initial_parameter_training function

Algorithm to Find Plausible Starting Values for Parameter Estimation

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.

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: Poisson (pois); generalized Poisson (genpois, only available for training_method="numerical"); normal (norm)).
• 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. Default is FALSE for the general log-likelihood, TRUE for the discrete log-likelihood (for distribution_class = "norm").
• discr_logL_eps: a single numerical value, used to approximate the discrete log-likelihood for a hidden Markov model based on nomal distributions (for "norm"). The default value is 0.5.

Returns

The function 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)`.
  

Details

From our experience, parameter estimation for long time-series of observations (T>1000) or observation values >1500 tend to be numerical unstable 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.)

Examples

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)

See Also

Baum_Welch_algorithm, direct_numerical_maximization, HMM_training

Author(s)

Vitali Witowski (2013).