AIC_HMM function

AIC Value for a Discrete Time Hidden Markov Model

Computes the Aikaike's information criterion and the Bayesian information criterion for a discrete time hidden Markov model, given a time-series of observations.

AIC_HMM(logL, m, k)

Arguments

• logL: logarithmized likelihood of the model
• m: integer number of states in the Markov chain of the model
• k: single integer value representing the number of parameters of the underlying distribution of the observation process (e.g. k=2 for the normal distribution (mean and standard deviation))

Returns

The AIC or BIC value of the fitted hidden Markov model.

Details

For a discrete-time hidden Markov model, AIC and BIC are as follows (MacDonald & Zucchini (2009, Paragraph 6.1 and A.2.3)):

where T indicates the length/size of the observation time-series and p

denotes the number of independent parameters of the model. In case of a HMM as provided by this package, where k and m are defined as in the arguments above, p can be calculated as follows:

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) 

# Assummptions (probability vector, transition matrix, 
# and distribution parameters)

delta <- c(0.25,0.25,0.25,0.25)
gamma <- 0.7 * diag(length(delta)) + rep(0.3 / length(delta))
distribution_class <- "pois"
distribution_theta <- list(lambda = c(4,9,17,25))

# log-likelihood
logL <- forward_backward_algorithm (x = x, 
                                    delta = delta, gamma=gamma,  
                                    distribution_class= distribution_class, 
                                    distribution_theta=distribution_theta)$logL
                                    
# the Poisson distribution has one paramter, hence k=1
AIC_HMM(logL = logL, m = length(delta), k = 1)
BIC_HMM(size = length(x) , logL = logL, m = length(delta), k = 1)

References

MacDonald, I. L., Zucchini, W. (2009) Hidden Markov Models for Time Series: An Introduction Using R, Boca Raton: Chapman & Hall.

See Also

BIC_HMM, HMM_training

Author(s)

Based on MacDonald & Zucchini (2009, Paragraph 6.1 and A.2.3). Implementation by Vitali Witowski (2013).