Generating Realizations of a Hidden Markov Model
This function generates a sequence of hidden states of a Markov chain and a corresponding parallel sequence of observations.
HMM_simulation( size, m, delta = rep(1/m, times = m), gamma = 0.8 * diag(m) + rep(0.2/m, times = m), distribution_class, distribution_theta, obs_range = c(NA, NA), obs_round = FALSE, obs_non_neg = FALSE, plotting = 0 )
size: length of the time-series of hidden states and observations (also T).
m: a (finite) number of states in the hidden Markov chain.
delta: a vector object containing values for the marginal probability distribution of the m states of the Markov chain at the time point t=1. Default is delta = rep(1 / m, times = m).
gamma: a matrix (ncol = nrow = m) containing values for the transition matrix of the hidden Markov chain. Default is gamma=0.8 * diag(m) + rep(0.2 / m, times = m)
distribution_class: a single character string object with the abbreviated name of the m observation distributions of the Markov dependent observation process. The following distributions are supported by this algorithm: Poisson (pois); generalized Poisson (genpois); normal (norm); geometric (geom).
distribution_theta: a list object containing the parameter values for the m observation distributions that are dependent on the hidden Markov state.
obs_range: a vector object specifying the range for the observations to be generated. For instance, the vector c(0,1500) allows only observations between 0 and 1500 to be generated by the HMM. Default value is FALSE. See Notes for further details.
obs_round: a logical object. TRUE if all generated observations are natural. Default value is FALSE. See Notes for further details.
obs_non_neg: a logical object. TRUE, if non negative observations are generated. Default value is FALSE. See Notes for further details.
plotting: a numeric value between 0 and 5 (generates different outputs). NA suppresses graphical output. Default value is 0.
0: output 1-5
1: summary of all results
2: generated time series of states of the hidden Markov chain
3: means (of the observation distributions, which depend on the states of the Markov chain) along the time series of states of the hidden Markov chain
4: observations along the time series of states of the hidden Markov chain
5: simulated observations
The function HMM_simulation returns a list containing the following components:
m states of the Markov chain at the time point t=1.m observation distributions that are dependent on the hidden Markov state.Some notes regarding the default values:
gamma:
The default setting assigns higher probabilities for remaining in a state than c hanging into another. obs_range:
Has to be used with caution. since it manipulates the results of the HMM. If a value for an observation at time t is generated outside the defined range, it will be regenerated as long as it falls into obs_range. It is possible just to define one boundary, e.g. obs_range=c(NA,2000) for observations lower than 2000, or obs_range=c(100,NA) for observations higher than 100. obs_round:
Has to be used with caution! Rounds each generated observation and hence manipulates the results of the HMM (important for the normal distribution based HMM). obs_ non_neg:
Has to be used with caution, since it manipulates the results of the HMM. If a negative value for an observation at a time t is generated, it will be re-generated as long as it is non-negative (important for the normal distribution based HMM).
# i.) Generating a HMM with Poisson-distributed data ----- Pois_HMM_data <- HMM_simulation(size = 300, m = 5, distribution_class = "pois", distribution_theta = list( lambda=c(10,15,25,35,55))) print(Pois_HMM_data) # ii.) Generating 6 physical activities with normally ----- # distributed accelerometer counts using a HMM. # Define number of time points (1440 counts equal 6 hours of # activity counts assuming an epoch length of 15 seconds). size <- 1440 # Define 6 possible physical activity ranges m <- 6 # Start with the lowest possible state # (in this case with the lowest physical activity) delta <- c(1, rep(0, times = (m - 1))) # Define transition matrix to generate according to a # specific activity gamma <- 0.935 * diag(m) + rep(0.065 / m, times = m) # Define parameters # (here: means and standard deviations for m=6 normal # distributions that define the distribution in # a phsycial acitivity level) distribution_theta <- list(mean = c(0,100,400,600,900,1200), sd = rep(x = 200, times = 6)) # Assume for each count an upper boundary of 2000 obs_range <-c(NA,2000) # Accelerometer counts shall not be negative obs_non_neg <-TRUE # Start simulation accelerometer_data <- HMM_simulation(size = size, m = m, delta = delta, gamma = gamma, distribution_class = "norm", distribution_theta = distribution_theta, obs_range = obs_range, obs_non_neg = obs_non_neg, plotting = 0) print(accelerometer_data)
AIC_HMM, BIC_HMM, HMM_training
Vitali Witowski (2013).