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

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

  ## 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)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(AdMit)
Loading required package: mvtnorm
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/AdMit/Mit.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Mit
> ### Title: Mixture of Student-t Distributions
> ### Aliases: dMit rMit
> ### Keywords: distribution
> 
> ### ** 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)
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>