R/distributions.R
In benmarwick/LaplacesDemon: Complete Environment for Bayesian Inference
###########################################################################
# Asymmetric Laplace Distribution #
# #
# These functions are similar to those in the VGAM package. #
###########################################################################
dalaplace <- function(x, location=0, scale=1, kappa=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(x), length(location), length(scale), length(kappa))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
logconst <- 0.5 * log(2) - log(scale) + log(kappa) - log1p(kappa^2)
exponent <- -(sqrt(2) / scale) * abs(x - location) *
ifelse(x >= location, kappa, 1/kappa)
dens <- logconst + exponent
if(log == FALSE) dens <- exp(logconst + exponent)
return(dens)
}
palaplace <- function(q, location=0, scale=1, kappa=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if((kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(q), length(location), length(scale), length(kappa))
q <- rep(q, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
exponent <- -(sqrt(2) / scale) * abs(q - location) *
ifelse(q >= location, kappa, 1/kappa)
temp <- exp(exponent) / (1 + kappa^2)
p <- 1 - temp
index1 <- (q < location)
p[index1] <- (kappa[index1])^2 * temp[index1]
return(p)
}
qalaplace <- function(p, location=0, scale=1, kappa=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(p), length(location), length(scale), length(kappa))
p <- rep(p, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
q <- p
temp <- kappa^2 / (1 + kappa^2)
index1 <- (p <= temp)
exponent <- p[index1] / temp[index1]
q[index1] <- location[index1] + (scale[index1] * kappa[index1]) *
log(exponent) / sqrt(2)
q[!index1] <- location[!index1] - (scale[!index1] / kappa[!index1]) *
(log1p((kappa[!index1])^2) + log1p(-p[!index1])) / sqrt(2)
q[p == 0] = -Inf
q[p == 1] = Inf
return(q)
}
ralaplace <- function(n, location=0, scale=1, kappa=1)
{
location <- rep(location, len=n)
scale <- rep(scale, len=n)
kappa <- rep(kappa, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
x <- location + scale *
log(runif(n)^kappa / runif(n)^(1/kappa)) / sqrt(2)
return(x)
}
###########################################################################
# Asymmetric Log-Laplace Distribution #
# #
# These functions are similar to those in the VGAM package. #
###########################################################################
dallaplace <- function(x, location=0, scale=1, kappa=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(x), length(location), length(scale), length(kappa))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
Alpha <- sqrt(2) * kappa / scale
Beta <- sqrt(2) / (scale * kappa)
Delta <- exp(location)
exponent <- ifelse(x >= Delta, -(Alpha+1),
(Beta-1)) * (log(x) - location)
dens <- -location + log(Alpha) + log(Beta) -
log(Alpha + Beta) + exponent
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pallaplace <- function(q, location=0, scale=1, kappa=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(q), length(location), length(scale), length(kappa))
q <- rep(q, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
Alpha <- sqrt(2) * kappa / scale
Beta <- sqrt(2) / (scale * kappa)
Delta <- exp(location)
temp <- Alpha + Beta
p <- (Alpha / temp) * (q / Delta)^(Beta)
p[q <= 0] <- 0
index1 <- (q >= Delta)
p[index1] <- (1 - (Beta/temp) * (Delta/q)^(Alpha))[index1]
return(p)
}
qallaplace <- function(p, location=0, scale=1, kappa=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(p), length(location), length(scale), length(kappa))
p <- rep(p, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
Alpha <- sqrt(2) * kappa / scale
Beta <- sqrt(2) / (scale * kappa)
Delta <- exp(location)
temp <- Alpha + Beta
q <- Delta * (p * temp / Alpha)^(1/Beta)
index1 <- (p > Alpha / temp)
q[index1] <- (Delta * ((1-p) * temp / Beta)^(-1/Alpha))[index1]
q[p == 0] <- 0
q[p == 1] <- Inf
return(q)
}
rallaplace <- function(n, location=0, scale=1, kappa=1)
{
location <- rep(location, len=n); scale <- rep(scale, len=n)
kappa <- rep(kappa, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
x <- exp(location) *
(runif(n)^kappa / runif(n)^(1/kappa))^(scale / sqrt(2))
return(x)
}
###########################################################################
# Bernoulli Distribution #
# #
# These functions are similar to those in the Rlab package. #
###########################################################################
dbern <- function(x, prob, log=FALSE)
{return(dbinom(x, 1, prob, log))}
pbern <- function(q, prob, lower.tail=TRUE, log.p=FALSE)
{return(pbinom(q, 1, prob, lower.tail, log.p))}
qbern <- function(p, prob, lower.tail=TRUE, log.p=FALSE)
{return(qbinom(p, 1, prob, lower.tail, log.p))}
rbern <- function(n, prob)
{return(rbinom(n, 1, prob))}
###########################################################################
# Categorical Distribution #
###########################################################################
dcat <- function(x, p, log=FALSE)
{
if(is.vector(x) & !is.matrix(p))
p <- matrix(p, length(x), length(p), byrow=TRUE)
if(is.matrix(x) & !is.matrix(p))
p <- matrix(p, nrow(x), length(p), byrow=TRUE)
if(is.vector(x) & {length(x) == 1}) {
temp <- rep(0, ncol(p))
temp[x] <- 1
x <- t(temp)}
else if(is.vector(x) & (length(x) > 1))
x <- as.indicator.matrix(x)
if(!identical(nrow(x), nrow(p)))
stop("The number of rows of x and p differ.")
if(!identical(ncol(x), ncol(p))) {
x.temp <- matrix(0, nrow(p), ncol(p))
x.temp[,as.numeric(colnames(x))] <- x
x <- x.temp}
dens <- x*p
if(log == TRUE) dens <- x*log(p)
dens <- as.vector(rowSums(dens))
return(dens)
}
qcat <- function(pr, p, lower.tail=TRUE, log.pr=FALSE)
{
if(!is.vector(pr)) pr <- as.vector(pr)
if(!is.vector(p)) p <- as.vector(p)
if(log.pr == FALSE) {
if(any(pr < 0) | any(pr > 1))
stop("pr must be in [0,1].")}
else if(any(!is.finite(pr)) | any(pr > 0))
stop("pr, as a log, must be in (-Inf,0].")
if(sum(p) != 1) stop("sum(p) must be 1.")
if(lower.tail == FALSE) pr <- 1 - pr
breaks <- c(0, cumsum(p))
if(log.pr == TRUE) breaks <- log(breaks)
breaks <- matrix(breaks, length(pr), length(breaks), byrow=TRUE)
x <- rowSums(pr > breaks)
return(x)
}
rcat <- function(n, p)
{
if(is.vector(p)) {
x <- as.vector(which(rmultinom(n, size=1, prob=p) == 1,
arr.ind=TRUE)[, "row"])}
else {
n <- nrow(p)
x <- apply(p, 1, function(x) {
as.vector(which(rmultinom(1, size=1, prob=x) == 1,
arr.ind=TRUE)[, "row"])})
}
return(x)
}
###########################################################################
# Dirichlet Distribution #
# #
# These functions are similar to those in the MCMCpack package. #
###########################################################################
ddirichlet <- function(x, alpha, log=FALSE)
{
if(missing(x)) stop("x is a required argument.")
if(missing(alpha)) stop("alpha is a required argument.")
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(alpha))
alpha <- matrix(alpha, nrow(x), length(alpha), byrow=TRUE)
if(any(rowSums(x) != 1)) x / rowSums(x)
if(any(x < 0)) stop("x must be non-negative.")
if(any(alpha <= 0)) stop("alpha must be positive.")
dens <- as.vector(lgamma(rowSums(alpha)) - rowSums(lgamma(alpha)) +
rowSums((alpha-1)*log(x)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rdirichlet <- function (n, alpha)
{
alpha <- rbind(alpha)
alpha.dim <- dim(alpha)
if(n > alpha.dim[1])
alpha <- matrix(alpha, n, alpha.dim[2], byrow=TRUE)
x <- matrix(rgamma(alpha.dim[2]*n, alpha), ncol=alpha.dim[2])
sm <- x %*% rep(1, alpha.dim[2])
return(x/as.vector(sm))
}
###########################################################################
# Generalized Poisson #
###########################################################################
dgpois <- function(x, lambda=0, omega=0, log=FALSE)
{
x <- as.vector(x); lambda <- as.vector(lambda)
omega <- as.vector(omega)
lambda.star <- (1 - omega)*lambda + omega*x
dens <- log(1 - omega) + log(lambda) + (x - 1)*log(lambda.star) -
lgamma(x + 1) - lambda.star
if(log == FALSE) dens <- exp(dens)
return(dens)
}
###########################################################################
# Half-Cauchy Distribution #
###########################################################################
dhalfcauchy <- function(x, scale=25, log=FALSE)
{
x <- as.vector(x); scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(scale))
x <- rep(x, len=NN); scale <- rep(scale, len=NN)
dens <- log(2*scale) - log(pi*{x*x + scale*scale})
if(log == FALSE) dens <- exp(dens)
return(dens)
}
phalfcauchy <- function(q, scale=25)
{
q <- as.vector(q); scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(scale))
q <- rep(q, len=NN); scale <- rep(scale, len=NN)
z <- {2/pi}*atan(q/scale)
return(z)
}
qhalfcauchy <- function(p, scale=25)
{
p <- as.vector(p); scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(scale))
p <- rep(p, len=NN); scale <- rep(scale, len=NN)
q <- scale*tan({pi*p}/2)
return(q)
}
rhalfcauchy <- function(n, scale=25)
{
scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
p <- runif(n, 0, 1)
x <- scale*tan({pi*p}/2)
return(x)
}
###########################################################################
# Half-Normal Distribution #
# #
# This half-normal distribution has mean=0 and is similar to the halfnorm #
# functions in package fdrtool. #
###########################################################################
dhalfnorm <- function(x, scale=sqrt(pi/2), log=FALSE)
{
dens <- log(2) + dnorm(x, mean=0, sd=sqrt(pi/2) / scale, log=TRUE)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
phalfnorm <- function(q, scale=sqrt(pi/2), lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(scale))
q <- rep(q, len=NN); scale <- rep(scale, len=NN)
p <- 2*pnorm(q, mean=0, sd=sqrt(pi/2) / scale) - 1
if(lower.tail == FALSE) p <- 1-p
if(log.p == TRUE) p <- log.p(p)
return(p)
}
qhalfnorm <- function(p, scale=sqrt(pi/2), lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(scale))
p <- rep(p, len=NN); scale <- rep(scale, len=NN)
if(log.p == TRUE) p <- exp(p)
if(lower.tail == FALSE) p <- 1-p
q <- qnorm((p+1)/2, mean=0, sd=sqrt(pi/2) / scale)
return(q)
}
rhalfnorm <- function(n, scale=sqrt(pi/2))
{
scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
x <- abs(rnorm(n, mean=0, sd=sqrt(pi/2) / scale))
return(x)
}
###########################################################################
# Half-t Distribution #
###########################################################################
dhalft <- function(x, scale=25, nu=1, log=FALSE)
{
x <- as.vector(x); scale <- as.vector(scale); nu <- as.vector(nu)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(scale), length(nu))
x <- rep(x, len=NN); scale <- rep(scale, len=NN)
nu <- rep(nu, len=NN)
dens <- (-(nu+1)/2)*log(1 + (1/nu)*(x/scale)*(x/scale))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
phalft <- function(q, scale=25, nu=1)
{
q <- as.vector(q); scale <- as.vector(scale); nu <- as.vector(nu)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(scale), length(nu))
q <- rep(q, len=NN); scale <- rep(scale, len=NN)
p <- ptrunc(q, "st", a=0, b=Inf, mu=0, sigma=scale, nu=nu)
return(p)
}
qhalft <- function(p, scale=25, nu=1)
{
p <- as.vector(p); scale <- as.vector(scale); nu <- as.vector(nu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(scale), length(nu))
p <- rep(p, len=NN); scale <- rep(scale, len=NN)
q <- rtrunc(p, "st", a=0, b=Inf, mu=0, sigma=scale, nu=nu)
return(q)
}
rhalft <- function(n, scale=25, nu=1)
{
scale <- rep(scale, len=n); nu <- rep(nu, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
x <- rtrunc(n, "st", a=0, b=Inf, mu=0, sigma=scale, nu=nu)
return(x)
}
###########################################################################
# Horseshoe Distribution #
###########################################################################
dhs <- function(x, lambda, tau, sigma, log=FALSE)
{
NN <- max(length(x), length(lambda))
x <- rep(x, len=NN); lambda <- rep(lambda, len=NN)
p.theta.lambda <- dnorm(x, 0, lambda, log=TRUE)
p.lambda.tau <- dhalfcauchy(lambda, tau, log=TRUE)
p.tau <- dhalfcauchy(tau, sigma, log=TRUE)
dens <- p.theta.lambda - {p.lambda.tau + p.tau} / NN
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rhs <- function(n, lambda, tau, sigma)
{
if(missing(tau)) tau <- rhalfcauchy(1, sigma)
if(missing(lambda)) lambda <- rhalfcauchy(n, tau)
theta <- rnorm(n, 0, lambda)
return(theta)
}
###########################################################################
# Inverse Beta Distribution #
###########################################################################
dinvbeta <- function(x, a, b, log=FALSE)
{
const <- lgamma(a + b) - lgamma(a) - lgamma(b)
dens <- const + (a - 1) * log(x) - (a + b) * log(1 + x)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvbeta <- function(n, a, b)
{
x <- rbeta(n, a, b)
x <- x / (1-x)
return(x)
}
###########################################################################
# Inverse Chi-Squared Distribution #
# #
# These functions are similar to those in the GeoR package. #
###########################################################################
dinvchisq <- function(x, df, scale=1/df, log=FALSE)
{
x <- as.vector(x); df <- as.vector(df); scale <- as.vector(scale)
if(any(x <= 0)) stop("x must be positive.")
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(df), length(scale))
x <- rep(x, len=NN); df <- rep(df, len=NN);
scale <- rep(scale, len=NN)
nu <- df / 2
dens <- nu*log(nu) - log(gamma(nu)) + nu*log(scale) -
(nu+1)*log(x) - (nu*scale/x)
if(log == FALSE) dens <- exp(dens)
}
rinvchisq <- function(n, df, scale=1/df)
{
df <- rep(df, len=n); scale <- rep(scale, len=n)
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
z <- rchisq(n, df=df)
z <- ifelse(z == 0, 1e-100, z)
x <- (df*scale) / z
return(x)
}
###########################################################################
# Inverse Gamma Distribution #
# #
# These functions are similar to those in the MCMCpack package. #
###########################################################################
dinvgamma <- function(x, shape=1, scale=1, log=FALSE)
{
x <- as.vector(x); shape <- as.vector(shape)
scale <- as.vector(scale)
if(any(shape <= 0) | any(scale <=0))
stop("The shape and scale parameters must be positive.")
