LassoDistribution: The Lasso Distribution

Description

Provides functions related to the Lasso distribution, including the normalizing constant, probability density function, cumulative distribution function, quantile function, and random number generation for given parameters a, b, and c. Additional utilities include the Mills ratio, expected value, and variance of the distribution. The package also implements modified versions of the Hans and Park–Casella Gibbs sampling algorithms for Bayesian Lasso regression.

Usage

zlasso(a, b, c, logarithm)
dlasso(x, a, b, c, logarithm)
plasso(q, a, b, c)
qlasso(p, a, b, c)
rlasso(n, a, b, c)
elasso(a, b, c)
vlasso(a, b, c)
mlasso(a, b, c)
MillsRatio(d)
Modified_Hans_Gibbs(X, y, a1, b1, u1, v1,
              nsamples, beta_init, lambda_init, sigma2_init, verbose)
Modified_PC_Gibbs(X, y, a1, b1, u1, v1, 
              nsamples, lambda_init, sigma2_init, verbose)

Value

zlasso, dlasso, plasso, qlasso, rlasso, elasso, vlasso, mlasso, MillsRatio: return the corresponding scalar or vector values related to the Lasso distribution and a numeric value representing the Mills ratio.
Modified_Hans_Gibbs: returns a list containing:

mBeta

Matrix of MCMC samples for the regression coefficients $\beta$, with nsamples rows and p columns.

vsigma2

Vector of MCMC samples for the error variance $\sigma^2$.

vlambda2

Vector of MCMC samples for the shrinkage parameter $\lambda^2$.

mA

Matrix of sampled values for parameter $a_j$ of the Lasso distribution for each $\beta_j$.

mB

Matrix of sampled values for parameter $b_j$ of the Lasso distribution for each $\beta_j$.

mC

Matrix of sampled values for parameter $c_j$ of the Lasso distribution for each $\beta_j$.

Modified_PC_Gibbs: returns a list containing:

mBeta: Matrix of MCMC samples for the regression coefficients $\beta$.

vsigma2

Vector of MCMC samples for the error variance $\sigma^2$.

vlambda2

Vector of MCMC samples for the shrinkage parameter $\lambda^2$.

mM

Matrix of estimated means of the full conditional distributions of each $\beta_j$.

mV

Matrix of estimated variances of the full conditional distributions of each $\beta_j$.

va_til

Vector of estimated shape parameters for the full conditional inverse-gamma distribution of $\sigma^2$.

vb_til

Vector of estimated rate parameters for the full conditional inverse-gamma distribution of $\sigma^2$.

vu_til

Vector of estimated shape parameters for the full conditional inverse-gamma distribution of $\lambda^2$.

vv_til

Vector of estimated rate parameters for the full conditional inverse-gamma distribution of $\lambda^2$.

Arguments

x, q: Vector of quantiles (vectorized).
p: Vector of probabilities.
a: Vector of precision parameter which must be non-negative.
b: Vector of off set parameter.
c: Vector of tuning parameter which must be non-negative values.
n: Number of observations.
logarithm: Logical. If TRUE, probabilities are returned on the log scale.
d: A scalar numeric value. Represents the point at which the Mills ratio is evaluated.
X: Design matrix (numeric matrix).
y: Response vector (numeric vector).
a1: Shape parameter of the prior on $\lambda^2$.
b1: Rate parameter of the prior on $\lambda^2$.
u1: Shape parameter of the prior on $\sigma^2$.
v1: Rate parameter of the prior on $\sigma^2$.
nsamples: Number of Gibbs samples to draw.
beta_init: Initial value for the model parameter $\beta$.
lambda_init: Initial value for the shrinkage parameter $\lambda^2$.
sigma2_init: Initial value for the error variance $\sigma^2$.
verbose: Integer. If greater than 0, progress is printed every verbose iterations during sampling. Set to 0 to suppress output.

Details

If $X \sim \text{Lasso}(a, b, c)$ then its density function is: $$ p(x;a,b,c) = Z^{-1} \exp\left(-\frac{1}{2} a x^2 + bx - c|x| \right) $$ where $x \in \mathbb{R}$, $a > 0$, $b \in \mathbb{R}$, $c > 0$, and $Z$ is the normalizing constant.

More details are included for the CDF, quantile function, and normalizing constant in the original documentation.

Examples

Run this code

a <- 2; b <- 1; c <- 3
x <- seq(-3, 3, length.out = 1000)
plot(x, dlasso(x, a, b, c, logarithm = FALSE), type = 'l')

r <- rlasso(1000, a, b, c)
hist(r, breaks = 50, probability = TRUE, col = "grey", border = "white")
lines(x, dlasso(x, a, b, c, logarithm = FALSE), col = "blue")

plasso(0, a, b, c)
qlasso(0.25, a, b, c)
elasso(a, b, c)
vlasso(a, b, c)
mlasso(a, b, c)
MillsRatio(2)




# The Modified_Hans_Gibbs() function uses the Lasso distribution to draw 
# samples from the full conditional distribution of the regression coefficients.

y <- 1:20
X <- matrix(c(1:20,12:31,7:26),20,3,byrow = TRUE)

a1 <- b1 <- u1 <- v1 <- 0.01
sigma2_init <- 1
lambda_init <- 0.1
beta_init <- rep(1, ncol(X))
nsamples <- 1000
verbose <- 100

Output_Hans <- Modified_Hans_Gibbs(
                X, y, a1, b1, u1, v1,
                nsamples, beta_init, lambda_init, sigma2_init, verbose
)

colMeans(Output_Hans$mBeta)
mean(Output_Hans$vlambda2)


Output_PC <- Modified_PC_Gibbs(
               X, y, a1, b1, u1, v1, 
               nsamples, lambda_init, sigma2_init, verbose)

colMeans(Output_PC$mBeta)
mean(Output_PC$vlambda2)