Log-likelihood function of an (H)HMM

This function computes the log-likelihood value of a (hierarchical) hidden Markov model for given observations and parameter values.

Usage

ll_hmm(
  parUncon,
  observations,
  controls = list(),
  hierarchy = FALSE,
  states = if (!hierarchy) 2 else c(2, 2),
  sdds = if (!hierarchy) "normal" else c("normal", "normal"),
  negative = FALSE,
  check_controls = TRUE
)

Arguments

parUncon

An object of class parUncon, which is a numeric vector with identified and unconstrained model parameters in the following order:

1. non-diagonal transition probabilities gammasUncon

2. expectations muUncon

3. standard deviations sigmaUncon (if any)

4. degrees of freedom dfUncon (if any)

5. fine-scale parameters for each coarse-scale state, in the same order (if any)

observations

A numeric vector of time-series data.

In the hierarchical case (hierarchy = TRUE), a matrix with coarse-scale data in the first column and corresponding fine-scale data in the rows.

controls

Either a list or an object of class fHMM_controls.

The list can contain the following elements, which are described in more detail below:

• hierarchy, defines an hierarchical HMM,

• states, defines the number of states,

• sdds, defines the state-dependent distributions,

• horizon, defines the time horizon,

• period, defines a flexible, periodic fine-scale time horizon,

• data, a list of controls that define the data,

• fit, a list of controls that define the model fitting

Either none, all, or selected elements can be specified.

Unspecified parameters are set to their default values.

Important: Specifications in controls always override individual specifications.

hierarchy

A logical, set to TRUE for an hierarchical HMM.

If hierarchy = TRUE, some of the other controls must be specified for the coarse-scale and the fine-scale layer.

By default, hierarchy = FALSE.

states

An integer, the number of states of the underlying Markov chain.

If hierarchy = TRUE, states must be a vector of length 2. The first entry corresponds to the coarse-scale layer, while the second entry corresponds to the fine-scale layer.

By default, states = 2 if hierarchy = FALSE and states = c(2, 2) if hierarchy = TRUE.

sdds

A character, specifying the state-dependent distribution. One of

• "normal" (the normal distribution),

• "lognormal" (the log-normal distribution),

• "t" (the t-distribution),

• "gamma" (the gamma distribution),

• "poisson" (the Poisson distribution).

The distribution parameters, i.e. the

• mean mu,

• standard deviation sigma (not for the Poisson distribution),

• degrees of freedom df (only for the t-distribution),

can be fixed via, e.g., "t(df = 1)" or "gamma(mu = 0, sigma = 1)". To fix different values of a parameter for different states, separate by "|", e.g. "poisson(mu = 1|2|3)".

If hierarchy = TRUE, sdds must be a vector of length 2. The first entry corresponds to the coarse-scale layer, while the second entry corresponds to the fine-scale layer.

By default, sdds = "normal" if hierarchy = FALSE and sdds = c("normal", "normal") if hierarchy = TRUE.

negative

Either TRUE to return the negative log-likelihood value (useful for optimization) or FALSE (default), else.

check_controls

Either TRUE to check the defined controls or FALSE to not check them (which saves computation time), else.

Value

The (negative) log-likelihood value.

Examples

### HMM log-likelihood 
controls <- set_controls(states = 2, sdds = "normal")
parameters <- fHMM_parameters(controls)
parUncon <- par2parUncon(parameters, controls)
observations <- 1:10
ll_hmm(parUncon, observations, controls)
#> [1] -268.9179

### HHMM log-likelihood 
controls <- set_controls(
  hierarchy = TRUE, states = c(2, 2), sdds = c("normal", "normal")
)
parameters <- fHMM_parameters(controls)
parUncon <- par2parUncon(parameters, controls)
observations <- matrix(dnorm(110), ncol = 11, nrow = 10)
ll_hmm(parUncon, observations, controls)
#> [1] -70.26436