# AIC_HMM: AIC and BIC Value for a Discrete Time Hidden Markov Model In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

## Description

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

## Usage

 ```1 2``` ```AIC_HMM(logL, m, k) BIC_HMM(size, m, k, logL) ```

## Arguments

 `size` length of the time-series of observations x (also `T`). `m` number of states in the Markov chain of the model. `k` single numeric 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)). `logL` logarithmized likelihood of the 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)):

AIC = -2 logL + 2p

BIC = -2 logL + p log(T),

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:

p = m^2 + km - 1.

## Value

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

## Author(s)

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

## References

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

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