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Normal mixture distribution

normal_mix.Rd

Finite mixture of normal components for real-valued bulk modeling. This topic provides the scalar mixture density, CDF, RNG, and quantile functions used by the bulk-only normal kernel in the package registry.

Usage

dNormMix(x, w, mean, sd, log = 0)

pNormMix(q, w, mean, sd, lower.tail = 1, log.p = 0)

rNormMix(n, w, mean, sd)

qNormMix(
  p,
  w,
  mean,
  sd,
  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.

mean, sd

Numeric vectors of length \(K\) giving component means and 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. qNormMix() returns a numeric vector with the same length as p.

Details

With weights \(w_k\), means \(\mu_k\), and standard deviations \(\sigma_k\), the mixture density is $$ f(x) = \sum_{k = 1}^K \tilde{w}_k \phi(x \mid \mu_k, \sigma_k^2), $$ where \(\tilde{w}_k = w_k / \sum_j w_j\). These uppercase functions are scalar NIMBLE-compatible building blocks. For vectorized R usage, use normal_lowercase().

If \(F_k\) denotes the \(k\)-th component CDF, then the mixture distribution function is $$ F(x) = \sum_{k=1}^K \tilde{w}_k F_k(x) = \sum_{k=1}^K \tilde{w}_k \Phi\left(\frac{x-\mu_k}{\sigma_k}\right). $$ Random generation first draws a component index with probability \(\tilde{w}_k\) and then generates from the corresponding normal law. The quantile function has no closed form for a general finite mixture, so qNormMix() solves \(F(x)=p\) numerically by bracketing the root and applying stats::uniroot().

The mixture mean is $$ E(X) = \sum_{k=1}^K \tilde{w}_k \mu_k, $$ which is the analytical mean used by the package when a normal-mixture draw contributes to a posterior predictive mean calculation.

Functions

  • dNormMix(): Normal mixture density

  • pNormMix(): Normal mixture distribution function

  • rNormMix(): Normal mixture random generation

  • qNormMix(): Normal mixture quantile function

See also

normal_mixgpd(), normal_gpd(), normal_lowercase(), build_nimble_bundle(), kernel_support_table().

Other normal kernel families: normal_gpd, normal_mixgpd

Examples

w <- c(0.60, 0.25, 0.15)
mean <- c(-1, 0.5, 2.0)
sd <- c(1.0, 0.7, 1.3)

dNormMix(0.5, w = w, mean = mean, sd = sd, log = FALSE)
#> [1] 0.2438468
pNormMix(0.5, w = w, mean = mean, sd = sd,
        lower.tail = TRUE, log.p = FALSE)
#> [1] 0.7035579
qNormMix(0.50, w = w, mean = mean, sd = sd)
#> [1] -0.2713211
qNormMix(0.95, w = w, mean = mean, sd = sd)
#> [1] 2.571684
replicate(10, rNormMix(1, w = w, mean = mean, sd = sd))
#>  [1] -0.3697656  1.6883524 -2.6346383 -1.6941589 -1.6695334  0.4361516
#>  [7] -0.3634372  0.8686264 -0.1064025  0.1765331