NN <- max(length(x), length(shape), length(scale))
x <- rep(x, len=NN); shape <- rep(shape, len=NN)
scale <- rep(scale, len=NN)
alpha <- shape; beta <- scale
dens <- alpha * log(beta) - lgamma(alpha) -
{alpha + 1} * log(x) - {beta/x}
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvgamma <- function(n, shape=1, scale=1)
{return(1 / rgamma(n=n, shape=shape, rate=scale))}
###########################################################################
# Inverse Gaussian Distribution #
###########################################################################
dinvgaussian <- function(x, mu, lambda, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); lambda <- as.vector(lambda)
if(any(x <= 0)) stop("x must be positive.")
if(any(mu <= 0)) stop("The mu parameter must be positive.")
if(any(lambda <= 0)) stop("The lambda parameter must be positive.")
NN <- max(length(x), length(mu), length(lambda))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
lambda <- rep(lambda, len=NN)
dens <- log(lambda / (2*pi*x^3)^0.5) -
((lambda*(x - mu)^2) / (2*mu^2*x))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvgaussian <- function(n, mu, lambda)
{
mu <- rep(mu, len=n); lambda <- rep(lambda, len=n)
if(any(mu <= 0)) stop("The mu parameter must be positive.")
if(any(lambda <= 0)) stop("The lambda parameter must be positive.")
nu <- rnorm(n)
y <- nu^2
x <- mu + ((mu^2*y)/(2*lambda)) - (mu/(2*lambda)) *
sqrt(4*mu*lambda*y + mu^2*y^2)
z <- runif(n)
x <- ifelse(z > (mu / (mu+x)), mu^2/x, x)
return(x)
}
###########################################################################
# Inverse Wishart Distribution #
###########################################################################
dinvwishart <- function(Sigma, nu, S, log=FALSE)
{
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(S), dim(Sigma)))
stop("The dimensions of Sigma and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Sigma)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -(nu*k/2)*log(2) - ((k*(k - 1))/4)*log(pi) - gamsum +
(nu/2)*log(det(S)) - ((nu + k + 1)/2)*log(det(Sigma)) -
0.5*tr(S %*% as.inverse(Sigma))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvwishart <- function(nu, S)
{return(as.inverse(rwishart(nu, as.inverse(S))))}
###########################################################################
# Inverse Wishart Distribution (Cholesky Parameterization) #
###########################################################################
dinvwishartc <- function(U, nu, S, log=FALSE)
{
if(missing(U)) stop("Upper triangular U is required.")
Sigma <- t(U) %*% U
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(S), dim(Sigma)))
stop("The dimensions of Sigma and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Sigma)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -(nu*k/2)*log(2) - ((k*(k - 1))/4)*log(pi) - gamsum +
(nu/2)*log(det(S)) - ((nu + k + 1)/2)*log(det(Sigma)) -
0.5*tr(S %*% as.inverse(Sigma))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvwishartc <- function(nu, S)
{return(chol(as.inverse(rwishart(nu, as.inverse(S)))))}
###########################################################################
# Laplace Distribution #
# #
# These functions are similar to those in the VGAM package. #
###########################################################################
dlaplace <- function(x, location=0, scale=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(location), length(scale))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
dens <- (-abs(x - location) / scale) - log(2 * scale)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
plaplace <- function(q, location=0, scale=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
z <- {q - location} / scale
NN <- max(length(q), length(location), length(scale))
q <- rep(q, len=NN); location <- rep(location, len=NN)
p <- ifelse(q < location, 0.5 * exp(z), 1 - 0.5 * exp(-z))
return(p)
}
qlaplace <- function(p, location=0, scale=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(location), length(scale))
p <- rep(p, len=NN); location <- rep(location, len=NN)
q <- location - sign(p - 0.5) * scale * log(2 * ifelse(p < 0.5,
p, 1 - p))
return(q)
}
rlaplace <- function(n, location=0, scale=1)
{
location <- rep(location, len=n); scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
r <- runif(n)
x <- location - sign(r - 0.5) * scale * log(2 * ifelse(r < 0.5,
r, 1 - r))
return(x)
}
###########################################################################
# Laplace Distribution (Precision Parameterization) #
###########################################################################
dlaplacep <- function(x, mu=0, tau=1, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(x), length(mu), length(tau))
x <- rep(x, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
dens <- log(tau/2) + (-tau*abs(x-mu))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
plaplacep <- function(q, mu=0, tau=1)
{
q <- as.vector(q); mu <- as.vector(mu)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(q), length(mu), length(tau))
q <- rep(q, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
p <- plaplace(q, mu, scale=1/tau)
return(p)
}
qlaplacep <- function(p, mu=0, tau=1)
{
p <- as.vector(p); mu <- as.vector(mu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(p), length(mu), length(tau))
p <- rep(p, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
q <- qlaplace(p, mu, scale=1/tau)
return(q)
}
rlaplacep <- function(n, mu=0, tau=1)
{
mu <- rep(mu, len=n); tau <- rep(tau, len=n)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
x <- rlaplace(n, mu, scale=1/tau)
return(x)
}
###########################################################################
# Log-Laplace Distribution #
###########################################################################
dllaplace <- function(x, location=0, scale=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(location), length(scale))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
Alpha <- sqrt(2) * scale
Beta <- sqrt(2) / scale
Delta <- exp(location)
exponent <- ifelse(x >= Delta, -(Alpha+1),
(Beta-1)) * (log(x) - location)
dens <- -location + log(Alpha) + log(Beta) -
log(Alpha + Beta) + exponent
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pllaplace <- function(q, location=0, scale=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(location), length(scale))
q <- rep(q, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
Alpha <- sqrt(2) * scale
Beta <- sqrt(2) / scale
Delta <- exp(location)
temp <- Alpha + Beta
p <- (Alpha / temp) * (q / Delta)^(Beta)
p[q <= 0] <- 0
index1 <- (q >= Delta)
p[index1] <- (1 - (Beta/temp) * (Delta/q)^(Alpha))[index1]
return(p)
}
qllaplace <- function(p, location=0, scale=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(location), length(scale))
p <- rep(p, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
Alpha <- sqrt(2) * scale
Beta <- sqrt(2) / scale
Delta <- exp(location)
temp <- Alpha + Beta
q <- Delta * (p * temp / Alpha)^(1/Beta)
index1 <- (p > Alpha / temp)
q[index1] <- (Delta * ((1-p) * temp / Beta)^(-1/Alpha))[index1]
q[p == 0] <- 0
q[p == 1] <- Inf
return(q)
}
rllaplace <- function(n, location=0, scale=1)
{
location <- rep(location, len=n); scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
x <- exp(location) * (runif(n) / runif(n))^(scale / sqrt(2))
return(x)
}
###########################################################################
# Log-Normal Distribution (Precision Parameterization) #
###########################################################################
dlnormp <- function(x, mu, tau, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(x), length(mu), length(tau))
x <- rep(x, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
dens <- log(sqrt(tau/(2*pi))) + log(1/x) + (-(tau/2)*(log(x-mu)^2))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
plnormp <- function(q, mu, tau, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(q), length(mu), length(tau))
q <- rep(q, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
p <- pnorm(q, mu, sqrt(1/tau), lower.tail, log.p)
return(p)
}
qlnormp <- function(p, mu, tau, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(p), length(mu), length(tau))
p <- rep(p, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
q <- qnorm(p, mu, sqrt(1/tau), lower.tail, log.p)
return(q)
}
rlnormp <- function(n, mu, tau)
{
mu <- rep(mu, len=n); tau <- rep(tau, len=n)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
x <- rnorm(n, mu, sqrt(1/tau))
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution #
###########################################################################
dmvc <- function(x, mu, S, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(S)) S <- diag(ncol(x))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
k <- nrow(S)
ss <- x - mu
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma(k/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi) + 0.5*log(det(S)) + ((1+k)/2)*log(1+z)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvc <- function(n=1, mu=rep(0,k), S)
{
mu <- rbind(mu)
if(missing(S)) S <- diag(ncol(mu))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
k <- ncol(S)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), S)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution (Cholesky Parameterization) #
###########################################################################
dmvcc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- nrow(U)
S <- t(U) %*% U
ss <- x - mu
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma(k/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi) + 0.5*log(det(S)) + ((1+k)/2)*log(1+z)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvcc <- function(n=1, mu=rep(0,k), U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
S <- t(U) %*% U
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution (Precision Parameterization) #
###########################################################################
dmvcp <- function(x, mu, Omega, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Omega)) Omega <- diag(ncol(x))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
k <- nrow(Omega)
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((1+k)/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi)) + 0.5*log(detOmega) + (-(1+k)/2)*log(1 + z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvcp <- function(n=1, mu, Omega)
{
mu <- rbind(mu)
if(missing(Omega)) Omega <- diag(ncol(mu))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
Sigma <- as.inverse(Omega)
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), Sigma)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution (Precision-Cholesky Parameterization) #
###########################################################################
dmvcpc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- nrow(U)
Omega <- t(U) %*% U
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((1+k)/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi)) + 0.5*log(detOmega) + (-(1+k)/2)*log(1 + z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvcpc <- function(n=1, mu, U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Laplace Distribution #
###########################################################################
dmvl <- function(x, mu, Sigma, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Sigma)) Sigma <- diag(ncol(x))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
Sigma <- as.symmetric.matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- log(2 / ((2*pi)^(k/2) * sqrt(det(Sigma)))) +
log((sqrt(pi / (2*sqrt(2*z))) * exp(-sqrt(2*z))) /
sqrt(z/2)^(k/2 - 1))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvl <- function(n, mu, Sigma)
{
mu <- rbind(mu)
if(missing(Sigma)) Sigma <- diag(ncol(mu))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
e <- matrix(rexp(n, 1), n, k)
z <- rmvn(n, rep(0, k), Sigma)
x <- mu + sqrt(e)*z
return(x)
}
###########################################################################
# Multivariate Laplace Distribution (Cholesky Parameterization) #
###########################################################################
dmvlc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
Sigma <- t(U) %*% U
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- log(2 / ((2*pi)^(k/2) * sqrt(det(Sigma)))) +
log((sqrt(pi / (2*sqrt(2*z))) * exp(-sqrt(2*z))) /
sqrt(z/2)^(k/2 - 1))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvlc <- function(n, mu, U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
e <- matrix(rexp(n, 1), n, k)
z <- rmvnc(n, rep(0, k), U)
x <- mu + sqrt(e)*z
return(x)
}
###########################################################################
# Multivariate Normal Distribution #
###########################################################################
dmvn <- function(x, mu, Sigma, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Sigma)) Sigma <- diag(ncol(x))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
Sigma <- as.symmetric.matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
d <- eigen(Sigma, symmetric=TRUE)$values
dens <- as.vector(-0.5 * (k * log(2 * pi) + sum(log(d))) - (0.5 * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvn <- function(n=1, mu=rep(0,k), Sigma)
{
mu <- rbind(mu)
if(missing(Sigma)) Sigma <- diag(ncol(mu))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% chol(Sigma)
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Normal Distribution (Cholesky Parameterization) #
###########################################################################
dmvnc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
Sigma <- t(U) %*% U
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
d <- eigen(Sigma, symmetric=TRUE)$values
dens <- as.vector(-0.5 * (k * log(2 * pi) + sum(log(d))) - (0.5 * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvnc <- function(n=1, mu=rep(0,k), U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% U
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Normal Distribution (Precision Parameterization) #
###########################################################################
dmvnp <- function(x, mu, Omega, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Omega)) Omega <- diag(ncol(x))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
k <- nrow(Omega)
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector((-k/2)*(log(2) + log(pi)) + 0.5*log(detOmega) -
0.5*z)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvnp <- function(n=1, mu=rep(0, k), Omega)
{
mu <- rbind(mu)
if(missing(Omega)) Omega <- diag(ncol(mu))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
k <- ncol(Omega)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% as.inverse(t(chol(Omega)))
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Normal Distribution (Precision-Cholesky Parameterization) #
###########################################################################
dmvnpc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
Omega <- t(U) %*% U
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector((-k/2)*(log(2) + log(pi)) + 0.5*log(detOmega) -
0.5*z)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvnpc <- function(n=1, mu=rep(0,k), U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
Sigma <- as.inverse(t(U) %*% U)
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% chol(Sigma)
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Polya Distribution #
###########################################################################
dmvpolya <- function(x, alpha, log=FALSE)
{
x <- as.vector(x)
alpha <- as.vector(alpha)
if(!identical(length(x), length(alpha)))
stop("x and alpha differ in length.")
dens <- (log(factorial(sum(x))) - sum(log(factorial(x)))) +
(log(factorial(sum(alpha)-1)) -
log(factorial(sum(x) + sum(alpha)-1))) +
(sum(log(factorial(x + alpha - 1)) - log(factorial(alpha - 1))))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvpolya <- function(n=1, alpha)
{
p <- rdirichlet(n, alpha)
x <- rcat(n,p)
return(x)
}
###########################################################################
# Multivariate Power Exponential Distribution #
###########################################################################
dmvpe <- function(x=c(0,0), mu=c(0,0), Sigma=diag(2), kappa=1, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Sigma)) Sigma <- diag(ncol(x))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
temp <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(((log(k)+lgamma(k/2)) - ((k/2)*log(pi) +
0.5*log(det(Sigma)) + lgamma(1 + k/(2*kappa)) +
(1 + k/(2*kappa))*log(2))) + kappa*(-0.5*temp))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvpe <- function(n, mu=c(0,0), Sigma=diag(2), kappa=1)
{
mu <- rbind(mu)
k <- ncol(mu)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(k != nrow(Sigma)) {
stop("mu and Sigma have non-conforming size.")}
ev <- eigen(Sigma, symmetric=TRUE)
if(!all(ev$values >= -sqrt(.Machine$double.eps) *
abs(ev$values[1]))) {
stop("Sigma must be positive-definite.")}
SigmaSqrt <- ev$vectors %*% diag(sqrt(ev$values),
length(ev$values)) %*% t(ev$vectors)
radius <- (rgamma(n, shape=k/(2*kappa), scale=1/2))^(1/(2*kappa))
runifsphere <- function(n, k)
{
p <- as.integer(k)
if(!is.integer(k))
stop("k must be an integer in [2,Inf)")
if(k < 2) stop("k must be an integer in [2,Inf).")
