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Laplace (double exponential) mixture distribution

laplace_mix.Rd

Finite mixture of Laplace components for real-valued bulk modeling. The scalar functions in this topic are the NIMBLE-compatible building blocks for Laplace-based kernels.

Usage

dLaplaceMix(x, w, location, scale, log = 0)

pLaplaceMix(q, w, location, scale, lower.tail = 1, log.p = 0)

rLaplaceMix(n, w, location, scale)

qLaplaceMix(
  p,
  w,
  location,
  scale,
  lower.tail = TRUE,
  log.p = FALSE,
  tol = 1e-10,
  maxiter = 200
)

Arguments

x

Numeric scalar giving the point at which the density is evaluated.

w

Numeric vector of mixture weights of length \(K\). The functions normalize w internally when needed.

location

Numeric vector of length \(K\) giving component locations.

scale

Numeric vector of length \(K\) giving component scales.

log

Logical; if TRUE, return the log-density.

q

Numeric scalar giving the point at which the distribution function is evaluated.

lower.tail

Logical; if TRUE (default), probabilities are \(P(X \le q)\).

log.p

Logical; if TRUE, probabilities are returned on the log scale.

n

Integer giving the number of draws. The RNG implementation supports n = 1.

p

Numeric scalar probability in \((0,1)\) for the quantile function.

tol

Numeric scalar tolerance passed to stats::uniroot.

maxiter

Integer maximum iterations for stats::uniroot.

Value

Density/CDF/RNG functions return numeric scalars. qLaplaceMix() returns a numeric vector with the same length as p.

Details

The mixture density is $$ f(x) = \sum_{k = 1}^K \tilde{w}_k f_{Lap}(x \mid \mu_k, b_k), $$ with normalized weights \(\tilde{w}_k\). For vectorized R usage, use laplace_lowercase().

Each component is a Laplace law with density \(f(x \mid \mu,b) = (2b)^{-1}\exp\{-|x-\mu|/b\}\). The mixture CDF is the corresponding weighted average of component CDFs, and random generation selects a component first and then samples from that component. As with the other finite mixtures in the package, the quantile has no closed form and is therefore obtained numerically.

The analytical mean of the mixture is simply $$ E(X) = \sum_{k=1}^K \tilde{w}_k \mu_k. $$ That is the formula used in downstream predictive mean calculations for Laplace-based fits.

Functions

  • dLaplaceMix(): Laplace mixture density

  • pLaplaceMix(): Laplace mixture distribution function

  • rLaplaceMix(): Laplace mixture random generation

  • qLaplaceMix(): Laplace mixture quantile function

See also

laplace_MixGpd(), laplace_gpd(), laplace_lowercase(), build_nimble_bundle(), kernel_support_table().

Other laplace kernel families: laplace_MixGpd, laplace_gpd

Examples

w <- c(0.50, 0.30, 0.20)
location <- c(-1, 0.5, 2.0)
scale <- c(1.0, 0.7, 1.4)

dLaplaceMix(0.8, w = w, location = location, scale = scale, log = FALSE)
#> [1] 0.2112312
pLaplaceMix(0.8, w = w, location = location, scale = scale,
           lower.tail = TRUE, log.p = FALSE)
#> [1] 0.7033967
qLaplaceMix(0.50, w = w, location = location, scale = scale)
#> [1] -0.02546085
qLaplaceMix(0.95, w = w, location = location, scale = scale)
#> [1] 3.18341
replicate(10, rLaplaceMix(1, w = w, location = location, scale = scale))
#>  [1] -0.1827557  0.3229609  2.5837559 -0.4809303 -0.5316669 -1.9789015
#>  [7] -1.7040411  0.5314815 -0.5180490  2.6544431