LaMa (version 1.0.0)

forward_p: Forward algorithm with (only) periodically varying transition probability matrix

Description

When the transition probability matrix only varies periodically (e.g. as a function of time of day), there are only \(L\) unique matrices if \(L\) is the period length (e.g. \(L=24\) for hourly data and time-of-day variation). Thus, it is much more efficient to only calculate these \(L\) matrices and index them by a time variable (e.g. time of day or day of year) instead of calculating such a matrix for each index in the data set (which would be redundant). This function allows for that, by only expecting a transition probability matrix for each time point in a period, and an integer valued (\(1, \dots, L\)) time variable that maps the data index to the according time.

Usage

forward_p(delta, Gamma, allprobs, tod)

Value

Log-likelihood for given data and parameters

Arguments

delta

Initial or periodically stationary distribution of length N

Gamma

Array of transition probability matrices of dimension c(N,N,L).

Here we use the definition \(\Pr(S_t=j \mid S_{t-1}=i) = \gamma_{ij}^{(t)}\) such that the transition probabilities between time point \(t-1\) and \(t\) are an element of \(\Gamma^{(t)}\).

allprobs

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

tod

(Integer valued) time variable in 1, ..., L, mapping the data index to a generalized 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.

Examples

## generating data from periodic 2-state HMM
mu = c(0, 6)
sigma = c(2, 4)
beta = matrix(c(-2,-2,1,-1, 1, -1),nrow=2)
delta = c(0.5, 0.5)
# simulation
n = 2000
s = x = rep(NA, n)
tod = rep(1:24, ceiling(2000/24))
s[1] = sample(1:2, 1, prob = delta)
x[1] = rnorm(1, mu[s[1]], sigma[s[1]])
# 24 unique t.p.m.s
Gamma = array(dim = c(2,2,24))
for(t in 1:24){
  G = diag(2)
  G[!G] = exp(beta[,1]+beta[,2]*sin(2*pi*t/24)+
    beta[,3]*cos(2*pi*t/24)) # trigonometric link
  Gamma[,,t] = G / rowSums(G)
}
for(t in 2:n){
  s[t] = sample(1:2, 1, prob = Gamma[s[t-1],,tod[t]])
  x[t] = rnorm(1, mu[s[t]], sigma[s[t]])
}
# we can also use function from LaMa to make building periodic tpms much easier
Gamma = tpm_p(1:24, 24, beta, degree = 1)

## negative log likelihood function
mllk = function(theta.star, x, tod){
  # parameter transformations for unconstraint optimization
  beta = matrix(theta.star[1:6], 2, 3)
  Gamma = tpm_p(tod=tod, L=24, beta=beta)
  delta = stationary_p(Gamma, t=tod[1])
  mu = theta.star[8:9]
  sigma = exp(theta.star[10:11])
  # 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_p(delta, Gamma, allprobs, tod)
}

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