Mnormal <- matrix(rnorm(n*k,0,1), nrow=n)
rownorms <- sqrt(rowSums(Mnormal^2))
unifsphere <- sweep(Mnormal,1,rownorms, "/")
return(unifsphere)
}
un <- runifsphere(n=n, k=k)
x <- mu + radius * un %*% SigmaSqrt
return(x)
}
###########################################################################
# Multivariate Power Exponential Distribution (Cholesky Parameterization) #
###########################################################################
dmvpec <- function(x=c(0,0), mu=c(0,0), U, kappa=1, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
Sigma <- t(U) %*% U
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
temp <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(((log(k)+lgamma(k/2)) - ((k/2)*log(pi) +
0.5*log(det(Sigma)) + lgamma(1 + k/(2*kappa)) +
(1 + k/(2*kappa))*log(2))) + kappa*(-0.5*temp))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvpec <- function(n, mu=c(0,0), U, kappa=1)
{
mu <- rbind(mu)
k <- ncol(mu)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(k != nrow(U)) {
stop("mu and U have non-conforming size.")}
Sigma <- t(U) %*% U
ev <- eigen(Sigma, symmetric=TRUE)
if(!all(ev$values >= -sqrt(.Machine$double.eps) *
abs(ev$values[1]))) {
stop("Sigma must be positive-definite.")}
SigmaSqrt <- ev$vectors %*% diag(sqrt(ev$values),
length(ev$values)) %*% t(ev$vectors)
radius <- (rgamma(n, shape=k/(2*kappa), scale=1/2))^(1/(2*kappa))
runifsphere <- function(n, k)
{
p <- as.integer(k)
if(!is.integer(k))
stop("k must be an integer in [2,Inf)")
if(k < 2) stop("k must be an integer in [2,Inf).")
Mnormal <- matrix(rnorm(n*k,0,1), nrow=n)
rownorms <- sqrt(rowSums(Mnormal^2))
unifsphere <- sweep(Mnormal,1,rownorms, "/")
return(unifsphere)
}
un <- runifsphere(n=n, k=k)
x <- mu + radius * un %*% SigmaSqrt
return(x)
}
###########################################################################
# Multivariate t Distribution #
###########################################################################
dmvt <- function(x, mu, S, df=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(S)) S <- diag(ncol(x))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(df > 10000)) return(dmvn(x, mu, S, log))
k <- nrow(S)
ss <- x - mu
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((df+k)/2) - lgamma(df/2) + (k/2)*df +
(k/2)*log(pi) + 0.5*log(det(S)) + ((df+k)/2)*log(1 + (1/df) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvt <- function(n=1, mu=rep(0,k), S, df=Inf)
{
mu <- rbind(mu)
if(missing(S)) S <- diag(ncol(mu))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
if(any(df <= 0)) stop("The df parameter must be positive.")
k <- ncol(S)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(df==Inf) x <- 1 else x <- rchisq(n,df) / df
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), S)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate t Distribution (Cholesky Parameterization) #
###########################################################################
dmvtc <- function(x, mu, U, df=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(df > 10000)) return(dmvnc(x, mu, U, log))
k <- nrow(U)
ss <- x - mu
S <- t(U) %*% U
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((df+k)/2) - lgamma(df/2) + (k/2)*df +
(k/2)*log(pi) + 0.5*log(det(S)) + ((df+k)/2)*log(1 + (1/df) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvtc <- function(n=1, mu=rep(0,k), U, df=Inf)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
if(any(df <= 0)) stop("The df parameter must be positive.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(df==Inf) x <- 1 else x <- rchisq(n,df) / df
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate t Distribution (Precision Parameterization) #
###########################################################################
dmvtp <- function(x, mu, Omega, nu=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Omega)) Omega <- diag(ncol(x))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
if(any(nu > 10000)) return(dmvnp(x, mu, Omega, log))
k <- ncol(Omega)
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((nu+k)/2) - (lgamma(nu/2) + (k/2)*log(nu) +
(k/2)*log(pi)) + 0.5*log(detOmega) +
(-(nu+k)/2)*log(1 + (1/nu) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvtp <- function(n=1, mu, Omega, nu=Inf)
{
mu <- rbind(mu)
if(missing(Omega)) Omega <- diag(ncol(mu))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
Sigma <- as.inverse(Omega)
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(nu == Inf) x <- 1 else x <- rchisq(n,nu) / nu
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), Sigma)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate t Distribution (Precision-Cholesky Parameterization) #
###########################################################################
dmvtpc <- function(x, mu, U, nu=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
if(any(nu > 10000)) return(dmvnpc(x, mu, U, log))
k <- ncol(U)
Omega <- t(U) %*% U
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((nu+k)/2) - (lgamma(nu/2) + (k/2)*log(nu) +
(k/2)*log(pi)) + 0.5*log(detOmega) +
(-(nu+k)/2)*log(1 + (1/nu) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvtpc <- function(n=1, mu, U, nu=Inf)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(nu == Inf) x <- 1 else x <- rchisq(n,nu) / nu
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnpc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Normal Distribution Mixture #
###########################################################################
dnormm <- function(x, p, mu, sigma, log=FALSE)
{
if(missing(x)) stop("x is a required argument.")
x <- as.vector(x)
n <- length(x)
if(missing(p)) stop("p is a required argument.")
p <- as.vector(p)
if(any(p <= 0) | any(p > 1)) stop("p must be in (0,1].")
if(sum(p) != 1) stop("p must sum to 1 for all components.")
m <- length(p)
p <- matrix(p, n, m, byrow=TRUE)
if(missing(mu)) stop("mu is a required argument.")
mu <- as.vector(mu)
if(!identical(m, length(mu)))
stop("p and mu differ in length.")
mu <- matrix(mu, n, m, byrow=TRUE)
if(missing(sigma)) stop("sigma is a required argument.")
sigma <- as.vector(sigma)
if(!identical(m, length(sigma)))
stop("p and sigma differ in length.")
sigma <- matrix(sigma, n, m, byrow=TRUE)
dens <- matrix(dnorm(x, mu, sigma, log=TRUE), n, m)
dens <- dens + log(p)
if(log == TRUE) dens <- apply(dens, 1, logadd)
else dens <- rowSums(exp(dens))
return(dens)
}
pnormm <- function(q, p, mu, sigma, lower.tail=TRUE, log.p=FALSE)
{
n <- length(q)
m <- length(p)
q <- matrix(q, n, m)
p <- matrix(p, n, m, byrow=TRUE)
mu <- matrix(mu, n, m, byrow=TRUE)
sigma <- matrix(sigma, n, m, byrow=TRUE)
cdf <- matrix(pnorm(q, mu, sigma, lower.tail=lower.tail,
log.p=log.p), n, m)
if(log.p == FALSE) cdf <- rowSums(cdf * p)
else stop("The log.p argument does not work yet.")
return(cdf)
}
rnormm <- function(n, p, mu, sigma)
{
if(missing(p)) stop("p is a required argument.")
p <- as.vector(p)
if(any(p <= 0) | any(p > 1)) stop("p must be in (0,1].")
if(sum(p) != 1) stop("p must sum to 1 for all components.")
m <- length(p)
p <- matrix(p, n, m, byrow=TRUE)
if(missing(mu)) stop("mu is a required argument.")
mu <- as.vector(mu)
if(!identical(m, length(mu)))
stop("p and mu differ in length.")
if(missing(sigma)) stop("sigma is a required argument.")
sigma <- as.vector(sigma)
if(!identical(m, length(sigma)))
stop("p and sigma differ in length.")
if(any(sigma <= 0)) stop("sigma must be positive.")
z <- rcat(n, p)
x <- rnorm(n, mean=mu[z], sd=sigma[z])
return(x)
}
###########################################################################
# Normal Distribution (Precision Parameterization) #
###########################################################################
dnormp <- function(x, mean=0, prec=1, log=FALSE)
{
#dens <- sqrt(prec/(2*pi)) * exp(-(prec/2)*(x-mu)^2)
dens <- dnorm(x, mean, sqrt(1/prec), log)
return(dens)
}
pnormp <- function(q, mean=0, prec=1, lower.tail=TRUE, log.p=FALSE)
{return(pnorm(q, mean=mean, sd=sqrt(1/prec), lower.tail, log.p))}
qnormp <- function(p, mean=0, prec=1, lower.tail=TRUE, log.p=FALSE)
{return(qnorm(p, mean=mean, sd=sqrt(1/prec), lower.tail, log.p))}
rnormp <- function(n, mean=0, prec=1)
{return(rnorm(n, mean=mean, sd=sqrt(1/prec)))}
###########################################################################
# Normal Distribution (Variance Parameterization) #
###########################################################################
dnormv <- function(x, mean=0, var=1, log=FALSE)
{
#dens <- (1/(sqrt(2*pi*var))) * exp(-((x-mu)^2/(2*var)))
dens <- dnorm(x, mean, sqrt(var), log)
return(dens)
}
pnormv <- function(q, mean=0, var=1, lower.tail=TRUE, log.p=FALSE)
{return(pnorm(q, mean=mean, sd=sqrt(var), lower.tail, log.p))}
qnormv <- function(p, mean=0, var=1, lower.tail=TRUE, log.p=FALSE)
{return(qnorm(p, mean=mean, sd=sqrt(var), lower.tail, log.p))}
rnormv <- function(n, mean=0, var=1)
{return(rnorm(n, mean=mean, sd=sqrt(var)))}
###########################################################################
# Pareto Distribution #
###########################################################################
dpareto <- function(x, alpha, log=FALSE)
{
x <- as.vector(x); alpha <- as.vector(alpha)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
NN <- max(length(x), length(alpha))
x <- rep(x, len=NN); alpha <- rep(alpha, len=NN)
dens <- ifelse(x < 1, -Inf, log(alpha) - (alpha + 1)*log(x))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
ppareto <- function(q, alpha)
{
q <- as.vector(q); alpha <- as.vector(alpha)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
NN <- max(length(q), length(alpha))
q <- rep(q, len=NN); alpha <- rep(alpha, len=NN)
p <- ifelse(q < 1, 0, 1 - 1/q^alpha)
return(p)
}
qpareto <- function(p, alpha)
{
p <- as.vector(p); alpha <- as.vector(alpha)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
NN <- max(length(p), length(alpha))
p <- rep(p, len=NN); alpha <- rep(alpha, len=NN)
q <- (1-p)^(-1/alpha)
return(q)
}
rpareto <- function(n, alpha)
{
alpha <- rep(alpha, len=n)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
x <- runif(n)^(-1/alpha)
return(x)
}
###########################################################################
# Power Exponential Distribution #
# #
# These functions are similar to those in the normalp package. #
###########################################################################
dpe <- function(x, mu=0, sigma=1, kappa=2, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); sigma <- as.vector(sigma)
kappa <- as.vector(kappa)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(x), length(mu), length(sigma), length(kappa))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); kappa <- rep(kappa, len=NN)
cost <- 2 * kappa^(1/kappa) * gamma(1 + 1/kappa) * sigma
expon1 <- (abs(x - mu))^kappa
expon2 <- kappa * sigma^kappa
dens <- log(1/cost) + (-expon1 / expon2)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
ppe <- function(q, mu=0, sigma=1, kappa=2, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu); sigma <- as.vector(sigma)
kappa <- as.vector(kappa)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(q), length(mu), length(sigma), length(kappa))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); kappa <- rep(kappa, len=NN)
z <- (q - mu) / sigma
zz <- abs(z)^kappa
p <- pgamma(zz, shape=1/kappa, scale=kappa)
p <- p / 2
p <- ifelse(z < 0, 0.5 - p, 0.5 + p)
if(lower.tail == FALSE) p <- 1 - p
if(log.p == TRUE) p <- log(p)
return(p)
}
qpe <- function(p, mu=0, sigma=1, kappa=2, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu); sigma <- as.vector(sigma)
kappa <- as.vector(kappa)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(p), length(mu), length(sigma), length(kappa))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); kappa <- rep(kappa, len=NN)
if(log.p == TRUE) p <- log(p)
if(lower.tail == FALSE) p <- 1 - p
zp <- ifelse(p < 0.5, 0.5 - p, p - 0.5)
zp <- 2 * zp
qg <- qgamma(zp, shape=1/kappa, scale=kappa)
z <- qg^(1/kappa)
z <- ifelse(p < 0.5, -z, z)
q <- mu + z * sigma
return(q)
}
rpe <- function(n, mu=0, sigma=1, kappa=2)
{
mu <- rep(mu, len=n); sigma <- rep(sigma, len=n)
kappa <- rep(kappa, len=n)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
qg <- rgamma(n, shape=1/kappa, scale=kappa)
z <- qg^(1/kappa)
z <- ifelse(runif(n) < 0.5, -z, z)
x <- mu + z * sigma
return(x)
}
###########################################################################
# Skew Discrete Laplace Distribution #
###########################################################################
dsdlaplace <- function(x, p, q, log=FALSE)
{
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(q < 0) || any(q > 1)) stop("q must be in [0,1].")
NN <- max(length(x), length(p), length(q))
x <- rep(x, len=NN); p <- rep(p, len=NN); q <- rep(q, len=NN)
dens <- ifelse(x >= 0, log(1-p) + log(1-q) - (log(1-p*q) +
x*log(p)), log(1-p) + log(1-q) - (log(1-p*q) + abs(x)*log(q)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
psdlaplace <- function(x, p, q)
{
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(q < 0) || any(q > 1)) stop("q must be in [0,1].")
NN <- max(length(x), length(p), length(q))
x <- rep(x, len=NN); p <- rep(p, len=NN); q <- rep(q, len=NN)
pr <- ifelse(x >= 0, 1-(1-q)*p^(floor(x)+1)/(1-p*q),
(1-p)*q^(-floor(x))/(1-p*q))
return(pr)
}
qsdlaplace <- function(prob, p, q)
{
if(any(prob < 0) || any(prob > 1)) stop("prob must be in [0,1].")
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(q < 0) || any(q > 1)) stop("q must be in [0,1].")
