Package 'relliptical'

Mean and Variance for Truncated Multivariate Elliptical Distributions

Description

This function approximates the mean vector and variance-covariance matrix for some specific truncated elliptical distributions. The argument dist sets the distribution to be used and accepts the values Normal, t, Laplace, PE, PVII, Slash, and CN, for the truncated Normal, Student-t, Power Exponential, Pearson VII, Slash, and Contaminated Normal distribution, respectively. Moments are computed through Monte Carlo method for the truncated variables and using properties of the conditional expectation for the non-truncated variables.

Usage

mvtelliptical(lower, upper = rep(Inf, length(lower)), mu = rep(0,
  length(lower)), Sigma = diag(length(lower)), dist = "Normal",
  nu = NULL, n = 10000, burn.in = 0, thinning = 3)

Arguments

lower	vector of lower truncation points of length $p$.
upper	vector of upper truncation points of length $p$.
mu	numeric vector of length $p$ representing the location parameter.
Sigma	numeric positive definite matrix with dimension $p$x$p$ representing the scale parameter.
dist	represents the truncated distribution to be used. The values are 'Normal', 't', 'Laplace', 'PE', 'PVII', 'Slash', and 'CN' for the truncated Normal, Student-t, Laplace, Power Exponential, Pearson VII, Slash, and Contaminated Normal distributions, respectively.
nu	additional parameter or vector of parameters depending on the density generating function. See Details.
n	number of Monte Carlo samples to be generated.
burn.in	number of samples to be discarded as a burn-in phase.
thinning	factor for reducing the autocorrelation of random points.

Details

This function also considers the univariate case. The argument nu is a parameter or vector of parameters depending on the density generating function (DGF). For the truncated Student-t, Power Exponential, and Slash distribution, nu is a positive number. For the truncated Pearson VII, nu is a vector with the first element greater than $p/2$ and the second element a positive number. For the truncated Contaminated Normal distribution, nu is a vector of length 2 assuming values between 0 and 1.

Value

It returns a list with three elements:

EY	the mean vector of length $p$.
EYY	the second moment matrix of dimensions $p$x$p$.
VarY	the variance-covariance matrix of dimensions $p$x$p$.

Note

The Normal distribution is a particular case of the Power Exponential distribution when nu = 1. The Student-t distribution with $\nu$ degrees of freedom results from the Pearson VII distribution when nu = (($\nu$+p)/2, $\nu$).

In the Student-t distribution, if nu >= 300, the Normal case is considered. For Student-t distribution, the algorithm also supports degrees of freedom nu <= 2. For Pearson VII distribution, the algorithm supports values of m <= (p+2)/2 (first element of nu).

Author(s)

Katherine L. Valeriano, Christian E. Galarza and Larissa A. Matos

References

Fang KT, Kotz S, Ng KW (2018). Symmetric multivariate and related distributions. Chapman and Hall/CRC.

Galarza CE, Matos LA, Castro LM, Lachos VH (2022). “Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution.” Journal of Multivariate Analysis, 189, 104944. doi:10.1016/j.jmva.2021.104944.

Valeriano KA, Galarza CE, Matos LA (2023). “Moments and random number generation for the truncated elliptical family of distributions.” Statistics and Computing, 33(1), 32.

See Also

rtelliptical

Examples

# Truncated Student-t distribution
set.seed(5678)
mu = c(0.1, 0.2, 0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1), nrow=length(mu),
               ncol=length(mu), byrow=TRUE)

# Example 1: one doubly truncated student-t (nu = 0.80) and Laplace
a = c(-0.8, -Inf, -Inf)
b = c(0.5, 0.6, Inf)
MC11 = mvtelliptical(a, b, mu, Sigma, "t", 0.80) # Student-t
MC12 = mvtelliptical(a, b, mu, Sigma, "Laplace") # Laplace

# Example 2: two doubly truncated student-t (nu = 0.80)
MC12 = mvtelliptical(a, b, mu, Sigma, "t", 0.80) # By default n=1e4

