initial_parameter_training {HMMpa}

Algorithm to Find Plausible Starting Values for Parameter Estimation

Description

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.

Usage

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

Arguments

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.

Details

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.)

Value

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).

Author(s)

Vitali Witowski (2013).

See Also

Baum_Welch_algorithm direct_numerical_maximization HMM_training

Examples

################################################################
### 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)