Forward algorithm with homogeneous transition probability matrix

Calculates the log-likelihood of a sequence of observations under a homogeneous hidden Markov model using the forward algorithm.

Usage

forward(
  delta,
  Gamma,
  allprobs,
  trackID = NULL,
  logspace = FALSE,
  ad = NULL,
  bw = NULL,
  report = TRUE
)

Arguments

delta

initial or stationary distribution of length N, or matrix of dimension c(k,N) for k independent tracks, if trackID is provided

Gamma

transition probability matrix of dimension c(N,N), or array of k transition probability matrices of dimension c(N,N,k), if trackID is provided

allprobs

matrix of state-dependent probabilities/ density values of dimension c(n, N)

trackID

optional vector of length n containing IDs that separate tracks.

If provided, the total log-likelihood will be the sum of each track's likelihood contribution. In this case, Gamma can be a matrix, leading to the same transition probabilities for each track, or an array of dimension c(N,N,k), with one (homogeneous) transition probability matrix for each track. Furthermore, instead of a single vector delta corresponding to the initial distribution, a delta matrix of initial distributions, of dimension c(k,N), can be provided, such that each track starts with it's own initial distribution.

logspace

logical, indicating whether the probabilities/ densities in the allprobs matrix are on log-scale. If so, internal computations are also done on log-scale which is numerically more robust when the entries are very small. Note that this is only supported when used in AD mode with RTMB.

ad

optional logical, indicating whether automatic differentiation with RTMB should be used. By default, the function determines this itself.

bw

optional integer, indicating the bandwidth for a banded approximation of the forward algorithm. This is for expert users only, if sparsity in the Hessian matrix w.r.t. observations is required.

report

logical, indicating whether delta, Gamma, allprobs, and potentially trackID should be reported from the fitted model. Defaults to TRUE, but only works if ad = TRUE, as it uses the RTMB package.

When there are multiple tracks, for compatibility with downstream functions like viterbi, stateprobs or pseudo_res, forward should only be called once with a trackID argument.

Value

log-likelihood for given data and parameters

See also

Other forward algorithms: forward2(), forward_g(), forward_g2(), forward_hsmm(), forward_ihsmm(), forward_p(), forward_phsmm()

Examples

## negative log likelihood function
nll = function(par, step) {
 # parameter transformations for unconstrained optimisation
 Gamma = tpm(par[1:2]) # multinomial logit link
 delta = stationary(Gamma) # stationary HMM
 mu = exp(par[3:4])
 sigma = exp(par[5:6])
 # calculate all state-dependent probabilities
 allprobs = matrix(1, length(step), 2)
 ind = which(!is.na(step))
 for(j in 1:2) allprobs[ind,j] = dgamma2(step[ind], mu[j], sigma[j])
 # simple forward algorithm to calculate log-likelihood
 -forward(delta, Gamma, allprobs)
}

## fitting an HMM to the trex data
par = c(-2,-2,            # initial tpm params (logit-scale)
        log(c(0.3, 2.5)), # initial means for step length (log-transformed)
        log(c(0.2, 1.5))) # initial sds for step length (log-transformed)
mod = nlm(nll, par, step = trex$step[1:1000])