# Truncated Pearson VII distribution
set.seed(9876)
MC21 = mvtelliptical(a, b, mu, Sigma, "PVII", c(1.90,0.80), n=1e6) # More precision
c(MC12$EY); c(MC21$EY)
MC12$VarY;  MC21$VarY

# Truncated Normal distribution
set.seed(1234)
MC31 = mvtelliptical(a, b, mu, Sigma, "Normal", n=1e4)
MC32 = mvtelliptical(a, b, mu, Sigma, "Normal", n=1e6) # More precision

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Sampling Random Numbers from Truncated Multivariate Elliptical Distributions

Description

This function generates observations from a truncated multivariate elliptical distribution with location parameter mu, scale matrix Sigma, lower and upper truncation points lower and upper via Slice Sampling algorithm with Gibbs sampler steps.

Usage

rtelliptical(n = 10000, mu = rep(0, length(lower)),
  Sigma = diag(length(lower)), lower, upper = rep(Inf, length(lower)),
  dist = "Normal", nu = NULL, expr = NULL, gFun = NULL,
  ginvFun = NULL, burn.in = 0, thinning = 1)

Arguments

n	number of observations to generate. Must be an integer >= 1.
mu	numeric vector of length $p$ representing the location parameter.
Sigma	numeric positive definite matrix with dimension $p$x$p$ representing the scale parameter.
lower	vector of lower truncation points of length $p$.
upper	vector of upper truncation points of length $p$.
dist	represents the truncated distribution to be used. The values are 'Normal', 't', 'Laplace', 'PE', 'PVII', 'Slash', and 'CN' for the truncated Normal, Student-t, Laplace, Power Exponential, Pearson VII, Slash, and Contaminated Normal distribution, respectively.
nu	additional parameter or vector of parameters depending on the density generating function. See Details.
expr	a character with the density generating function. See Details.
gFun	an R function with the density generating function. See Details.
ginvFun	an R function with the inverse of the density generating function defined in gFun. See Details.
burn.in	number of samples to be discarded as a burn-in phase.
thinning	factor for reducing the autocorrelation of random points.

Details

The dist argument represents the truncated distribution to be used. The values are Normal, t, 't', PE, PVII, Slash, and CN, for the truncated Normal, Student-t, Laplace, Power Exponential, Pearson VII, Slash, and Contaminated Normal distribution, respectively.

The argument nu is a parameter or vector of parameters depending on the density generating function (DGF). For the truncated Student-t, Power Exponential, and Slash distribution, nu is a positive number. For the truncated Pearson VII, nu is a vector with the first element greater than $p/2$ and the second element a positive number. For the truncated Contaminated Normal distribution, nu is a vector of length 2 assuming values between 0 and 1.

This function also allows generating random numbers from other truncated elliptical distributions not specified in the dist argument, by supplying the density generating function (DGF) through arguments either expr or gFun. The DGF must be a non-negative and strictly decreasing function on (0, Inf). The easiest way is to provide the DGF expression to argument expr as a character. The notation used in expr needs to be understood by package Ryacas0, and the environment of R. For instance, for the DGF $g(t)=e^{-t}$, the user must provide expr = "exp(1)^(-t)". See that the function must depend only on variable $t$, and any additional parameter must be passed as a fixed value. For this case, when a character expression is provided to expr, the algorithm tries to compute a closed-form expression for the inverse function of $g(t)$, however, this is not always possible (a warning message is returned). See example 2.

If it was no possible to generate random samples by passing a character expression to expr, the user may provide a custom R function to the gFun argument. By default, its inverse function is approximated numerically, however, the user may also provide its inverse to the ginvFun argument to gain some computational time. When gFun is provided, arguments dist and expr are ignored.

Value

It returns a matrix of dimensions $n$x$p$ with the random points sampled.

Note

The Normal distribution is a particular case of the Power Exponential distribution when nu = 1. The Student-t distribution with $\nu$ degrees of freedom results from the Pearson VII distribution when nu = (($\nu$+p)/2, $\nu$).

Author(s)

Katherine L. Valeriano, Christian E. Galarza and Larissa A. Matos

References

Fang KT, Kotz S, Ng KW (2018). Symmetric multivariate and related distributions. Chapman and Hall/CRC.

