Mixture of Student-t Distributions

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Description

Density function or random generation for an adaptive mixture of Student-t distributions

Usage

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dMit(theta, mit = list(), log = TRUE)
rMit(N = 1, mit = list())

Arguments

theta

matrix (of size Nxd, where N,d>=1) of real values.

mit

list containing information on the mixture approximation (see *Details*).

log

logical; if log = TRUE, returns (natural) logarithm values of the density. Default: log = TRUE.

N

number of draws (positive integer number).

Details

dMit returns the density values while rMit generates draws from a mixture of Student-t distributions.

The argument mit is a list containing information on the adaptive mixture of Student-t distributions. The following components must be provided:

p

vector (of length H) of mixture probabilities.

mu

matrix (of size Hxd) containing the vectors of modes (in row) of the mixture components.

Sigma

matrix (of size Hxd*d) containing the scale matrices (in row) of the mixture components.

df

degrees of freedom parameter of the Student-t components (integer number not smaller than one).

where H (>=1) is the number of components and d (>=1) is the dimension of the mixture approximation. Typically, mit is estimated by the function AdMit. If the mit = list(), a Student-t distribution located at rep(0,d) with scale matrix diag(d) and one degree of freedom parameter is used.

Value

Vector (of length N of density values, or matrix (of size Nxd) of random draws, where d (>=1) is the dimension of the mixture approximation.

Note

Further details and examples of the R package AdMit can be found in Ardia, Hoogerheide, van Dijk (2009a,b). See also the package vignette by typing vignette("AdMit") and the files ‘AdMitJSS.txt’ and ‘AdMitRnews.txt’ in the ‘/doc’ package's folder.

Please cite the package in publications. Use citation("AdMit").

Author(s)

David Ardia

References

Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009a). AdMit: Adaptive Mixture of Student-t Distributions. The R Journal 1(1), pp.25–30. http://journal.r-project.org/2009-1/

Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009b). Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit. Journal of Statistical Software 29(3), pp.1–32. http://www.jstatsoft.org/v29/i03/

See Also

AdMit for fitting an adaptive mixture of Student-t distributions to a given function KERNEL, AdMitIS for importance sampling using an adaptive mixture of Student-t distributions as the importance density, AdMitMH for the independence chain Metropolis-Hastings using an adaptive mixture of Student-t distributions as the candidate density.

Examples

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  ## NB : Low number of draws for speedup. Consider using more draws!
  ## One dimensional two components mixture of Student-t distributions
  mit <- list(p = c(0.5, 0.5),
              mu = matrix(c(-2.0, 0.5), 2, 1, byrow = TRUE),
              Sigma = matrix(0.1, 2),
              df = 10)
  ## Generate draws from the mixture
  hist(rMit(1e4, mit = mit), nclass = 100, freq = FALSE)
  x <- seq(from = -5.0, to = 5.0, by = 0.01)
  ## Add the density to the histogram
  lines(x, dMit(x, mit = mit, log = FALSE), col = "red", lwd = 2)

  ## Two dimensional (one component mixture) Student-t distribution
  mit <- list(p = 1,
              mu = matrix(0.0, 1.0, 2.0),
              Sigma = matrix(c(1.0, 0.0, 0.0, 1.0), 1, 4),
              df = 10)
  ## Function used to plot the mixture in two dimensions
  dMitPlot <- function(x1, x2, mit = mit)
  {
    dMit(cbind(x1, x2), mit = mit, log = FALSE)
  }
  x1 <- x2 <- seq(from = -10.0, to = 10.0, by = 0.1)
  thexlim <- theylim <- range(x1)
  z <- outer(x1, x2, FUN = dMitPlot, mit = mit)
  ## Contour plot of the mixture
  contour(x1, x2, z, nlevel = 20, las = 1, 
          col = rainbow(20),
          xlim = thexlim, ylim = theylim)
  par(new = TRUE)
  ## Generate draws from the mixture
  plot(rMit(1e4, mit = mit), pch = 20, cex = 0.3, 
            xlim = thexlim, ylim = theylim, col = "red", las = 1)