Log-Likelihood of a VLMC

This function evaluates the log-likelihood of a VLMC fitted on a discrete time series. When the optional argument newdata is provided, the function evaluates instead the log-likelihood for this (new) discrete time series.

Usage

loglikelihood(
  vlmc,
  newdata,
  initial = c("truncated", "specific", "extended"),
  ignore,
  ...
)

# S3 method for vlmc
loglikelihood(
  vlmc,
  newdata,
  initial = c("truncated", "specific", "extended"),
  ignore,
  ...
)

# S3 method for vlmc_cpp
loglikelihood(
  vlmc,
  newdata,
  initial = c("truncated", "specific", "extended"),
  ignore,
  ...
)

Arguments

vlmc: the vlmc representation.
newdata: an optional discrete time series.
initial: specifies the likelihood function, more precisely the way the first few observations for which contexts cannot be calculated are integrated in the likelihood. Defaults to "truncated". See below for details.
ignore: specifies the number of initial values for which the loglikelihood will not be computed. The minimal number depends on the likelihood function as detailed below.
...: additional parameters for loglikelihood.

Value

an object of class logLikMixVLMC and logLik. This is a number, the log-likelihood of the (CO)VLMC with the following attributes:

df: the number of parameters used by the VLMC for this likelihood calculation
nobs: the number of observations included in this likelihood calculation
initial: the value of the initial parameter used to compute this likelihood

Details

The definition of the likelihood function depends on the value of the initial parameters, see the section below as well as the dedicated vignette: vignette("likelihood", package = "mixvlmc").

For VLMC objects, the method loglikelihood.vlmc will be used. For VLMC with covariables, loglikelihood.covlmc will instead be called. For more informations on loglikelihood methods, use methods(loglikelihood) and their associated documentation.

likelihood calculation

In a (CO)VLMC of depth()=k, we need k past values in order to compute the context of a given observation. As a consequence, in a time series x, the contexts of x[1] to x[k] are unknown. Depending on the value of initial different likelihood functions are used to tackle this difficulty:

initial=="truncated": the likelihood is computed using only x[(k+1):length(x)]
initial=="specific": the likelihood is computed on the full time series using a specific context for the initial values, x[1] to x[k]. Each of the specific context is unique, leading to a perfect likelihood of 1 (0 in log scale). Thus the numerical value of the likelihood is identical as the one obtained with initial=="truncated" but it is computed on length(x) with a model with more parameters than in this previous case.
initial=="extended" (default): the likelihood is computed on the full time series using an extended context matching for the initial values, x[1] to x[k]. This can be seen as a compromised between the two other possibilities: the relaxed context matching needs in general to turn internal nodes of the context tree into actual context, increasing the number of parameters, but not as much as with "specific". However, the likelihood of say x[1] with an empty context is generally not 1 and thus the full likelihood is smaller than the one computed with "specific".

In all cases, the ignore first values of the time series are not included in the computed likelihood, but still used to compute contexts. If ignore is not specified, it is set to the minimal possible value, that is k for the truncated likelihood and 0 for the other ones. If it is specified, it must be larger or equal to k for truncated.

See the dedicated vignette for a more mathematically oriented discussion: vignette("likelihood", package = "mixvlmc").

Examples