Forward algorithm for hidden semi-Markov models with periodically inhomogeneous state durations and/ or conditional transition probabilities

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. This function can be used to fit HSMMs where the state-duration distribution and/ or the conditional transition probabilities vary with covariates. 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 can be used to fit HSMMs where the state-duration distribution and/ or the conditional transition probabilities vary periodically. In the special case of periodic variation (as compared to arbitrary covariate influence), this version is to be preferred over forward_ihsmm because it computes the correct periodically stationary distribution and no observations are lost for the approximation.

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

Usage

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

Arguments

dm

list of length N containing matrices (or vectors) of dwell-time probability mass functions (PMFs) for each state.

If the dwell-time PMFs are constant, the vectors are the PMF of the dwell-time distribution fixed in time. The vector lengths correspond to the approximating state aggregate sizes, hence there should be little probablity mass not covered by these.

If the dwell-time PMFs are inhomogeneous, the matrices need to have L rows, where L is the cycle length. The number of columns again correpond to the size of the approximating state aggregates.

omega

matrix of dimension c(N,N) or array of dimension c(N,N,L) of conditional transition probabilites, also called embedded transition probability matrix

It contains the transition probabilities given the current state is left. Hence, the diagonal elements need to be zero and the rows need to sum to one. Such a matrix can be constructed using tpm_emb and an array using tpm_emb_g.

allprobs

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

tod

(Integer valued) variable for cycle indexing in 1, ..., L, mapping the data index to a generalised time of day (length n). For half-hourly data L = 48. It could, however, also be day of year for daily data and L = 365.

trackID

optional vector of length n containing IDs

If provided, the total log-likelihood will be the sum of each track's likelihood contribution. 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, instead of this, the stationary distribution is computed corresponding to the first approximating t.p.m. of each track is computed. Contrary to the homogeneous case, this is not theoretically motivated but just for convenience.

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

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

Examples

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