Normal

Normal mixture kernel

A normal component is parameterized by () and (>0): [ f(y,)=!(-). ]

A finite normal mixture with (J) components has density [ f(y)=_{j=1}^J w_j,(y_j,j^2), {j=1}^J w_j=1. ]

Parameter mapping (math () code): (_j) mean[j], (_j) sd[j], (w_j) w[j].

Normal mixture with GPD tail

In the spliced version, the bulk distribution is the normal mixture below (u), and a GPD tail is attached above (u). The tail parameters are (>0) and (), applied to exceedances (x-u).

Tail mapping (math () code): (u) threshold, () tail_scale, () tail_shape.

Without GPD (mixture kernel)

Code

grid <- seq(-4, 5, length.out = 400)
normal_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.3, 0.1), mean = c(-1, 0.5, 2), sd = c(2, 0.6, 1.1)),
  list(label = "Mix B", w = c(0.5, 0.3, 0.2), mean = c(-1.2, 0.3, 1.5), sd = c(0.9, 0.7, 1.0)),
  list(label = "Mix C", w = c(0.4, 0.35, 0.25), mean = c(-0.5, 2, 2.5), sd = c(0.7, 0.6, 1.2))
)

example <- normal_sets[[1]]

Code

dNormMix(0, w = example$w, mean = example$mean, sd = example$sd)

[1] 0.2535206

Code

dNormMix(0, w = example$w, mean = example$mean, sd = example$sd, log = TRUE)

[1] -1.37231

Code

pNormMix(0, w = example$w, mean = example$mean, sd = example$sd)

[1] 0.4790278

Code

pNormMix(0, w = example$w, mean = example$mean, sd = example$sd, lower.tail = FALSE)

[1] 0.5209722

Code

pNormMix(0, w = example$w, mean = example$mean, sd = example$sd, log.p = TRUE)

[1] -0.7359966

Code

q_vec(qNormMix, c(0.25, 0.5, 0.75), w = example$w, mean = example$mean,
      sd = example$sd)

[1] -1.42337727  0.08046623  0.94945934

Code

q_vec(qNormMix, c(0.25, 0.5, 0.75), w = example$w, mean = example$mean,
      sd = example$sd, lower.tail = FALSE)

[1]  0.94945934  0.08046623 -1.42337727

Code

q_vec(qNormMix, c(log(0.25), log(0.5), log(0.75)), w = example$w,
      mean = example$mean, sd = example$sd, log.p = TRUE)

[1] -1.42337727  0.08046623  0.94945934

Code

draw_many(rNormMix, list(w = example$w, mean = example$mean, sd = example$sd))

[1] -1.6524667  1.0808085  2.4561056 -2.6409368  0.3231677

Code

df_norm <- do.call(rbind, lapply(normal_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dNormMix, list(w = ps$w, mean = ps$mean, sd = ps$sd)), label = ps$label)
}))

ggplot(df_norm, aes(x = x, y = density, color = label)) +
  geom_line(linewidth = 1) +
  labs(title = "Normal mixtures (bulk)", x = "x", y = "density") +
  theme_minimal() + theme(legend.position = "top")

With GPD tail

Spliced density uses the same mixture for \(x<u\) and attaches \(f_{GPD}\) above \(u\) with continuity. CDF combines mixture CDF up to \(u\) and GPD exceedance beyond \(u\); quantiles invert this spliced CDF; RNG draws bulk vs tail by the CDF mass at \(u\).

Code

normal_gpd_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.4), mean = c(-1, 2), sd = c(0.5, 0.8), threshold = 1.8, tail_scale = 0.4, tail_shape = 0.25),
  list(label = "Mix B", w = c(0.5, 0.5), mean = c(0, 1), sd = c(0.6, 0.6), threshold = 1.5, tail_scale = 0.3, tail_shape = 0.2),
  list(label = "Mix C", w = c(0.7, 0.3), mean = c(0.5, 2.5), sd = c(0.4, 1.0), threshold = 2.0, tail_scale = 0.5, tail_shape = 0.15)
)

example <- normal_gpd_sets[[1]]

Code

dNormMixGpd(2, w = example$w, mean = example$mean, sd = example$sd, threshold = example$threshold, tail_scale = example$tail_scale, tail_shape = example$tail_shape)

[1] 0.3322395

Code

pNormMixGpd(2, w = example$w, mean = example$mean, sd = example$sd, threshold = example$threshold, tail_scale = example$tail_scale, tail_shape = example$tail_shape)

[1] 0.8504922

Code

q_vec(qNormMixGpd, c(0.5, 0.9), w = example$w, mean = example$mean, sd = example$sd, threshold = example$threshold, tail_scale = example$tail_scale, tail_shape = example$tail_shape)

[1] -0.5173894  2.1903912

Code

draw_many(rNormMixGpd, example)

[1]  3.9237227  3.3651944  1.0818744 -1.1496076 -0.8738883

Code

df_norm_gpd <- do.call(rbind, lapply(normal_gpd_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dNormMixGpd, list(w = ps$w, mean = ps$mean, sd = ps$sd, threshold = ps$threshold, tail_scale = ps$tail_scale, tail_shape = ps$tail_shape)), label = ps$label)
}))

ggplot(df_norm_gpd, aes(x = x, y = density, color = label)) +
  geom_line(linewidth = 1) +
  labs(title = "Normal mixtures with GPD tail (different thresholds)", x = "x", y = "density") +
  theme_minimal() + theme(legend.position = "top")

Prereqs

Required packages and data for this page are listed in the setup chunks above.

Outputs

This page renders model fits, diagnostics, and summary artifacts generated by package APIs.

Interpretation

Canonical concept page: Introduction With Gpd Kernel
Treat this page as an application/example view and use the canonical page for core definitions.

Continue to the linked canonical concept page, then return for implementation-specific details.

Normal

Normal mixture kernel

Normal mixture with GPD tail

Without GPD (mixture kernel)

With GPD tail

Prereqs

Outputs

Interpretation

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