forward_g {LaMa}

General forward algorithm with time-varying transition probability matrix

Description

General forward algorithm with time-varying transition probability matrix

Usage

forward_g(delta, Gamma, allprobs, trackInd = NULL)

Arguments

delta

Initial distribution of length N, or matrix of dimension c(k,N) for k independent tracks, if trackInd is provided.

Gamma

Array of transition probability matrices of dimension c(N,N,n-1), as in a time series of length n, there are only n-1 transitions. If you provide an array of dimension c(N,N,n), the first slice will be ignored.

If the elements of \Gamma^{(t)} depend on covariate values at t or covariates t+1 is your choice in the calculation of the array, prior to using this function. When conducting the calculation by using tpm_g(), the choice comes down to including the covariate matrix Z[-1,] oder Z[-n,].

If trackInd is provided, Gamma needs to be an array of dimension c(N,N,n), matching the number of rows of allprobs. For each track, the transition matrix at the beginning will be ignored. If the parameters for Gamma are pooled across tracks or not, depends on your calculation of Gamma. If pooled, you can use tpm_g(Z, beta) to calculate the entire array of transition matrices when Z is of dimension c(n,p).

This function can also be used to fit continuous-time HMMs, where each array entry is the Markov semigroup \Gamma(\Delta t) = \exp(Q \Delta t) and Q is the generator of the continuous-time Markov chain.

allprobs

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

trackInd

Optional vector of length k containing the indices that correspond to the beginning of separate tracks. If provided, the total log-likelihood will be the sum of each track's likelihood contribution. In this case, Gamma needs to be an array of dimension c(N,N,n), matching the number of rows of allprobs. For each track, the transition matrix at the beginning of the track will be ignored (as there is no transition between tracks). 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.

Value

Log-likelihood for given data and parameters

Examples

## generating data from inhomogeneous 2-state HMM
mu = c(0, 6)
sigma = c(2, 4)
beta = matrix(c(-2,-2,0.5,-0.5),nrow=2)
delta = c(0.5, 0.5)
# simulation
n = 2000
s = x = rep(NA, n)
z = rnorm(n, 0, 2)
s[1] = sample(1:2, 1, prob = delta)
x[1] = rnorm(1, mu[s[1]], sigma[s[1]])
for(t in 2:n){
  Gamma = diag(2)
  Gamma[!Gamma] = exp(beta[,1]+beta[,2]*z[t])
  Gamma = Gamma / rowSums(Gamma)
  s[t] = sample(1:2, 1, prob = Gamma[s[t-1],])
  x[t] = rnorm(1, mu[s[t]], sigma[s[t]])
}

## negative log likelihood function
mllk = function(theta.star, x, z){
  # parameter transformations for unconstraint optimization
  beta = matrix(theta.star[1:4], 2, 2)
  Gamma = tpm_g(Z = z, beta = beta)
  delta = c(plogis(theta.star[5]), 1-plogis(theta.star[5]))
  mu = theta.star[6:7]
  sigma = exp(theta.star[8:9])
  # calculate all state-dependent probabilities
  allprobs = matrix(1, length(x), 2)
  for(j in 1:2){ allprobs[,j] = dnorm(x, mu[j], sigma[j]) }
  # return negative for minimization
  -forward_g(delta, Gamma, allprobs)
}

## fitting an HMM to the data
theta.star = c(-2,-2,1,-1,0,0,5,log(2),log(3))
mod = nlm(mllk, theta.star, x = x, z = z)