Ho HJ, Lin TI, Chen HY, Wang WL (2012). “Some results on the truncated multivariate t distribution.” Journal of Statistical Planning and Inference, 142(1), 25–40. doi:10.1016/j.jspi.2011.06.006.

Neal RM (2003). “Slice sampling.” Annals of statistics, 705–741.

Robert CP, Casella G (2010). Introducing Monte Carlo Methods with R, volume 18. New York: Springer.

Valeriano KA, Galarza CE, Matos LA (2023). “Moments and random number generation for the truncated elliptical family of distributions.” Statistics and Computing, 33(1), 32.

See Also

mvtelliptical

Examples

library(ggplot2)
library(ggExtra)
library(gridExtra)

# Example 1: Sampling from the Truncated Normal distribution
set.seed(1234)
mu  = c(0, 1)
Sigma = matrix(c(1,0.70,0.70,3), 2, 2)
lower = c(-2, -3)
upper = c(3, 3)
sample1 = rtelliptical(5e4, mu, Sigma, lower, upper, dist="Normal")

# Histogram and density for variable 1
ggplot(data.frame(sample1), aes(x=X1)) +
   geom_histogram(aes(y=after_stat(density)), colour="black", fill="grey", bins=15) +
   geom_density(color="red") + labs(x=bquote(X[1]), y="Density") + theme_bw()

# Histogram and density for variable 2
ggplot(data.frame(sample1), aes(x=X2)) +
   geom_histogram(aes(y=after_stat(density)), colour="black", fill="grey", bins=15) +
   geom_density(color="red") + labs(x=bquote(X[2]), y="Density") + theme_bw()


# Example 2: Sampling from the Truncated Logistic distribution

# Function for plotting the sample autocorrelation using ggplot2
acf.plot = function(samples){
 p = ncol(samples); n = nrow(samples); q1 = qnorm(0.975)/sqrt(n); acf1 = list(p)
 for (i in 1:p){
   bacfdf = with(acf(samples[,i], plot=FALSE), data.frame(lag, acf))
   acf1[[i]] = ggplot(data=bacfdf, aes(x=lag,y=acf)) + geom_hline(aes(yintercept=0)) +
     geom_segment(aes(xend=lag, yend=0)) + labs(x="Lag", y="ACF", subtitle=bquote(X[.(i)])) +
     geom_hline(yintercept=c(q1,-q1), color="red", linetype="twodash") + theme_bw()
 }
 return (acf1)
}

set.seed(5678)
mu  = c(0, 0)
Sigma = matrix(c(1,0.70,0.70,1), 2, 2)
lower = c(-2, -2)
upper = c(3, 2)
# Sample autocorrelation with no thinning
sample2 = rtelliptical(10000, mu, Sigma, lower, upper, expr="exp(1)^(-t)/(1+exp(1)^(-t))^2")
grid.arrange(grobs=acf.plot(sample2), top="Logistic distribution with no thinning", nrow=1)

# Sample autocorrelation with thinning = 3
sample3 = rtelliptical(10000, mu, Sigma, lower, upper, expr="exp(1)^(-t)/(1+exp(1)^(-t))^2",
                       thinning=3)
grid.arrange(grobs=acf.plot(sample3), top="Logistic distribution with thinning = 3", nrow=1)


# Example 3: Sampling from the Truncated Kotz-type distribution
set.seed(5678)
mu  = c(0, 0)
Sigma = matrix(c(1,-0.5,-0.5,1), 2, 2)
lower = c(-2, -2)
upper = c(3, 2)
sample4 = rtelliptical(2000, mu, Sigma, lower, upper, gFun=function(t){t^(-1/2)*exp(-2*t^(1/4))})
f1 = ggplot(data.frame(sample4), aes(x=X1,y=X2)) + geom_point(size=0.50) +
     labs(x=expression(X[1]), y=expression(X[2]), subtitle="Kotz(2,1/4,1/2)") + theme_bw()
ggMarginal(f1, type="histogram", fill="grey")

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