NN <- max(length(prob), length(p), length(q))
prob <- rep(prob, len=NN); p <- rep(p, len=NN); q <- rep(q, len=NN)
x <- numeric(NN)
for (i in 1:NN) {
k <- 0
if(prob[i] >= psdlaplace(k, p[i], q[i])) {
while(prob[i] >= psdlaplace(k, p[i], q[i])) {
k <- k + 1}}
else if(prob[i] < psdlaplace(k, p[i], q[i])) {
while(prob[i] < psdlaplace(k, p[i], q[i])) {
k <- k - 1}
k <- k + 1}
x[i] <- k
}
return(x)
}
rsdlaplace <- function(n, p, q)
{
if(length(p) > 1) stop("p must have a length of 1.")
if(length(q) > 1) stop("q must have a length of 1.")
if((p < 0) || (p > 1)) stop("p must be in [0,1].")
if((q < 0) || (q > 1)) stop("q must be in [0,1].")
u <- runif(n)
return(qsdlaplace(u,p,q))
}
###########################################################################
# Skew-Laplace Distribution #
###########################################################################
dslaplace <- function(x, mu, alpha, beta, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu)
alpha <- as.vector(alpha); beta <- as.vector(beta)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
NN <- max(length(x), length(mu), length(alpha), length(beta))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
alpha <- rep(alpha, len=NN); beta <- rep(beta, len=NN)
ab <- alpha + beta
belowMu <- log(1/ab) + ((x - mu)/alpha)
aboveMu <- log(1/ab) + ((mu - x)/beta)
dens <- ifelse(x <= mu, belowMu, aboveMu)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pslaplace <- function(q, mu, alpha, beta)
{
q <- as.vector(q); mu <- as.vector(mu)
alpha <- as.vector(alpha); beta <- as.vector(beta)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
NN <- max(length(q), length(mu), length(alpha), length(beta))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
alpha <- rep(alpha, len=NN); beta <- rep(beta, len=NN)
ab <- alpha + beta
belowMu <- (alpha/ab) * exp((q - mu)/alpha)
aboveMu <- 1 - (beta/ab) * exp((mu - q)/beta)
p <- ifelse(q < mu, belowMu, aboveMu)
return(p)
}
qslaplace <- function(p, mu, alpha, beta)
{
p <- as.vector(p); mu <- as.vector(mu)
alpha <- as.vector(alpha); beta <- as.vector(beta)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
NN <- max(length(p), length(mu), length(alpha), length(beta))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
alpha <- rep(alpha, len=NN); beta <- rep(beta, len=NN)
ab <- alpha + beta
belowMu <- alpha*log(p*ab/alpha) + mu
aboveMu <- mu - beta*log(ab*(1 - p)/beta)
q <- ifelse(p < alpha/ab, belowMu, aboveMu)
return(q)
}
rslaplace <- function(n, mu, alpha, beta)
{
#mu <- rep(mu, len=n); alpha <- rep(alpha, len=n)
#beta <- rep(beta, len=n)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
ab <- alpha + beta
y <- rexp(n,1)
probs <- c(alpha,beta) / ab
signs <- sample(c(-1,1), n, replace=TRUE, prob=probs)
mult <- ifelse(signs < 0, signs*alpha, signs*beta)
x <- mult*y + mu
return(x)
}
###########################################################################
# Stick-Breaking Prior Distribution #
###########################################################################
dStick <- function(theta, gamma, log=FALSE)
{
dens <- sum(dbeta(theta, 1, gamma, log=log))
return(dens)
}
rStick <- function(M, gamma)
{
return(Stick(rbeta(M,1,gamma)))
}
###########################################################################
# Student t Distribution (3-parameter) #
# #
# The pst and qst functions are similar to the TF functions in the #
# gamlss.dist package, but dst and rst have been refined. #
###########################################################################
dst <- function(x, mu=0, sigma=1, nu=10, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu)
sigma <- as.vector(sigma); nu <- as.vector(nu)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
else if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(x), length(mu), length(sigma), length(nu))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); nu <- rep(nu, len=NN)
const <- lgamma((nu+1)/2) - lgamma(nu/2) - log(sqrt(pi*nu) * sigma)
dens <- const + log((1 + (1/nu)*((x-mu)/sigma)^2)^(-(nu+1)/2))
if(log == FALSE) dens <- exp(dens)
return(dens)
#return({1/sigma} * dt({x-mu}/sigma, df=nu, log)) #Deprecated
}
pst <- function(q, mu=0, sigma=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu)
sigma <- as.vector(sigma); nu <- as.vector(nu)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(q), length(mu), length(sigma), length(nu))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); nu <- rep(nu, len=NN)
if(length(nu) > 1) p <- ifelse(nu > 1000000,
pnorm(q, mu, sigma, lower.tail=lower.tail, log.p=log.p),
pt({q-mu}/sigma, df=nu, lower.tail=lower.tail, log.p=log.p))
else p <- if(nu > 1000000) {pnorm(q, mu, sigma,
lower.tail=lower.tail, log.p=log.p)}
else {pt({q-mu}/sigma, df=nu, lower.tail=lower.tail,
log.p=log.p)}
return(p)
}
qst <- function(p, mu=0, sigma=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu)
sigma <- as.vector(sigma); nu <- as.vector(nu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(p), length(mu), length(sigma), length(nu))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); nu <- rep(nu, len=NN)
if(length(nu) > 1) q <- ifelse(nu > 1000000,
qnorm(p, mu, sigma, lower.tail=lower.tail, log.p=log.p),
mu + sigma * qt(p, df=nu, lower.tail=lower.tail))
else q <- if(nu > 1000000) {qnorm(p, mu, sigma,
lower.tail=lower.tail, log.p=log.p)}
else {mu + sigma * qt(p, df=nu, lower.tail=lower.tail)}
return(q)
}
rst <- function(n, mu=0, sigma=1, nu=10)
{
mu <- rep(mu, len=n); sigma <- rep(sigma, len=n); nu <- rep(nu, len=n)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
n <- ceiling(n)
y <- rnorm(n)
z <- rchisq(n, nu)
x <- mu + sigma*y*sqrt(nu/z)
return(x)
}
###########################################################################
# Student t Distribution (Precision Parameterization) #
###########################################################################
dstp <- function(x, mu=0, tau=1, nu=10, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu)
tau <- as.vector(tau); nu <- as.vector(nu)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(x), length(mu), length(tau), length(nu))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
tau <- rep(tau, len=NN); nu <- rep(nu, len=NN)
dens <- ifelse(nu > 1000000,
dnorm(x, mu, sqrt(1/tau), log=TRUE),
(lgamma((nu+1)/2) - lgamma(nu/2)) + 0.5*log(tau/(nu*pi)) +
(-(nu+1)/2)*log(1 + (tau/nu)*(x-mu)^2))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pstp <- function(q, mu=0, tau=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu)
tau <- as.vector(tau); nu <- as.vector(nu)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(q), length(mu), length(tau), length(nu))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
tau <- rep(tau, len=NN); nu <- rep(nu, len=NN)
p <- pst(q, mu, sqrt(1/tau), nu, lower.tail, log.p)
return(p)
}
qstp <- function(p, mu=0, tau=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu)
tau <- as.vector(tau); nu <- as.vector(nu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(p), length(mu), length(tau), length(nu))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
tau <- rep(tau, len=NN); nu <- rep(nu, len=NN)
q <- qst(p, mu, sqrt(1/tau), nu, lower.tail, log.p)
return(q)
}
rstp <- function(n, mu=0, tau=1, nu=10)
{
mu <- rep(mu, len=n); tau <- rep(tau, len=n); nu <- rep(nu, len=n)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
x <- rst(n, mu, sqrt(1/tau), nu)
n <- ceiling(n)
p <- runif(n)
x <- qst(p, mu=mu, sqrt(1/tau), nu=nu)
return(x)
}
###########################################################################
# Truncated Distribution #
# #
# These functions are similar to those from Nadarajah, S. and Kotz, S. #
# (2006). ``R Programs for Computing Truncated Distributions''. Journal #
# of Statistical Software, 16, Code Snippet 2, 1-8. These functions have #
# been corrected to work with log-densities. #
###########################################################################
dtrunc <- function(x, spec, a=-Inf, b=Inf, log=FALSE, ...)
{
if(a >= b) stop("Lower bound a is not less than upper bound b.")
if(any(x < a) | any(x > b))
stop("At least one instance of (x < a) or (x > b) found.")
dens <- rep(0, length(x))
g <- get(paste("d", spec, sep=""), mode="function")
G <- get(paste("p", spec, sep=""), mode="function")
if(log == TRUE) {
dens <- g(x, log=TRUE, ...) - log(G(b, ...) - G(a, ...))
}
else {
dens <- g(x, ...) / (G(b, ...) - G(a, ...))}
return(dens)
}
extrunc <- function(spec, a=-Inf, b=Inf, ...)
{
f <- function(x) x * dtrunc(x, spec, a=a, b=b, log=FALSE, ...)
return(integrate(f, lower=a, upper=b)$value)
}
ptrunc <- function(x, spec, a=-Inf, b=Inf, ...)
{
if(a >= b) stop("Lower bound a is not less than upper bound b.")
if(any(x < a) | any(x > b))
stop("At least one instance of (x < a) or (x > b) found.")
p <- x
aa <- rep(a, length(x))
bb <- rep(b, length(x))
G <- get(paste("p", spec, sep=""), mode="function")
p <- G(apply(cbind(apply(cbind(x, bb), 1, min), aa), 1, max), ...)
p <- p - G(aa, ...)
p <- p / {G(bb, ...) - G(aa, ...)}
return(p)
}
qtrunc <- function(p, spec, a=-Inf, b=Inf, ...)
{
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(a >= b) stop("Lower bound a is not less than upper bound b.")
q <- p
G <- get(paste("p", spec, sep=""), mode="function")
Gin <- get(paste("q", spec, sep=""), mode="function")
q <- Gin(G(a, ...) + p*{G(b, ...) - G(a, ...)}, ...)
return(q)
}
rtrunc <- function(n, spec, a=-Inf, b=Inf, ...)
{
if(a >= b) stop("Lower bound a is not less than upper bound b.")
x <- u <- runif(n)
x <- qtrunc(u, spec, a=a, b=b,...)
return(x)
}
vartrunc <- function(spec, a=-Inf, b=Inf, ...)
{
ex <- extrunc(spec, a=a, b=b, ...)
f <- function(x) {
{x - ex}^2 * dtrunc(x, spec, a=a, b=b, log=FALSE, ...)}
sigma2 <- integrate(f, lower=a, upper=b)$value
return(sigma2)
}
###########################################################################
# Wishart Distribution #
###########################################################################
dwishart <- function(Omega, nu, S, log=FALSE)
{
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(Omega), dim(S)))
stop("The dimensions of Omega and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Omega)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -((nu*k)/2) * log(2) - ((k*(k-1))/4) * log(pi) - gamsum -
(nu/2) * log(det(S)) + ((nu-k-1)/2) * log(det(Omega)) -
(tr(as.inverse(S) %*% Omega)/2)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rwishart <- function(nu, S)
{
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(nu < nrow(S)) {
stop("The nu parameter is less than the dimension of S.")}
k <- nrow(S)
Z <- matrix(0, k, k)
x <- rchisq(k, nu:{nu - k + 1})
x <- ifelse(x == 0, 1e-100, x)
diag(Z) <- sqrt(x)
if(k > 1) {
kseq <- 1:(k-1)
Z[rep(k*kseq, kseq) +
unlist(lapply(kseq, seq))] <- rnorm(k*{k-1}/2)}
return(crossprod(Z %*% chol(S)))
}
###########################################################################
# Wishart Distribution (Cholesky Parameterization) #
###########################################################################
dwishartc <- function(U, nu, S, log=FALSE)
{
if(missing(U)) stop("Upper triangular U is required.")
Omega <- t(U) %*% U
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(Omega), dim(S)))
stop("The dimensions of Omega and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Omega)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -((nu*k)/2) * log(2) - ((k*(k-1))/4) * log(pi) - gamsum -
(nu/2) * log(det(S)) + ((nu-k-1)/2) * log(det(Omega)) -
(tr(as.inverse(S) %*% Omega)/2)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rwishartc <- function(nu, S)
{
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(nu < nrow(S)) {
stop("The nu parameter is less than the dimension of S.")}
k <- nrow(S)
Z <- matrix(0, k, k)
x <- rchisq(k, nu:{nu - k + 1})
x <- ifelse(x == 0, 1e-100, x)
diag(Z) <- sqrt(x)
if(k > 1) {
kseq <- 1:(k-1)
Z[rep(k*kseq, kseq) +
unlist(lapply(kseq, seq))] <- rnorm(k*{k-1}/2)}
return(chol(crossprod(Z %*% chol(S))))
}
###########################################################################
# Zellner's g-Prior #
###########################################################################
dhyperg <- function(g, alpha=3, log=FALSE)
{
if(g <= 0) stop("The g parameter must be positive.")
if(alpha <= 0) stop("The alpha parameter must be positive.")
dens <- log((alpha - 2)/2) -(alpha/2)*log(1 + g)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
dzellner <- function(beta, g, sigma, X, log=FALSE)
{
if(g <= 0) stop("The g parameter must be positive.")
if(sigma <= 0) stop("The sigma parameter must be positive.")
dens <- dmvn(beta, rep(0, length(beta)),
g*sigma*sigma*as.inverse(t(X) %*% X), log=log)
return(dens)
}
#End
benmarwick/LaplacesDemon documentation built on May 12, 2019, 12:59 p.m.
