initial_parameter_training: Algorithm to Find Plausible Starting Values for Parameter...
In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

Description Usage Arguments Details Value Author(s) See Also Examples

The function computes plausible starting values for both the Baum-Welch algorithm and the algorithm for directly maximizing the log-Likelihood. Plausible starting values can potentially diminish problems of (i) numerical instability and (ii) not finding the global optimum.

1 2	initial_parameter_training(x, m, distribution_class, n = 100, discr_logL = FALSE, discr_logL_eps = 0.5)

`x`	a vector object containing the time-series of observations that are assumed to be realizations of the (hidden Markov state dependent) observation process of the model.
`m`	a (finite) number of states in the hidden Markov chain.
`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`).
`n`	a single numerical value specifying the number of samples to find the best starting value for the training algorithm. Default value is `100`.
`discr_logL`	a logical object. `TRUE`, if the discrete log-likelihood shall be calculated (for `distribution_class="norm"` instead of the general log-likelihood. Default is `FALSE`.
`discr_logL_eps`	discrete log-likelihood for a hidden Markov model based on nomal distributions (for `distribution_class="norm"`). The default value is `0.5`.

From our experience, parameter estimation for long time-series of observations (T>1000) or observation values >1500 tend to be numerical instable and does not necessarily find a global maximum. Both problems can eventually be diminished with plausible starting values. Basically, the idea behind initial_parameter_training is to sample randomly n sets of m observations from the time-series x, as means (E) of the state-dependent distributions. This n samplings of E, therefore induce n sets of parameters (distribution_theta) for the HMM without running a (slow) parameter estimation algorithm. Furthermore, initial_parameter_training calculates the log-Likelihood for all those n sets of parameters. The set of parameters with the best Likelihood are outputted as plausible starting values. (Additionally to the n sets of randomly chosen observations as means, the m quantiles of the observations are also checked as plausible means within this algorithm.)

initial_parameter_training returns a list containing the following components:

`m`	input number of states in the hidden Markov chain.
`k`	a single numerical value representing the number of parameters of the defined distribution class of the observation process.
`logL`	logarithmized likelihood of the model evaluated at the HMM with given starting values (`delta, gamma, distribution theta`) induced by `E`.
`E`	randomly choosen means of the observation time-series `x`, used for the observation distributions, for which the induced parameters (`delta, gamma, distribution theta`) produce the largest Likelihood.
`distribution_theta`	a list object containing the plausible starting values for the parameters of the `m` observation distributions that are dependent on the hidden Markov state.
`delta`	a vector object containing plausible starting values for the marginal probability distribution of the `m` states of the Markov chain at the time point `t=1`. At the moment: `delta = rep(1/m, times=m)`.
`gamma`	a matrix (`nrow=ncol=m`) containing the plausible starting values for the transition matrix of the hidden Markov chain. At the moment: `gamma = 0.8 * diag(m) + rep(0.2/m, times=m)`.

Vitali Witowski (2013).

Baum_Welch_algorithm direct_numerical_maximization HMM_training

################################################################
### Fictitious observations ####################################
################################################################

x <- c(1,16,19,34,22,6,3,5,6,3,4,1,4,3,5,7,9,8,11,11,
  14,16,13,11,11,10,12,19,23,25,24,23,20,21,22,22,18,7,
  5,3,4,3,2,3,4,5,4,2,1,3,4,5,4,5,3,5,6,4,3,6,4,8,9,12,
  9,14,17,15,25,23,25,35,29,36,34,36,29,41,42,39,40,43,
  37,36,20,20,21,22,23,26,27,28,25,28,24,21,25,21,20,21,
  11,18,19,20,21,13,19,18,20,7,18,8,15,17,16,13,10,4,9,
  7,8,10,9,11,9,11,10,12,12,5,13,4,6,6,13,8,9,10,13,13,
  11,10,5,3,3,4,9,6,8,3,5,3,2,2,1,3,5,11,2,3,5,6,9,8,5,
  2,5,3,4,6,4,8,15,12,16,20,18,23,18,19,24,23,24,21,26,
  36,38,37,39,45,42,41,37,38,38,35,37,35,31,32,30,20,39,
  40,33,32,35,34,36,34,32,33,27,28,25,22,17,18,16,10,9,
  5,12,7,8,8,9,19,21,24,20,23,19,17,18,17,22,11,12,3,9,
  10,4,5,13,3,5,6,3,5,4,2,5,1,2,4,4,3,2,1) 


### Finding plausibel starting values for the parameter estimation 
### for a generealized-Pois-HMM with m=4 states
m <- 4 


plausible_starting_values <- 
   initial_parameter_training(x = x, 
     m = m, 
     distribution_class = "genpois", 
     n=100)

print(plausible_starting_values)