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Lognormal mixture distribution

lognormal_mix.Rd

Finite mixture of lognormal components for positive-support bulk modeling. The scalar functions in this topic are the NIMBLE-compatible building blocks for the lognormal bulk kernel family.

Usage

dLognormalMix(x, w, meanlog, sdlog, log = 0)

pLognormalMix(q, w, meanlog, sdlog, lower.tail = 1, log.p = 0)

rLognormalMix(n, w, meanlog, sdlog)

qLognormalMix(
  p,
  w,
  meanlog,
  sdlog,
  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.

meanlog, sdlog

Numeric vectors of length \(K\) giving component log-means and log-standard deviations.

log

Integer flag 0/1; if 1, return the log-density.

q

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

lower.tail

Integer flag 0/1; if 1 (default), probabilities are \(P(X \le q)\).

log.p

Integer flag 0/1; if 1, 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 number of iterations for stats::uniroot.

Value

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

Details

The mixture density is $$ f(x) = \sum_{k = 1}^K \tilde{w}_k f_{LN}(x \mid \mu_k, \sigma_k), \qquad x > 0, $$ with normalized weights \(\tilde{w}_k\). For vectorized R usage, use lognormal_lowercase().

Each component satisfies \(\log X_k \sim N(\mu_k,\sigma_k^2)\), so the mixture CDF is $$ F(x) = \sum_{k=1}^K \tilde{w}_k \Phi\left(\frac{\log x-\mu_k}{\sigma_k}\right), \qquad x>0. $$ Random generation proceeds by drawing a component index with probability \(\tilde{w}_k\) and then sampling from the corresponding lognormal law. Because a finite mixture of lognormals does not admit a closed-form inverse CDF, qLognormalMix() computes quantiles by numerical inversion.

The analytical mixture mean is $$ E(X) = \sum_{k=1}^K \tilde{w}_k \exp(\mu_k + \sigma_k^2/2), $$ which is the expression used by the package whenever an ordinary predictive mean exists.

Functions

  • dLognormalMix(): Lognormal mixture density

  • pLognormalMix(): Lognormal mixture distribution function

  • rLognormalMix(): Lognormal mixture random generation

  • qLognormalMix(): Lognormal mixture quantile function

See also

lognormal_mixgpd(), lognormal_gpd(), lognormal_lowercase(), build_nimble_bundle(), kernel_support_table().

Other lognormal kernel families: lognormal_gpd, lognormal_mixgpd

Examples

w <- c(0.60, 0.25, 0.15)
meanlog <- c(-0.2, 0.6, 1.2)
sdlog <- c(0.4, 0.3, 0.5)

dLognormalMix(2.0, w = w, meanlog = meanlog, sdlog = sdlog, log = FALSE)
#> [1] 0.2189383
pLognormalMix(2.0, w = w, meanlog = meanlog, sdlog = sdlog,
             lower.tail = TRUE, log.p = FALSE)
#> [1] 0.7711135
qLognormalMix(0.50, w = w, meanlog = meanlog, sdlog = sdlog)
#> [1] 1.151134
qLognormalMix(0.95, w = w, meanlog = meanlog, sdlog = sdlog)
#> [1] 4.147585
replicate(10, rLognormalMix(1, w = w, meanlog = meanlog, sdlog = sdlog))
#>  [1] 4.1293830 0.6873494 1.3316464 0.6494833 0.9846285 1.6379267 3.3315677
#>  [8] 0.7021537 2.5439287 2.1535620