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###########################################################################
# Asymmetric Laplace Distribution #
# #
# These functions are similar to those in the VGAM package. #
###########################################################################
dalaplace <- function(x, location=0, scale=1, kappa=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(x), length(location), length(scale), length(kappa))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
logconst <- 0.5 * log(2) - log(scale) + log(kappa) - log1p(kappa^2)
exponent <- -(sqrt(2) / scale) * abs(x - location) *
ifelse(x >= location, kappa, 1/kappa)
dens <- logconst + exponent
if(log == FALSE) dens <- exp(logconst + exponent)
return(dens)
}
palaplace <- function(q, location=0, scale=1, kappa=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if((kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(q), length(location), length(scale), length(kappa))
q <- rep(q, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
exponent <- -(sqrt(2) / scale) * abs(q - location) *
ifelse(q >= location, kappa, 1/kappa)
temp <- exp(exponent) / (1 + kappa^2)
p <- 1 - temp
index1 <- (q < location)
p[index1] <- (kappa[index1])^2 * temp[index1]
return(p)
}
qalaplace <- function(p, location=0, scale=1, kappa=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(p), length(location), length(scale), length(kappa))
p <- rep(p, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
q <- p
temp <- kappa^2 / (1 + kappa^2)
index1 <- (p <= temp)
exponent <- p[index1] / temp[index1]
q[index1] <- location[index1] + (scale[index1] * kappa[index1]) *
log(exponent) / sqrt(2)
q[!index1] <- location[!index1] - (scale[!index1] / kappa[!index1]) *
(log1p((kappa[!index1])^2) + log1p(-p[!index1])) / sqrt(2)
q[p == 0] = -Inf
q[p == 1] = Inf
return(q)
}
ralaplace <- function(n, location=0, scale=1, kappa=1)
{
location <- rep(location, len=n)
scale <- rep(scale, len=n)
kappa <- rep(kappa, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
x <- location + scale *
log(runif(n)^kappa / runif(n)^(1/kappa)) / sqrt(2)
return(x)
}
###########################################################################
# Asymmetric Log-Laplace Distribution #
# #
# These functions are similar to those in the VGAM package. #
###########################################################################
dallaplace <- function(x, location=0, scale=1, kappa=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(x), length(location), length(scale), length(kappa))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
Alpha <- sqrt(2) * kappa / scale
Beta <- sqrt(2) / (scale * kappa)
Delta <- exp(location)
exponent <- ifelse(x >= Delta, -(Alpha+1),
(Beta-1)) * (log(x) - location)
dens <- -location + log(Alpha) + log(Beta) -
log(Alpha + Beta) + exponent
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pallaplace <- function(q, location=0, scale=1, kappa=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(q), length(location), length(scale), length(kappa))
q <- rep(q, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
Alpha <- sqrt(2) * kappa / scale
Beta <- sqrt(2) / (scale * kappa)
Delta <- exp(location)
temp <- Alpha + Beta
p <- (Alpha / temp) * (q / Delta)^(Beta)
p[q <= 0] <- 0
index1 <- (q >= Delta)
p[index1] <- (1 - (Beta/temp) * (Delta/q)^(Alpha))[index1]
return(p)
}
qallaplace <- function(p, location=0, scale=1, kappa=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale); kappa <- as.vector(kappa)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(p), length(location), length(scale), length(kappa))
p <- rep(p, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN); kappa <- rep(kappa, len=NN)
Alpha <- sqrt(2) * kappa / scale
Beta <- sqrt(2) / (scale * kappa)
Delta <- exp(location)
temp <- Alpha + Beta
q <- Delta * (p * temp / Alpha)^(1/Beta)
index1 <- (p > Alpha / temp)
q[index1] <- (Delta * ((1-p) * temp / Beta)^(-1/Alpha))[index1]
q[p == 0] <- 0
q[p == 1] <- Inf
return(q)
}
rallaplace <- function(n, location=0, scale=1, kappa=1)
{
location <- rep(location, len=n); scale <- rep(scale, len=n)
kappa <- rep(kappa, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
x <- exp(location) *
(runif(n)^kappa / runif(n)^(1/kappa))^(scale / sqrt(2))
return(x)
}
###########################################################################
# Bernoulli Distribution #
# #
# These functions are similar to those in the Rlab package. #
###########################################################################
dbern <- function(x, prob, log=FALSE)
{return(dbinom(x, 1, prob, log))}
pbern <- function(q, prob, lower.tail=TRUE, log.p=FALSE)
{return(pbinom(q, 1, prob, lower.tail, log.p))}
qbern <- function(p, prob, lower.tail=TRUE, log.p=FALSE)
{return(qbinom(p, 1, prob, lower.tail, log.p))}
rbern <- function(n, prob)
{return(rbinom(n, 1, prob))}
###########################################################################
# Categorical Distribution #
###########################################################################
dcat <- function(x, p, log=FALSE)
{
if(is.vector(x) & !is.matrix(p))
p <- matrix(p, length(x), length(p), byrow=TRUE)
if(is.matrix(x) & !is.matrix(p))
p <- matrix(p, nrow(x), length(p), byrow=TRUE)
if(is.vector(x) & {length(x) == 1}) {
temp <- rep(0, ncol(p))
temp[x] <- 1
x <- t(temp)}
else if(is.vector(x) & (length(x) > 1))
x <- as.indicator.matrix(x)
if(!identical(nrow(x), nrow(p)))
stop("The number of rows of x and p differ.")
if(!identical(ncol(x), ncol(p))) {
x.temp <- matrix(0, nrow(p), ncol(p))
x.temp[,as.numeric(colnames(x))] <- x
x <- x.temp}
dens <- x*p
if(log == TRUE) dens <- x*log(p)
dens <- as.vector(rowSums(dens))
return(dens)
}
qcat <- function(pr, p, lower.tail=TRUE, log.pr=FALSE)
{
if(!is.vector(pr)) pr <- as.vector(pr)
if(!is.vector(p)) p <- as.vector(p)
if(log.pr == FALSE) {
if(any(pr < 0) | any(pr > 1))
stop("pr must be in [0,1].")}
else if(any(!is.finite(pr)) | any(pr > 0))
stop("pr, as a log, must be in (-Inf,0].")
if(sum(p) != 1) stop("sum(p) must be 1.")
if(lower.tail == FALSE) pr <- 1 - pr
breaks <- c(0, cumsum(p))
if(log.pr == TRUE) breaks <- log(breaks)
breaks <- matrix(breaks, length(pr), length(breaks), byrow=TRUE)
x <- rowSums(pr > breaks)
return(x)
}
rcat <- function(n, p)
{
if(is.vector(p)) {
x <- as.vector(which(rmultinom(n, size=1, prob=p) == 1,
arr.ind=TRUE)[, "row"])}
else {
n <- nrow(p)
x <- apply(p, 1, function(x) {
as.vector(which(rmultinom(1, size=1, prob=x) == 1,
arr.ind=TRUE)[, "row"])})
}
return(x)
}
###########################################################################
# Dirichlet Distribution #
# #
# These functions are similar to those in the MCMCpack package. #
###########################################################################
ddirichlet <- function(x, alpha, log=FALSE)
{
if(missing(x)) stop("x is a required argument.")
if(missing(alpha)) stop("alpha is a required argument.")
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(alpha))
alpha <- matrix(alpha, nrow(x), length(alpha), byrow=TRUE)
if(any(rowSums(x) != 1)) x / rowSums(x)
if(any(x < 0)) stop("x must be non-negative.")
if(any(alpha <= 0)) stop("alpha must be positive.")
dens <- as.vector(lgamma(rowSums(alpha)) - rowSums(lgamma(alpha)) +
rowSums((alpha-1)*log(x)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rdirichlet <- function (n, alpha)
{
alpha <- rbind(alpha)
alpha.dim <- dim(alpha)
if(n > alpha.dim[1])
alpha <- matrix(alpha, n, alpha.dim[2], byrow=TRUE)
x <- matrix(rgamma(alpha.dim[2]*n, alpha), ncol=alpha.dim[2])
sm <- x %*% rep(1, alpha.dim[2])
return(x/as.vector(sm))
}
###########################################################################
# Generalized Poisson #
###########################################################################
dgpois <- function(x, lambda=0, omega=0, log=FALSE)
{
x <- as.vector(x); lambda <- as.vector(lambda)
omega <- as.vector(omega)
lambda.star <- (1 - omega)*lambda + omega*x
dens <- log(1 - omega) + log(lambda) + (x - 1)*log(lambda.star) -
lgamma(x + 1) - lambda.star
if(log == FALSE) dens <- exp(dens)
return(dens)
}
###########################################################################
# Half-Cauchy Distribution #
###########################################################################
dhalfcauchy <- function(x, scale=25, log=FALSE)
{
x <- as.vector(x); scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(scale))
x <- rep(x, len=NN); scale <- rep(scale, len=NN)
dens <- log(2*scale) - log(pi*{x*x + scale*scale})
if(log == FALSE) dens <- exp(dens)
return(dens)
}
phalfcauchy <- function(q, scale=25)
{
q <- as.vector(q); scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(scale))
q <- rep(q, len=NN); scale <- rep(scale, len=NN)
z <- {2/pi}*atan(q/scale)
return(z)
}
qhalfcauchy <- function(p, scale=25)
{
p <- as.vector(p); scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(scale))
p <- rep(p, len=NN); scale <- rep(scale, len=NN)
q <- scale*tan({pi*p}/2)
return(q)
}
rhalfcauchy <- function(n, scale=25)
{
scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
p <- runif(n, 0, 1)
x <- scale*tan({pi*p}/2)
return(x)
}
###########################################################################
# Half-Normal Distribution #
# #
# This half-normal distribution has mean=0 and is similar to the halfnorm #
# functions in package fdrtool. #
###########################################################################
dhalfnorm <- function(x, scale=sqrt(pi/2), log=FALSE)
{
dens <- log(2) + dnorm(x, mean=0, sd=sqrt(pi/2) / scale, log=TRUE)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
phalfnorm <- function(q, scale=sqrt(pi/2), lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(scale))
q <- rep(q, len=NN); scale <- rep(scale, len=NN)
p <- 2*pnorm(q, mean=0, sd=sqrt(pi/2) / scale) - 1
if(lower.tail == FALSE) p <- 1-p
if(log.p == TRUE) p <- log.p(p)
return(p)
}
qhalfnorm <- function(p, scale=sqrt(pi/2), lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(scale))
p <- rep(p, len=NN); scale <- rep(scale, len=NN)
if(log.p == TRUE) p <- exp(p)
if(lower.tail == FALSE) p <- 1-p
q <- qnorm((p+1)/2, mean=0, sd=sqrt(pi/2) / scale)
return(q)
}
rhalfnorm <- function(n, scale=sqrt(pi/2))
{
scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
x <- abs(rnorm(n, mean=0, sd=sqrt(pi/2) / scale))
return(x)
}
###########################################################################
# Half-t Distribution #
###########################################################################
dhalft <- function(x, scale=25, nu=1, log=FALSE)
{
x <- as.vector(x); scale <- as.vector(scale); nu <- as.vector(nu)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(scale), length(nu))
x <- rep(x, len=NN); scale <- rep(scale, len=NN)
nu <- rep(nu, len=NN)
dens <- (-(nu+1)/2)*log(1 + (1/nu)*(x/scale)*(x/scale))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
phalft <- function(q, scale=25, nu=1)
{
q <- as.vector(q); scale <- as.vector(scale); nu <- as.vector(nu)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(scale), length(nu))
q <- rep(q, len=NN); scale <- rep(scale, len=NN)
p <- ptrunc(q, "st", a=0, b=Inf, mu=0, sigma=scale, nu=nu)
return(p)
}
qhalft <- function(p, scale=25, nu=1)
{
p <- as.vector(p); scale <- as.vector(scale); nu <- as.vector(nu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(scale), length(nu))
p <- rep(p, len=NN); scale <- rep(scale, len=NN)
q <- rtrunc(p, "st", a=0, b=Inf, mu=0, sigma=scale, nu=nu)
return(q)
}
rhalft <- function(n, scale=25, nu=1)
{
scale <- rep(scale, len=n); nu <- rep(nu, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
x <- rtrunc(n, "st", a=0, b=Inf, mu=0, sigma=scale, nu=nu)
return(x)
}
###########################################################################
# Horseshoe Distribution #
###########################################################################
dhs <- function(x, lambda, tau, sigma, log=FALSE)
{
NN <- max(length(x), length(lambda))
x <- rep(x, len=NN); lambda <- rep(lambda, len=NN)
p.theta.lambda <- dnorm(x, 0, lambda, log=TRUE)
p.lambda.tau <- dhalfcauchy(lambda, tau, log=TRUE)
p.tau <- dhalfcauchy(tau, sigma, log=TRUE)
dens <- p.theta.lambda - {p.lambda.tau + p.tau} / NN
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rhs <- function(n, lambda, tau, sigma)
{
if(missing(tau)) tau <- rhalfcauchy(1, sigma)
if(missing(lambda)) lambda <- rhalfcauchy(n, tau)
theta <- rnorm(n, 0, lambda)
return(theta)
}
###########################################################################
# Inverse Beta Distribution #
###########################################################################
dinvbeta <- function(x, a, b, log=FALSE)
{
const <- lgamma(a + b) - lgamma(a) - lgamma(b)
dens <- const + (a - 1) * log(x) - (a + b) * log(1 + x)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvbeta <- function(n, a, b)
{
x <- rbeta(n, a, b)
x <- x / (1-x)
return(x)
}
###########################################################################
# Inverse Chi-Squared Distribution #
# #
# These functions are similar to those in the GeoR package. #
###########################################################################
dinvchisq <- function(x, df, scale=1/df, log=FALSE)
{
x <- as.vector(x); df <- as.vector(df); scale <- as.vector(scale)
if(any(x <= 0)) stop("x must be positive.")
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(df), length(scale))
x <- rep(x, len=NN); df <- rep(df, len=NN);
scale <- rep(scale, len=NN)
nu <- df / 2
dens <- nu*log(nu) - log(gamma(nu)) + nu*log(scale) -
(nu+1)*log(x) - (nu*scale/x)
if(log == FALSE) dens <- exp(dens)
}
rinvchisq <- function(n, df, scale=1/df)
{
df <- rep(df, len=n); scale <- rep(scale, len=n)
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
z <- rchisq(n, df=df)
z <- ifelse(z == 0, 1e-100, z)
x <- (df*scale) / z
return(x)
}
###########################################################################
# Inverse Gamma Distribution #
# #
# These functions are similar to those in the MCMCpack package. #
###########################################################################
dinvgamma <- function(x, shape=1, scale=1, log=FALSE)
{
x <- as.vector(x); shape <- as.vector(shape)
scale <- as.vector(scale)
if(any(shape <= 0) | any(scale <=0))
stop("The shape and scale parameters must be positive.")
NN <- max(length(x), length(shape), length(scale))
x <- rep(x, len=NN); shape <- rep(shape, len=NN)
scale <- rep(scale, len=NN)
alpha <- shape; beta <- scale
dens <- alpha * log(beta) - lgamma(alpha) -
{alpha + 1} * log(x) - {beta/x}
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvgamma <- function(n, shape=1, scale=1)
{return(1 / rgamma(n=n, shape=shape, rate=scale))}
###########################################################################
# Inverse Gaussian Distribution #
###########################################################################
dinvgaussian <- function(x, mu, lambda, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); lambda <- as.vector(lambda)
if(any(x <= 0)) stop("x must be positive.")
if(any(mu <= 0)) stop("The mu parameter must be positive.")
if(any(lambda <= 0)) stop("The lambda parameter must be positive.")
NN <- max(length(x), length(mu), length(lambda))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
lambda <- rep(lambda, len=NN)
dens <- log(lambda / (2*pi*x^3)^0.5) -
((lambda*(x - mu)^2) / (2*mu^2*x))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvgaussian <- function(n, mu, lambda)
{
mu <- rep(mu, len=n); lambda <- rep(lambda, len=n)
if(any(mu <= 0)) stop("The mu parameter must be positive.")
if(any(lambda <= 0)) stop("The lambda parameter must be positive.")
nu <- rnorm(n)
y <- nu^2
x <- mu + ((mu^2*y)/(2*lambda)) - (mu/(2*lambda)) *
sqrt(4*mu*lambda*y + mu^2*y^2)
z <- runif(n)
x <- ifelse(z > (mu / (mu+x)), mu^2/x, x)
return(x)
}
###########################################################################
# Inverse Wishart Distribution #
###########################################################################
dinvwishart <- function(Sigma, nu, S, log=FALSE)
{
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(S), dim(Sigma)))
stop("The dimensions of Sigma and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Sigma)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -(nu*k/2)*log(2) - ((k*(k - 1))/4)*log(pi) - gamsum +
(nu/2)*log(det(S)) - ((nu + k + 1)/2)*log(det(Sigma)) -
0.5*tr(S %*% as.inverse(Sigma))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvwishart <- function(nu, S)
{return(as.inverse(rwishart(nu, as.inverse(S))))}
###########################################################################
# Inverse Wishart Distribution (Cholesky Parameterization) #
###########################################################################
dinvwishartc <- function(U, nu, S, log=FALSE)
{
if(missing(U)) stop("Upper triangular U is required.")
