General forward algorithm with time-varying transition probability matrix
Calculates the log-likelihood of a sequence of observations under a hidden Markov model with time-varying transition probabilities using the forward algorithm .
forward_g(delta, Gamma, allprobs, trackID = NULL, ad = NULL, report = TRUE)
delta: initial or stationary distribution of length N, or matrix of dimension c(k,N) for k independent tracks, if trackID 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 an array of dimension c(N,N,n) for a single track is provided, the first slice will be ignored.
If the elements of 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 and is the generator of the continuous-time Markov chain.
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 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.
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.
log-likelihood for given data and parameters
## Simple usage Gamma = array(c(0.9, 0.2, 0.1, 0.8), dim = c(2,2,10)) delta = c(0.5, 0.5) allprobs = matrix(0.5, 10, 2) forward_g(delta, Gamma, allprobs) ## Full model fitting example ## negative log likelihood function nll = function(par, step, Z) { # parameter transformations for unconstrained optimisation beta = matrix(par[1:6], nrow = 2) Gamma = tpm_g(Z, beta) # multinomial logit link for each time point delta = stationary(Gamma[,,1]) # stationary HMM mu = exp(par[7:8]) sigma = exp(par[9:10]) # 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_g(delta, Gamma, allprobs) } ## fitting an HMM to the trex data par = c(-1.5,-1.5, # initial tpm intercepts (logit-scale) rep(0, 4), # initial tpm slopes 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$step[1:500], Z = trigBasisExp(trex$tod[1:500]))
Other forward algorithms: forward(), forward_hsmm(), forward_ihsmm(), forward_p(), forward_phsmm()