Mit {AdMit}

Mixture of Student-t Distributions

Description

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

Usage

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

Arguments

theta

matrix (of size N \times d, where N,d \geq 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 H \times d) containing the vectors of modes (in row) of the mixture components.

Sigma

matrix (of size H \times d^2) 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 (\geq 1) is the number of components and d (\geq 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 (\geq 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").

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. R Journal 1(1), pp.25-30. doi: 10.32614/RJ-2009-003

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. doi: 10.18637/jss.v029.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

  ## 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)