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Forward algorithm with homogeneous transition probability matrix

forward.Rd

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, ad = 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

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.

ad

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

report

logical, indicating whether delta, Gamma and allprobs should be reported from the fitted model. Defaults to TRUE, but only works if ad = TRUE.

Value

log-likelihood for given data and parameters

See also

Other forward algorithms: forward_g(), 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])