Sigma <- t(U) %*% U
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(S), dim(Sigma)))
stop("The dimensions of Sigma and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Sigma)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -(nu*k/2)*log(2) - ((k*(k - 1))/4)*log(pi) - gamsum +
(nu/2)*log(det(S)) - ((nu + k + 1)/2)*log(det(Sigma)) -
0.5*tr(S %*% as.inverse(Sigma))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rinvwishartc <- function(nu, S)
{return(chol(as.inverse(rwishart(nu, as.inverse(S)))))}
###########################################################################
# Laplace Distribution #
# #
# These functions are similar to those in the VGAM package. #
###########################################################################
dlaplace <- function(x, location=0, scale=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(location), length(scale))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
dens <- (-abs(x - location) / scale) - log(2 * scale)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
plaplace <- function(q, location=0, scale=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
z <- {q - location} / scale
NN <- max(length(q), length(location), length(scale))
q <- rep(q, len=NN); location <- rep(location, len=NN)
p <- ifelse(q < location, 0.5 * exp(z), 1 - 0.5 * exp(-z))
return(p)
}
qlaplace <- function(p, location=0, scale=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(location), length(scale))
p <- rep(p, len=NN); location <- rep(location, len=NN)
q <- location - sign(p - 0.5) * scale * log(2 * ifelse(p < 0.5,
p, 1 - p))
return(q)
}
rlaplace <- function(n, location=0, scale=1)
{
location <- rep(location, len=n); scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
r <- runif(n)
x <- location - sign(r - 0.5) * scale * log(2 * ifelse(r < 0.5,
r, 1 - r))
return(x)
}
###########################################################################
# Laplace Distribution (Precision Parameterization) #
###########################################################################
dlaplacep <- function(x, mu=0, tau=1, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(x), length(mu), length(tau))
x <- rep(x, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
dens <- log(tau/2) + (-tau*abs(x-mu))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
plaplacep <- function(q, mu=0, tau=1)
{
q <- as.vector(q); mu <- as.vector(mu)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(q), length(mu), length(tau))
q <- rep(q, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
p <- plaplace(q, mu, scale=1/tau)
return(p)
}
qlaplacep <- function(p, mu=0, tau=1)
{
p <- as.vector(p); mu <- as.vector(mu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(p), length(mu), length(tau))
p <- rep(p, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
q <- qlaplace(p, mu, scale=1/tau)
return(q)
}
rlaplacep <- function(n, mu=0, tau=1)
{
mu <- rep(mu, len=n); tau <- rep(tau, len=n)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
x <- rlaplace(n, mu, scale=1/tau)
return(x)
}
###########################################################################
# Log-Laplace Distribution #
###########################################################################
dllaplace <- function(x, location=0, scale=1, log=FALSE)
{
x <- as.vector(x); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(x), length(location), length(scale))
x <- rep(x, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
Alpha <- sqrt(2) * scale
Beta <- sqrt(2) / scale
Delta <- exp(location)
exponent <- ifelse(x >= Delta, -(Alpha+1),
(Beta-1)) * (log(x) - location)
dens <- -location + log(Alpha) + log(Beta) -
log(Alpha + Beta) + exponent
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pllaplace <- function(q, location=0, scale=1)
{
q <- as.vector(q); location <- as.vector(location)
scale <- as.vector(scale)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(q), length(location), length(scale))
q <- rep(q, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
Alpha <- sqrt(2) * scale
Beta <- sqrt(2) / scale
Delta <- exp(location)
temp <- Alpha + Beta
p <- (Alpha / temp) * (q / Delta)^(Beta)
p[q <= 0] <- 0
index1 <- (q >= Delta)
p[index1] <- (1 - (Beta/temp) * (Delta/q)^(Alpha))[index1]
return(p)
}
qllaplace <- function(p, location=0, scale=1)
{
p <- as.vector(p); location <- as.vector(location)
scale <- as.vector(scale)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(scale <= 0)) stop("The scale parameter must be positive.")
NN <- max(length(p), length(location), length(scale))
p <- rep(p, len=NN); location <- rep(location, len=NN)
scale <- rep(scale, len=NN)
Alpha <- sqrt(2) * scale
Beta <- sqrt(2) / scale
Delta <- exp(location)
temp <- Alpha + Beta
q <- Delta * (p * temp / Alpha)^(1/Beta)
index1 <- (p > Alpha / temp)
q[index1] <- (Delta * ((1-p) * temp / Beta)^(-1/Alpha))[index1]
q[p == 0] <- 0
q[p == 1] <- Inf
return(q)
}
rllaplace <- function(n, location=0, scale=1)
{
location <- rep(location, len=n); scale <- rep(scale, len=n)
if(any(scale <= 0)) stop("The scale parameter must be positive.")
x <- exp(location) * (runif(n) / runif(n))^(scale / sqrt(2))
return(x)
}
###########################################################################
# Log-Normal Distribution (Precision Parameterization) #
###########################################################################
dlnormp <- function(x, mu, tau, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(x), length(mu), length(tau))
x <- rep(x, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
dens <- log(sqrt(tau/(2*pi))) + log(1/x) + (-(tau/2)*(log(x-mu)^2))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
plnormp <- function(q, mu, tau, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(q), length(mu), length(tau))
q <- rep(q, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
p <- pnorm(q, mu, sqrt(1/tau), lower.tail, log.p)
return(p)
}
qlnormp <- function(p, mu, tau, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu); tau <- as.vector(tau)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(tau <= 0)) stop("The tau parameter must be positive.")
NN <- max(length(p), length(mu), length(tau))
p <- rep(p, len=NN); mu <- rep(mu, len=NN); tau <- rep(tau, len=NN)
q <- qnorm(p, mu, sqrt(1/tau), lower.tail, log.p)
return(q)
}
rlnormp <- function(n, mu, tau)
{
mu <- rep(mu, len=n); tau <- rep(tau, len=n)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
x <- rnorm(n, mu, sqrt(1/tau))
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution #
###########################################################################
dmvc <- function(x, mu, S, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(S)) S <- diag(ncol(x))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
k <- nrow(S)
ss <- x - mu
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma(k/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi) + 0.5*log(det(S)) + ((1+k)/2)*log(1+z)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvc <- function(n=1, mu=rep(0,k), S)
{
mu <- rbind(mu)
if(missing(S)) S <- diag(ncol(mu))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
k <- ncol(S)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), S)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution (Cholesky Parameterization) #
###########################################################################
dmvcc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- nrow(U)
S <- t(U) %*% U
ss <- x - mu
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma(k/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi) + 0.5*log(det(S)) + ((1+k)/2)*log(1+z)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvcc <- function(n=1, mu=rep(0,k), U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
S <- t(U) %*% U
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution (Precision Parameterization) #
###########################################################################
dmvcp <- function(x, mu, Omega, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Omega)) Omega <- diag(ncol(x))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
k <- nrow(Omega)
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((1+k)/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi)) + 0.5*log(detOmega) + (-(1+k)/2)*log(1 + z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvcp <- function(n=1, mu, Omega)
{
mu <- rbind(mu)
if(missing(Omega)) Omega <- diag(ncol(mu))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
Sigma <- as.inverse(Omega)
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), Sigma)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Cauchy Distribution (Precision-Cholesky Parameterization) #
###########################################################################
dmvcpc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- nrow(U)
Omega <- t(U) %*% U
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((1+k)/2) - (lgamma(0.5) + log(1^(k/2)) +
(k/2)*log(pi)) + 0.5*log(detOmega) + (-(1+k)/2)*log(1 + z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvcpc <- function(n=1, mu, U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
x <- rchisq(n,1)
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate Laplace Distribution #
###########################################################################
dmvl <- function(x, mu, Sigma, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Sigma)) Sigma <- diag(ncol(x))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
Sigma <- as.symmetric.matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- log(2 / ((2*pi)^(k/2) * sqrt(det(Sigma)))) +
log((sqrt(pi / (2*sqrt(2*z))) * exp(-sqrt(2*z))) /
sqrt(z/2)^(k/2 - 1))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvl <- function(n, mu, Sigma)
{
mu <- rbind(mu)
if(missing(Sigma)) Sigma <- diag(ncol(mu))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
e <- matrix(rexp(n, 1), n, k)
z <- rmvn(n, rep(0, k), Sigma)
x <- mu + sqrt(e)*z
return(x)
}
###########################################################################
# Multivariate Laplace Distribution (Cholesky Parameterization) #
###########################################################################
dmvlc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
Sigma <- t(U) %*% U
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- log(2 / ((2*pi)^(k/2) * sqrt(det(Sigma)))) +
log((sqrt(pi / (2*sqrt(2*z))) * exp(-sqrt(2*z))) /
sqrt(z/2)^(k/2 - 1))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvlc <- function(n, mu, U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
e <- matrix(rexp(n, 1), n, k)
z <- rmvnc(n, rep(0, k), U)
x <- mu + sqrt(e)*z
return(x)
}
###########################################################################
# Multivariate Normal Distribution #
###########################################################################
dmvn <- function(x, mu, Sigma, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Sigma)) Sigma <- diag(ncol(x))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
Sigma <- as.symmetric.matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
d <- eigen(Sigma, symmetric=TRUE)$values
dens <- as.vector(-0.5 * (k * log(2 * pi) + sum(log(d))) - (0.5 * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvn <- function(n=1, mu=rep(0,k), Sigma)
{
mu <- rbind(mu)
if(missing(Sigma)) Sigma <- diag(ncol(mu))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% chol(Sigma)
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Normal Distribution (Cholesky Parameterization) #
###########################################################################
dmvnc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
Sigma <- t(U) %*% U
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
d <- eigen(Sigma, symmetric=TRUE)$values
dens <- as.vector(-0.5 * (k * log(2 * pi) + sum(log(d))) - (0.5 * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvnc <- function(n=1, mu=rep(0,k), U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% U
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Normal Distribution (Precision Parameterization) #
###########################################################################
dmvnp <- function(x, mu, Omega, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Omega)) Omega <- diag(ncol(x))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
k <- nrow(Omega)
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector((-k/2)*(log(2) + log(pi)) + 0.5*log(detOmega) -
0.5*z)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvnp <- function(n=1, mu=rep(0, k), Omega)
{
mu <- rbind(mu)
if(missing(Omega)) Omega <- diag(ncol(mu))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
k <- ncol(Omega)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% as.inverse(t(chol(Omega)))
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Normal Distribution (Precision-Cholesky Parameterization) #
###########################################################################
dmvnpc <- function(x, mu, U, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
k <- ncol(U)
Omega <- t(U) %*% U
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector((-k/2)*(log(2) + log(pi)) + 0.5*log(detOmega) -
0.5*z)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvnpc <- function(n=1, mu=rep(0,k), U)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
Sigma <- as.inverse(t(U) %*% U)
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
z <- matrix(rnorm(n*k),n,k) %*% chol(Sigma)
x <- mu + z
return(x)
}
###########################################################################
# Multivariate Polya Distribution #
###########################################################################
dmvpolya <- function(x, alpha, log=FALSE)
{
x <- as.vector(x)
alpha <- as.vector(alpha)
if(!identical(length(x), length(alpha)))
stop("x and alpha differ in length.")
dens <- (log(factorial(sum(x))) - sum(log(factorial(x)))) +
(log(factorial(sum(alpha)-1)) -
log(factorial(sum(x) + sum(alpha)-1))) +
(sum(log(factorial(x + alpha - 1)) - log(factorial(alpha - 1))))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvpolya <- function(n=1, alpha)
{
p <- rdirichlet(n, alpha)
x <- rcat(n,p)
return(x)
}
###########################################################################
# Multivariate Power Exponential Distribution #
###########################################################################
dmvpe <- function(x=c(0,0), mu=c(0,0), Sigma=diag(2), kappa=1, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Sigma)) Sigma <- diag(ncol(x))
if(!is.matrix(Sigma)) Sigma <- matrix(Sigma)
if(!is.positive.definite(Sigma))
stop("Matrix Sigma is not positive-definite.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
temp <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(((log(k)+lgamma(k/2)) - ((k/2)*log(pi) +
0.5*log(det(Sigma)) + lgamma(1 + k/(2*kappa)) +
(1 + k/(2*kappa))*log(2))) + kappa*(-0.5*temp))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvpe <- function(n, mu=c(0,0), Sigma=diag(2), kappa=1)
{
mu <- rbind(mu)
k <- ncol(mu)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(k != nrow(Sigma)) {
stop("mu and Sigma have non-conforming size.")}
ev <- eigen(Sigma, symmetric=TRUE)
if(!all(ev$values >= -sqrt(.Machine$double.eps) *
abs(ev$values[1]))) {
stop("Sigma must be positive-definite.")}
SigmaSqrt <- ev$vectors %*% diag(sqrt(ev$values),
length(ev$values)) %*% t(ev$vectors)
radius <- (rgamma(n, shape=k/(2*kappa), scale=1/2))^(1/(2*kappa))
runifsphere <- function(n, k)
{
p <- as.integer(k)
if(!is.integer(k))
stop("k must be an integer in [2,Inf)")
if(k < 2) stop("k must be an integer in [2,Inf).")
Mnormal <- matrix(rnorm(n*k,0,1), nrow=n)
rownorms <- sqrt(rowSums(Mnormal^2))
unifsphere <- sweep(Mnormal,1,rownorms, "/")
return(unifsphere)
}
un <- runifsphere(n=n, k=k)
x <- mu + radius * un %*% SigmaSqrt
return(x)
}
###########################################################################
# Multivariate Power Exponential Distribution (Cholesky Parameterization) #
###########################################################################
dmvpec <- function(x=c(0,0), mu=c(0,0), U, kappa=1, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
Sigma <- t(U) %*% U
k <- nrow(Sigma)
Omega <- as.inverse(Sigma)
ss <- x - mu
temp <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(((log(k)+lgamma(k/2)) - ((k/2)*log(pi) +
0.5*log(det(Sigma)) + lgamma(1 + k/(2*kappa)) +
(1 + k/(2*kappa))*log(2))) + kappa*(-0.5*temp))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvpec <- function(n, mu=c(0,0), U, kappa=1)
{
mu <- rbind(mu)
k <- ncol(mu)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(k != nrow(U)) {
stop("mu and U have non-conforming size.")}
Sigma <- t(U) %*% U
ev <- eigen(Sigma, symmetric=TRUE)
if(!all(ev$values >= -sqrt(.Machine$double.eps) *
abs(ev$values[1]))) {
stop("Sigma must be positive-definite.")}
SigmaSqrt <- ev$vectors %*% diag(sqrt(ev$values),
length(ev$values)) %*% t(ev$vectors)
radius <- (rgamma(n, shape=k/(2*kappa), scale=1/2))^(1/(2*kappa))
runifsphere <- function(n, k)
{
p <- as.integer(k)
if(!is.integer(k))
stop("k must be an integer in [2,Inf)")
if(k < 2) stop("k must be an integer in [2,Inf).")
