HMM_simulation: Generating Realizations of a Hidden Markov Model

In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

Generating Realizations of a Hidden Markov Model

Description

This function generates a sequence of hidden states of a Markov chain and a corresponding parallel sequence of observations.

Usage

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
)

Arguments

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

Value

The function HMM_simulation returns a list containing the following components:

The function HMM_simulation returns a list containing the following components:

size

length of the generated time-series of hidden states and observations.

m

input number of states in the hidden Markov chain.

delta

a vector object containing the chosen values for the marginal probability distribution of the m states of the Markov chain at the time point t=1.

gamma

a matrix containing the chosen values for the transition matrix of the hidden Markov chain.

distribution_class

a single character string object with the abbreviated name of the chosen observation distributions of the Markov dependent observation process.

distribution_theta

a list object containing the chosen values for the parameters of the m observation distributions that are dependent on the hidden Markov state.

markov_chain

a vector object containing the generated sequence of states of the hidden Markov chain of the HMM.

means_along_markov_chain

a vector object containing the sequence of means (of the state dependent distributions) corresponding to the generated sequence of states.

observations

a vector object containing the generated sequence of (state dependent) observations of the HMM.

Note

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).

Author(s)

Vitali Witowski (2013).

See Also

AIC_HMM, BIC_HMM, HMM_training

Examples

# 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)
 

HMMpa documentation built on April 3, 2025, 10:29 p.m.