tpm_phsmm2 function

Build all transition probability matrices of an periodic-HSMM-approximating HMM

Hidden semi-Markov models (HSMMs) are a flexible extension of HMMs. 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 computes the transition matrices of a periodically inhomogeneos HSMMs.

tpm_phsmm2(omega, dm, eps = 1e-10)

Arguments

• omega: embedded transition probability matrix

  Either a matrix of dimension c(N,N) for homogeneous conditional transition probabilities, or an array of dimension c(N,N,L) for inhomogeneous conditional transition probabilities.

• dm: state dwell-time distributions arranged in a list of length(N)

  Each list element needs to be a matrix of dimension c(L, N_i), where each row t is the (approximate) probability mass function of state i at time t.

• eps: rounding value: If an entry of the transition probabily matrix is smaller, than it is rounded to zero.

Returns

array of dimension c(N,N,L), containing the extended-state-space transition probability matrices of the approximating HMM for each time point of the cycle.

Examples

N = 3
L = 24
# time-varying mean dwell times
Lambda = exp(matrix(rnorm(L*N, 2, 0.5), nrow = L))
sizes = c(25, 25, 25) # approximating chain with 75 states
# state dwell-time distributions
dm = list()
for(i in 1:3){
  dmi = matrix(nrow = L, ncol = sizes[i])
  for(t in 1:L){
    dmi[t,] = dpois(1:sizes[i]-1, Lambda[t,i])
  }
  dm[[i]] = dmi
}

## homogeneous conditional transition probabilites
# diagonal elements are zero, rowsums are one
omega = matrix(c(0,0.5,0.5,0.2,0,0.8,0.7,0.3,0), nrow = N, byrow = TRUE)

# calculating extended-state-space t.p.m.s
Gamma = tpm_phsmm(omega, dm)

## inhomogeneous conditional transition probabilites
# omega can be an array
omega = array(rep(omega,L), dim = c(N,N,L))
omega[1,,4] = c(0, 0.2, 0.8) # small change for inhomogeneity

# calculating extended-state-space t.p.m.s
Gamma = tpm_phsmm(omega, dm)