1–16.
Langrock, Roland, and Walter Zucchini. 2011. “Hidden Markov Models with Arbitrary State Dwell-Time Distributions.” Computational Statistics & Data Analysis 55 (1): 715–24.
Pohle, Jennifer, Timo Adam, and Larissa T Beumer. 2022. “Flexible Estimation of the State Dwell-Time Distribution in Hidden Semi-Markov Models.” Computational Statistics & Data Analysis 172: 107479.
Zucchini, Walter, Iain L. MacDonald, and Roland Langrock. 2016. Hidden Markov Models for Time Series: An Introduction Using R. Boca Raton: Chapman & Hall/CRC.