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The Lasso Distribution

Source: R/LassoDistribution.R
LassoDistribution.Rd

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, beta_init, a1, b1, u1, v1,
              nsamples, lambda_init, sigma2_init, thin, verbose,
              tune_lambda2, rao_blackwellization)
Modified_PC_Gibbs(X, y, a1, b1, u1, v1, 
              nsamples, lambda_init, sigma2_init, thin, verbose)

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 \(\sigma^2\).

thin

Thinning interval for the MCMC chain. Only every `thin`-th draw is stored. Default is 1 (no thinning).

verbose

Integer. If greater than 0, progress is printed every verbose iterations during sampling. Set to 0 to suppress output.

tune_lambda2

Logical; if TRUE (default), the tuning parameter \(\lambda^2\) is estimated during sampling.

rao_blackwellization

Logical; if TRUE, Rao–Blackwellization is applied to improve posterior estimation. Default is FALSE.

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 \(\lambda^2\).

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.

See also

normalize for preprocessing input data before applying the samplers.

Examples

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)
#> [1] 0.3739435
qlasso(0.25, a, b, c)
#> [1] -0.08945799
elasso(a, b, c)
#> [1] 0.1218306
vlasso(a, b, c)
#> [1] 0.1287739
mlasso(a, b, c)
#> [1] 0
MillsRatio(2)
#> [1] 0.4213692




# 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
tune_lambda2 <- TRUE
rao_blackwellization <- FALSE

Output_Hans <- Modified_Hans_Gibbs(
                X, y, beta_init, a1, b1, u1, v1,
                nsamples, lambda_init, sigma2_init, 
                verbose, tune_lambda2, rao_blackwellization
)
#> iter: 0 lambda2: 0.01 sigma2: 87.1593
#> iter: 1 lambda2: 0.01 sigma2: 95.9609
#> iter: 2 lambda2: 0.01 sigma2: 50.9426
#> iter: 3 lambda2: 0.01 sigma2: 52.1754
#> iter: 4 lambda2: 0.01 sigma2: 80.7908
#> iter: 5 lambda2: 0.01 sigma2: 74.9386
#> iter: 6 lambda2: 0.01 sigma2: 69.402
#> iter: 7 lambda2: 0.01 sigma2: 50.4434
#> iter: 8 lambda2: 0.01 sigma2: 41.5339
#> iter: 9 lambda2: 0.01 sigma2: 39.48

colMeans(Output_Hans$mBeta)
#> [1] -0.12758493  0.08436887  0.10115044
mean(Output_Hans$vlambda2)
#> [1] 0.001


Output_PC <- Modified_PC_Gibbs(
               X, y, a1, b1, u1, v1, 
               nsamples, lambda_init, sigma2_init, verbose)
#> iter: 0

colMeans(Output_PC$mBeta)
#> [1]  0.03753301 -0.02313929  0.04647570
mean(Output_PC$vlambda2)
#> [1] 1.66784