Forward algorithm for homogeneous hidden semi-Markov models

Calculates the (approximate) log-likelihood of a sequence of observations under a homogeneous hidden semi-Markov model using a modified forward algorithm.

Usage

forward_hsmm(
  dm,
  omega,
  allprobs,
  trackID = NULL,
  delta = NULL,
  eps = 1e-10,
  report = TRUE
)

Arguments

dm

list of length N containing vectors of dwell-time probability mass functions (PMFs) for each state. The vector lengths correspond to the approximating state aggregate sizes, hence there should be little probablity mass not covered by these.

omega

matrix of dimension c(N,N) of conditional transition probabilites, also called embedded transition probability matrix.

Contains the transition probabilities given that the current state is left. Hence, the diagonal elements need to be zero and the rows need to sum to one. Can be constructed using tpm_emb.

allprobs

matrix of state-dependent probabilities/ density values of dimension c(n, N) which will automatically be converted to the appropriate dimension.

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, dm can be a nested list, where the top layer contains k dm lists as described above. omega can then also be an array of dimension c(N,N,k) with one conditional 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.

delta

optional vector of initial state probabilities of length N

By default, the stationary distribution is computed (which is typically recommended).

eps

small value to avoid numerical issues in the approximating transition matrix construction. Usually, this should not be changed.

report

logical, indicating whether initial distribution, approximating transition probability matrix and allprobs matrix should be reported from the fitted model. Defaults to TRUE.

Value

log-likelihood for given data and parameters

Details

Hidden semi-Markov models (HSMMs) are a flexible extension of HMMs, where the state duration distribution is explicitly modelled by a distribution on the positive integers. For direct numerical maximum likelhood estimation, HSMMs can be represented as HMMs on an enlarged state space (of size \(M\)) and with structured transition probabilities.

This function is designed to be used with automatic differentiation based on the R package RTMB. It will be very slow without it!

Examples

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