Mnormal <- matrix(rnorm(n*k,0,1), nrow=n)
rownorms <- sqrt(rowSums(Mnormal^2))
unifsphere <- sweep(Mnormal,1,rownorms, "/")
return(unifsphere)
}
un <- runifsphere(n=n, k=k)
x <- mu + radius * un %*% SigmaSqrt
return(x)
}
###########################################################################
# Multivariate t Distribution #
###########################################################################
dmvt <- function(x, mu, S, df=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(S)) S <- diag(ncol(x))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(df > 10000)) return(dmvn(x, mu, S, log))
k <- nrow(S)
ss <- x - mu
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((df+k)/2) - lgamma(df/2) + (k/2)*df +
(k/2)*log(pi) + 0.5*log(det(S)) + ((df+k)/2)*log(1 + (1/df) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvt <- function(n=1, mu=rep(0,k), S, df=Inf)
{
mu <- rbind(mu)
if(missing(S)) S <- diag(ncol(mu))
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.definite(S))
stop("Matrix S is not positive-definite.")
if(any(df <= 0)) stop("The df parameter must be positive.")
k <- ncol(S)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(df==Inf) x <- 1 else x <- rchisq(n,df) / df
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), S)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate t Distribution (Cholesky Parameterization) #
###########################################################################
dmvtc <- function(x, mu, U, df=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
if(any(df <= 0)) stop("The df parameter must be positive.")
if(any(df > 10000)) return(dmvnc(x, mu, U, log))
k <- nrow(U)
ss <- x - mu
S <- t(U) %*% U
Omega <- as.inverse(S)
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((df+k)/2) - lgamma(df/2) + (k/2)*df +
(k/2)*log(pi) + 0.5*log(det(S)) + ((df+k)/2)*log(1 + (1/df) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvtc <- function(n=1, mu=rep(0,k), U, df=Inf)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
if(any(df <= 0)) stop("The df parameter must be positive.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(df==Inf) x <- 1 else x <- rchisq(n,df) / df
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate t Distribution (Precision Parameterization) #
###########################################################################
dmvtp <- function(x, mu, Omega, nu=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(Omega)) Omega <- diag(ncol(x))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
if(any(nu > 10000)) return(dmvnp(x, mu, Omega, log))
k <- ncol(Omega)
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((nu+k)/2) - (lgamma(nu/2) + (k/2)*log(nu) +
(k/2)*log(pi)) + 0.5*log(detOmega) +
(-(nu+k)/2)*log(1 + (1/nu) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvtp <- function(n=1, mu, Omega, nu=Inf)
{
mu <- rbind(mu)
if(missing(Omega)) Omega <- diag(ncol(mu))
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
Sigma <- as.inverse(Omega)
k <- ncol(Sigma)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(nu == Inf) x <- 1 else x <- rchisq(n,nu) / nu
x <- ifelse(x == 0, 1e-100, x)
z <- rmvn(n, rep(0,k), Sigma)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Multivariate t Distribution (Precision-Cholesky Parameterization) #
###########################################################################
dmvtpc <- function(x, mu, U, nu=Inf, log=FALSE)
{
if(!is.matrix(x)) x <- rbind(x)
if(!is.matrix(mu)) mu <- rep(mu, each=nrow(x))
if(missing(U)) stop("Upper triangular U is required.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
if(any(nu > 10000)) return(dmvnpc(x, mu, U, log))
k <- ncol(U)
Omega <- t(U) %*% U
detOmega <- det(Omega)
ss <- x - mu
z <- rowSums({ss %*% Omega} * ss)
dens <- as.vector(lgamma((nu+k)/2) - (lgamma(nu/2) + (k/2)*log(nu) +
(k/2)*log(pi)) + 0.5*log(detOmega) +
(-(nu+k)/2)*log(1 + (1/nu) * z))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rmvtpc <- function(n=1, mu, U, nu=Inf)
{
mu <- rbind(mu)
if(missing(U)) stop("Upper triangular U is required.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
k <- ncol(U)
if(n > nrow(mu)) mu <- matrix(mu, n, k, byrow=TRUE)
if(nu == Inf) x <- 1 else x <- rchisq(n,nu) / nu
x <- ifelse(x == 0, 1e-100, x)
z <- rmvnpc(n, rep(0,k), U)
x <- mu + z/sqrt(x)
return(x)
}
###########################################################################
# Normal Distribution Mixture #
###########################################################################
dnormm <- function(x, p, mu, sigma, log=FALSE)
{
if(missing(x)) stop("x is a required argument.")
x <- as.vector(x)
n <- length(x)
if(missing(p)) stop("p is a required argument.")
p <- as.vector(p)
if(any(p <= 0) | any(p > 1)) stop("p must be in (0,1].")
if(sum(p) != 1) stop("p must sum to 1 for all components.")
m <- length(p)
p <- matrix(p, n, m, byrow=TRUE)
if(missing(mu)) stop("mu is a required argument.")
mu <- as.vector(mu)
if(!identical(m, length(mu)))
stop("p and mu differ in length.")
mu <- matrix(mu, n, m, byrow=TRUE)
if(missing(sigma)) stop("sigma is a required argument.")
sigma <- as.vector(sigma)
if(!identical(m, length(sigma)))
stop("p and sigma differ in length.")
sigma <- matrix(sigma, n, m, byrow=TRUE)
dens <- matrix(dnorm(x, mu, sigma, log=TRUE), n, m)
dens <- dens + log(p)
if(log == TRUE) dens <- apply(dens, 1, logadd)
else dens <- rowSums(exp(dens))
return(dens)
}
pnormm <- function(q, p, mu, sigma, lower.tail=TRUE, log.p=FALSE)
{
n <- length(q)
m <- length(p)
q <- matrix(q, n, m)
p <- matrix(p, n, m, byrow=TRUE)
mu <- matrix(mu, n, m, byrow=TRUE)
sigma <- matrix(sigma, n, m, byrow=TRUE)
cdf <- matrix(pnorm(q, mu, sigma, lower.tail=lower.tail,
log.p=log.p), n, m)
if(log.p == FALSE) cdf <- rowSums(cdf * p)
else stop("The log.p argument does not work yet.")
return(cdf)
}
rnormm <- function(n, p, mu, sigma)
{
if(missing(p)) stop("p is a required argument.")
p <- as.vector(p)
if(any(p <= 0) | any(p > 1)) stop("p must be in (0,1].")
if(sum(p) != 1) stop("p must sum to 1 for all components.")
m <- length(p)
p <- matrix(p, n, m, byrow=TRUE)
if(missing(mu)) stop("mu is a required argument.")
mu <- as.vector(mu)
if(!identical(m, length(mu)))
stop("p and mu differ in length.")
if(missing(sigma)) stop("sigma is a required argument.")
sigma <- as.vector(sigma)
if(!identical(m, length(sigma)))
stop("p and sigma differ in length.")
if(any(sigma <= 0)) stop("sigma must be positive.")
z <- rcat(n, p)
x <- rnorm(n, mean=mu[z], sd=sigma[z])
return(x)
}
###########################################################################
# Normal Distribution (Precision Parameterization) #
###########################################################################
dnormp <- function(x, mean=0, prec=1, log=FALSE)
{
#dens <- sqrt(prec/(2*pi)) * exp(-(prec/2)*(x-mu)^2)
dens <- dnorm(x, mean, sqrt(1/prec), log)
return(dens)
}
pnormp <- function(q, mean=0, prec=1, lower.tail=TRUE, log.p=FALSE)
{return(pnorm(q, mean=mean, sd=sqrt(1/prec), lower.tail, log.p))}
qnormp <- function(p, mean=0, prec=1, lower.tail=TRUE, log.p=FALSE)
{return(qnorm(p, mean=mean, sd=sqrt(1/prec), lower.tail, log.p))}
rnormp <- function(n, mean=0, prec=1)
{return(rnorm(n, mean=mean, sd=sqrt(1/prec)))}
###########################################################################
# Normal Distribution (Variance Parameterization) #
###########################################################################
dnormv <- function(x, mean=0, var=1, log=FALSE)
{
#dens <- (1/(sqrt(2*pi*var))) * exp(-((x-mu)^2/(2*var)))
dens <- dnorm(x, mean, sqrt(var), log)
return(dens)
}
pnormv <- function(q, mean=0, var=1, lower.tail=TRUE, log.p=FALSE)
{return(pnorm(q, mean=mean, sd=sqrt(var), lower.tail, log.p))}
qnormv <- function(p, mean=0, var=1, lower.tail=TRUE, log.p=FALSE)
{return(qnorm(p, mean=mean, sd=sqrt(var), lower.tail, log.p))}
rnormv <- function(n, mean=0, var=1)
{return(rnorm(n, mean=mean, sd=sqrt(var)))}
###########################################################################
# Pareto Distribution #
###########################################################################
dpareto <- function(x, alpha, log=FALSE)
{
x <- as.vector(x); alpha <- as.vector(alpha)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
NN <- max(length(x), length(alpha))
x <- rep(x, len=NN); alpha <- rep(alpha, len=NN)
dens <- ifelse(x < 1, -Inf, log(alpha) - (alpha + 1)*log(x))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
ppareto <- function(q, alpha)
{
q <- as.vector(q); alpha <- as.vector(alpha)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
NN <- max(length(q), length(alpha))
q <- rep(q, len=NN); alpha <- rep(alpha, len=NN)
p <- ifelse(q < 1, 0, 1 - 1/q^alpha)
return(p)
}
qpareto <- function(p, alpha)
{
p <- as.vector(p); alpha <- as.vector(alpha)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
NN <- max(length(p), length(alpha))
p <- rep(p, len=NN); alpha <- rep(alpha, len=NN)
q <- (1-p)^(-1/alpha)
return(q)
}
rpareto <- function(n, alpha)
{
alpha <- rep(alpha, len=n)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
x <- runif(n)^(-1/alpha)
return(x)
}
###########################################################################
# Power Exponential Distribution #
# #
# These functions are similar to those in the normalp package. #
###########################################################################
dpe <- function(x, mu=0, sigma=1, kappa=2, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu); sigma <- as.vector(sigma)
kappa <- as.vector(kappa)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(x), length(mu), length(sigma), length(kappa))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); kappa <- rep(kappa, len=NN)
cost <- 2 * kappa^(1/kappa) * gamma(1 + 1/kappa) * sigma
expon1 <- (abs(x - mu))^kappa
expon2 <- kappa * sigma^kappa
dens <- log(1/cost) + (-expon1 / expon2)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
ppe <- function(q, mu=0, sigma=1, kappa=2, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu); sigma <- as.vector(sigma)
kappa <- as.vector(kappa)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(q), length(mu), length(sigma), length(kappa))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); kappa <- rep(kappa, len=NN)
z <- (q - mu) / sigma
zz <- abs(z)^kappa
p <- pgamma(zz, shape=1/kappa, scale=kappa)
p <- p / 2
p <- ifelse(z < 0, 0.5 - p, 0.5 + p)
if(lower.tail == FALSE) p <- 1 - p
if(log.p == TRUE) p <- log(p)
return(p)
}
qpe <- function(p, mu=0, sigma=1, kappa=2, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu); sigma <- as.vector(sigma)
kappa <- as.vector(kappa)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
NN <- max(length(p), length(mu), length(sigma), length(kappa))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); kappa <- rep(kappa, len=NN)
if(log.p == TRUE) p <- log(p)
if(lower.tail == FALSE) p <- 1 - p
zp <- ifelse(p < 0.5, 0.5 - p, p - 0.5)
zp <- 2 * zp
qg <- qgamma(zp, shape=1/kappa, scale=kappa)
z <- qg^(1/kappa)
z <- ifelse(p < 0.5, -z, z)
q <- mu + z * sigma
return(q)
}
rpe <- function(n, mu=0, sigma=1, kappa=2)
{
mu <- rep(mu, len=n); sigma <- rep(sigma, len=n)
kappa <- rep(kappa, len=n)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(kappa <= 0)) stop("The kappa parameter must be positive.")
qg <- rgamma(n, shape=1/kappa, scale=kappa)
z <- qg^(1/kappa)
z <- ifelse(runif(n) < 0.5, -z, z)
x <- mu + z * sigma
return(x)
}
###########################################################################
# Skew Discrete Laplace Distribution #
###########################################################################
dsdlaplace <- function(x, p, q, log=FALSE)
{
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(q < 0) || any(q > 1)) stop("q must be in [0,1].")
NN <- max(length(x), length(p), length(q))
x <- rep(x, len=NN); p <- rep(p, len=NN); q <- rep(q, len=NN)
dens <- ifelse(x >= 0, log(1-p) + log(1-q) - (log(1-p*q) +
x*log(p)), log(1-p) + log(1-q) - (log(1-p*q) + abs(x)*log(q)))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
psdlaplace <- function(x, p, q)
{
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(q < 0) || any(q > 1)) stop("q must be in [0,1].")
NN <- max(length(x), length(p), length(q))
x <- rep(x, len=NN); p <- rep(p, len=NN); q <- rep(q, len=NN)
pr <- ifelse(x >= 0, 1-(1-q)*p^(floor(x)+1)/(1-p*q),
(1-p)*q^(-floor(x))/(1-p*q))
return(pr)
}
qsdlaplace <- function(prob, p, q)
{
if(any(prob < 0) || any(prob > 1)) stop("prob must be in [0,1].")
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(q < 0) || any(q > 1)) stop("q must be in [0,1].")
NN <- max(length(prob), length(p), length(q))
prob <- rep(prob, len=NN); p <- rep(p, len=NN); q <- rep(q, len=NN)
x <- numeric(NN)
for (i in 1:NN) {
k <- 0
if(prob[i] >= psdlaplace(k, p[i], q[i])) {
while(prob[i] >= psdlaplace(k, p[i], q[i])) {
k <- k + 1}}
else if(prob[i] < psdlaplace(k, p[i], q[i])) {
while(prob[i] < psdlaplace(k, p[i], q[i])) {
k <- k - 1}
k <- k + 1}
x[i] <- k
}
return(x)
}
rsdlaplace <- function(n, p, q)
{
if(length(p) > 1) stop("p must have a length of 1.")
if(length(q) > 1) stop("q must have a length of 1.")
if((p < 0) || (p > 1)) stop("p must be in [0,1].")
if((q < 0) || (q > 1)) stop("q must be in [0,1].")
u <- runif(n)
return(qsdlaplace(u,p,q))
}
###########################################################################
# Skew-Laplace Distribution #
###########################################################################
dslaplace <- function(x, mu, alpha, beta, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu)
alpha <- as.vector(alpha); beta <- as.vector(beta)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
NN <- max(length(x), length(mu), length(alpha), length(beta))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
alpha <- rep(alpha, len=NN); beta <- rep(beta, len=NN)
ab <- alpha + beta
belowMu <- log(1/ab) + ((x - mu)/alpha)
aboveMu <- log(1/ab) + ((mu - x)/beta)
dens <- ifelse(x <= mu, belowMu, aboveMu)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pslaplace <- function(q, mu, alpha, beta)
{
q <- as.vector(q); mu <- as.vector(mu)
alpha <- as.vector(alpha); beta <- as.vector(beta)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
NN <- max(length(q), length(mu), length(alpha), length(beta))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
alpha <- rep(alpha, len=NN); beta <- rep(beta, len=NN)
ab <- alpha + beta
belowMu <- (alpha/ab) * exp((q - mu)/alpha)
aboveMu <- 1 - (beta/ab) * exp((mu - q)/beta)
p <- ifelse(q < mu, belowMu, aboveMu)
return(p)
}
qslaplace <- function(p, mu, alpha, beta)
{
p <- as.vector(p); mu <- as.vector(mu)
alpha <- as.vector(alpha); beta <- as.vector(beta)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
NN <- max(length(p), length(mu), length(alpha), length(beta))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
alpha <- rep(alpha, len=NN); beta <- rep(beta, len=NN)
ab <- alpha + beta
belowMu <- alpha*log(p*ab/alpha) + mu
aboveMu <- mu - beta*log(ab*(1 - p)/beta)
q <- ifelse(p < alpha/ab, belowMu, aboveMu)
return(q)
}
rslaplace <- function(n, mu, alpha, beta)
{
#mu <- rep(mu, len=n); alpha <- rep(alpha, len=n)
#beta <- rep(beta, len=n)
if(any(alpha <= 0)) stop("The alpha parameter must be positive.")
if(any(beta <= 0)) stop("The beta parameter must be positive.")
ab <- alpha + beta
y <- rexp(n,1)
probs <- c(alpha,beta) / ab
signs <- sample(c(-1,1), n, replace=TRUE, prob=probs)
mult <- ifelse(signs < 0, signs*alpha, signs*beta)
x <- mult*y + mu
return(x)
}
###########################################################################
# Stick-Breaking Prior Distribution #
###########################################################################
dStick <- function(theta, gamma, log=FALSE)
{
dens <- sum(dbeta(theta, 1, gamma, log=log))
return(dens)
}
rStick <- function(M, gamma)
{
return(Stick(rbeta(M,1,gamma)))
}
###########################################################################
# Student t Distribution (3-parameter) #
# #
# The pst and qst functions are similar to the TF functions in the #
# gamlss.dist package, but dst and rst have been refined. #
###########################################################################
dst <- function(x, mu=0, sigma=1, nu=10, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu)
sigma <- as.vector(sigma); nu <- as.vector(nu)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
else if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(x), length(mu), length(sigma), length(nu))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); nu <- rep(nu, len=NN)
const <- lgamma((nu+1)/2) - lgamma(nu/2) - log(sqrt(pi*nu) * sigma)
dens <- const + log((1 + (1/nu)*((x-mu)/sigma)^2)^(-(nu+1)/2))
if(log == FALSE) dens <- exp(dens)
return(dens)
#return({1/sigma} * dt({x-mu}/sigma, df=nu, log)) #Deprecated
}
pst <- function(q, mu=0, sigma=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu)
sigma <- as.vector(sigma); nu <- as.vector(nu)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(q), length(mu), length(sigma), length(nu))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); nu <- rep(nu, len=NN)
if(length(nu) > 1) p <- ifelse(nu > 1000000,
pnorm(q, mu, sigma, lower.tail=lower.tail, log.p=log.p),
pt({q-mu}/sigma, df=nu, lower.tail=lower.tail, log.p=log.p))
else p <- if(nu > 1000000) {pnorm(q, mu, sigma,
lower.tail=lower.tail, log.p=log.p)}
else {pt({q-mu}/sigma, df=nu, lower.tail=lower.tail,
log.p=log.p)}
return(p)
}
qst <- function(p, mu=0, sigma=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu)
sigma <- as.vector(sigma); nu <- as.vector(nu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(p), length(mu), length(sigma), length(nu))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
sigma <- rep(sigma, len=NN); nu <- rep(nu, len=NN)
if(length(nu) > 1) q <- ifelse(nu > 1000000,
qnorm(p, mu, sigma, lower.tail=lower.tail, log.p=log.p),
mu + sigma * qt(p, df=nu, lower.tail=lower.tail))
else q <- if(nu > 1000000) {qnorm(p, mu, sigma,
lower.tail=lower.tail, log.p=log.p)}
else {mu + sigma * qt(p, df=nu, lower.tail=lower.tail)}
return(q)
}
rst <- function(n, mu=0, sigma=1, nu=10)
{
mu <- rep(mu, len=n); sigma <- rep(sigma, len=n); nu <- rep(nu, len=n)
if(any(sigma <= 0)) stop("The sigma parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
n <- ceiling(n)
y <- rnorm(n)
z <- rchisq(n, nu)
x <- mu + sigma*y*sqrt(nu/z)
return(x)
}
###########################################################################
# Student t Distribution (Precision Parameterization) #
###########################################################################
dstp <- function(x, mu=0, tau=1, nu=10, log=FALSE)
{
x <- as.vector(x); mu <- as.vector(mu)
tau <- as.vector(tau); nu <- as.vector(nu)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(x), length(mu), length(tau), length(nu))
x <- rep(x, len=NN); mu <- rep(mu, len=NN)
tau <- rep(tau, len=NN); nu <- rep(nu, len=NN)
dens <- ifelse(nu > 1000000,
dnorm(x, mu, sqrt(1/tau), log=TRUE),
(lgamma((nu+1)/2) - lgamma(nu/2)) + 0.5*log(tau/(nu*pi)) +
(-(nu+1)/2)*log(1 + (tau/nu)*(x-mu)^2))
if(log == FALSE) dens <- exp(dens)
return(dens)
}
pstp <- function(q, mu=0, tau=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
q <- as.vector(q); mu <- as.vector(mu)
tau <- as.vector(tau); nu <- as.vector(nu)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(q), length(mu), length(tau), length(nu))
q <- rep(q, len=NN); mu <- rep(mu, len=NN)
tau <- rep(tau, len=NN); nu <- rep(nu, len=NN)
p <- pst(q, mu, sqrt(1/tau), nu, lower.tail, log.p)
return(p)
}
qstp <- function(p, mu=0, tau=1, nu=10, lower.tail=TRUE, log.p=FALSE)
{
p <- as.vector(p); mu <- as.vector(mu)
tau <- as.vector(tau); nu <- as.vector(nu)
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
NN <- max(length(p), length(mu), length(tau), length(nu))
p <- rep(p, len=NN); mu <- rep(mu, len=NN)
tau <- rep(tau, len=NN); nu <- rep(nu, len=NN)
q <- qst(p, mu, sqrt(1/tau), nu, lower.tail, log.p)
return(q)
}
rstp <- function(n, mu=0, tau=1, nu=10)
{
mu <- rep(mu, len=n); tau <- rep(tau, len=n); nu <- rep(nu, len=n)
if(any(tau <= 0)) stop("The tau parameter must be positive.")
if(any(nu <= 0)) stop("The nu parameter must be positive.")
x <- rst(n, mu, sqrt(1/tau), nu)
n <- ceiling(n)
p <- runif(n)
x <- qst(p, mu=mu, sqrt(1/tau), nu=nu)
return(x)
}
###########################################################################
# Truncated Distribution #
# #
# These functions are similar to those from Nadarajah, S. and Kotz, S. #
# (2006). ``R Programs for Computing Truncated Distributions''. Journal #
# of Statistical Software, 16, Code Snippet 2, 1-8. These functions have #
# been corrected to work with log-densities. #
###########################################################################
dtrunc <- function(x, spec, a=-Inf, b=Inf, log=FALSE, ...)
{
if(a >= b) stop("Lower bound a is not less than upper bound b.")
if(any(x < a) | any(x > b))
stop("At least one instance of (x < a) or (x > b) found.")
dens <- rep(0, length(x))
g <- get(paste("d", spec, sep=""), mode="function")
G <- get(paste("p", spec, sep=""), mode="function")
if(log == TRUE) {
dens <- g(x, log=TRUE, ...) - log(G(b, ...) - G(a, ...))
}
else {
dens <- g(x, ...) / (G(b, ...) - G(a, ...))}
return(dens)
}
extrunc <- function(spec, a=-Inf, b=Inf, ...)
{
f <- function(x) x * dtrunc(x, spec, a=a, b=b, log=FALSE, ...)
return(integrate(f, lower=a, upper=b)$value)
}
ptrunc <- function(x, spec, a=-Inf, b=Inf, ...)
{
if(a >= b) stop("Lower bound a is not less than upper bound b.")
if(any(x < a) | any(x > b))
stop("At least one instance of (x < a) or (x > b) found.")
p <- x
aa <- rep(a, length(x))
bb <- rep(b, length(x))
G <- get(paste("p", spec, sep=""), mode="function")
p <- G(apply(cbind(apply(cbind(x, bb), 1, min), aa), 1, max), ...)
p <- p - G(aa, ...)
p <- p / {G(bb, ...) - G(aa, ...)}
return(p)
}
qtrunc <- function(p, spec, a=-Inf, b=Inf, ...)
{
if(any(p < 0) || any(p > 1)) stop("p must be in [0,1].")
if(a >= b) stop("Lower bound a is not less than upper bound b.")
q <- p
G <- get(paste("p", spec, sep=""), mode="function")
Gin <- get(paste("q", spec, sep=""), mode="function")
q <- Gin(G(a, ...) + p*{G(b, ...) - G(a, ...)}, ...)
return(q)
}
rtrunc <- function(n, spec, a=-Inf, b=Inf, ...)
{
if(a >= b) stop("Lower bound a is not less than upper bound b.")
x <- u <- runif(n)
x <- qtrunc(u, spec, a=a, b=b,...)
return(x)
}
vartrunc <- function(spec, a=-Inf, b=Inf, ...)
{
ex <- extrunc(spec, a=a, b=b, ...)
f <- function(x) {
{x - ex}^2 * dtrunc(x, spec, a=a, b=b, log=FALSE, ...)}
sigma2 <- integrate(f, lower=a, upper=b)$value
return(sigma2)
}
###########################################################################
# Wishart Distribution #
###########################################################################
dwishart <- function(Omega, nu, S, log=FALSE)
{
if(!is.matrix(Omega)) Omega <- matrix(Omega)
if(!is.positive.definite(Omega))
stop("Matrix Omega is not positive-definite.")
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(Omega), dim(S)))
stop("The dimensions of Omega and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Omega)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -((nu*k)/2) * log(2) - ((k*(k-1))/4) * log(pi) - gamsum -
(nu/2) * log(det(S)) + ((nu-k-1)/2) * log(det(Omega)) -
(tr(as.inverse(S) %*% Omega)/2)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rwishart <- function(nu, S)
{
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(nu < nrow(S)) {
stop("The nu parameter is less than the dimension of S.")}
k <- nrow(S)
Z <- matrix(0, k, k)
x <- rchisq(k, nu:{nu - k + 1})
x <- ifelse(x == 0, 1e-100, x)
diag(Z) <- sqrt(x)
if(k > 1) {
kseq <- 1:(k-1)
Z[rep(k*kseq, kseq) +
unlist(lapply(kseq, seq))] <- rnorm(k*{k-1}/2)}
return(crossprod(Z %*% chol(S)))
}
###########################################################################
# Wishart Distribution (Cholesky Parameterization) #
###########################################################################
dwishartc <- function(U, nu, S, log=FALSE)
{
if(missing(U)) stop("Upper triangular U is required.")
Omega <- t(U) %*% U
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(!identical(dim(Omega), dim(S)))
stop("The dimensions of Omega and S differ.")
if(nu < nrow(S))
stop("The nu parameter is less than the dimension of S.")
k <- nrow(Omega)
gamsum <- 0
for (i in 1:k) {gamsum <- gamsum + lgamma((nu + 1 - i)/2)}
dens <- -((nu*k)/2) * log(2) - ((k*(k-1))/4) * log(pi) - gamsum -
(nu/2) * log(det(S)) + ((nu-k-1)/2) * log(det(Omega)) -
(tr(as.inverse(S) %*% Omega)/2)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
rwishartc <- function(nu, S)
{
if(!is.matrix(S)) S <- matrix(S)
if(!is.positive.semidefinite(S))
stop("Matrix S is not positive-semidefinite.")
if(nu < nrow(S)) {
stop("The nu parameter is less than the dimension of S.")}
k <- nrow(S)
Z <- matrix(0, k, k)
x <- rchisq(k, nu:{nu - k + 1})
x <- ifelse(x == 0, 1e-100, x)
diag(Z) <- sqrt(x)
if(k > 1) {
kseq <- 1:(k-1)
Z[rep(k*kseq, kseq) +
unlist(lapply(kseq, seq))] <- rnorm(k*{k-1}/2)}
return(chol(crossprod(Z %*% chol(S))))
}
###########################################################################
# Zellner's g-Prior #
###########################################################################
dhyperg <- function(g, alpha=3, log=FALSE)
{
if(g <= 0) stop("The g parameter must be positive.")
if(alpha <= 0) stop("The alpha parameter must be positive.")
dens <- log((alpha - 2)/2) -(alpha/2)*log(1 + g)
if(log == FALSE) dens <- exp(dens)
return(dens)
}
dzellner <- function(beta, g, sigma, X, log=FALSE)
{
if(g <= 0) stop("The g parameter must be positive.")
if(sigma <= 0) stop("The sigma parameter must be positive.")
dens <- dmvn(beta, rep(0, length(beta)),
g*sigma*sigma*as.inverse(t(X) %*% X), log=log)
return(dens)
}
#End
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