Lognormal

Lognormal mixture kernel

A lognormal component is defined by (Y (,^2)), i.e., [ f(y,)=!(-),y>0. ]

A finite lognormal mixture has density [ f(y)=_{j=1}^J w_j,(y_j,_j^2),y>0. ]

Parameter mapping (math () code): (_j) meanlog[j], (_j) sdlog[j], (w_j) w[j].

Lognormal mixture with GPD tail

The *MixGpd variant uses the same splicing idea: lognormal mixture below (u), GPD tail above (u).

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

Without GPD (mixture kernel)

Code

grid <- seq(0, 8, length.out = 400)
logn_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.3, 0.1), meanlog = c(0.0, 0.3, 0.6), sdlog = c(0.4, 0.5, 0.6)),
  list(label = "Mix B", w = c(0.5, 0.3, 0.2), meanlog = c(0.1, 0.4, 0.7), sdlog = c(0.35, 0.45, 0.55)),
  list(label = "Mix C", w = c(0.4, 0.35, 0.25), meanlog = c(0.2, 0.5, 2), sdlog = c(0.3, 0.4, 0.5))
)

example <- logn_sets[[1]]

Code

dLognormalMix(1, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog)

[1] 0.8386766

Code

dLognormalMix(1, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog, log = TRUE)

[1] -0.1759301

Code

pLognormalMix(1, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog)

[1] 0.3981415

Code

pLognormalMix(1, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog, lower.tail = FALSE)

[1] 0.6018585

Code

pLognormalMix(1, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog, log.p = TRUE)

[1] -0.9209479

Code

q_vec(qLognormalMix, c(0.25, 0.5, 0.75), w = example$w,
      meanlog = example$meanlog, sdlog = example$sdlog)

[1] 0.8282429 1.1278180 1.5752146

Code

q_vec(qLognormalMix, c(0.25, 0.5, 0.75), w = example$w,
      meanlog = example$meanlog, sdlog = example$sdlog, lower.tail = FALSE)

[1] 1.5752146 1.1278180 0.8282429

Code

q_vec(qLognormalMix, c(log(0.25), log(0.5), log(0.75)), w = example$w,
      meanlog = example$meanlog, sdlog = example$sdlog, log.p = TRUE)

[1] 0.8282429 1.1278180 1.5752146

Code

draw_many(rLognormalMix, list(w = example$w, meanlog = example$meanlog, sdlog = example$sdlog))

[1] 0.8776623 1.1036500 2.3368056 0.7202281 1.1649053

Code

df_logn <- do.call(rbind, lapply(logn_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dLognormalMix, list(w = ps$w, meanlog = ps$meanlog, sdlog = ps$sdlog)), label = ps$label)
}))

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

With GPD tail

Code

logn_gpd_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.4), meanlog = c(0, 1), sdlog = c(0.3, 0.5), threshold = 2.5, tail_scale = 0.5, tail_shape = 0.2),
  list(label = "Mix B", w = c(0.5, 0.5), meanlog = c(0.5, 1.2), sdlog = c(0.4, 0.4), threshold = 2.0, tail_scale = 0.4, tail_shape = 0.15),
  list(label = "Mix C", w = c(0.7, 0.3), meanlog = c(0.8, 1.5), sdlog = c(0.25, 0.6), threshold = 3.0, tail_scale = 0.6, tail_shape = 0.18)
)

example <- logn_gpd_sets[[1]]

Code

dLognormalMixGpd(2.5, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog, threshold = example$threshold, tail_scale = example$tail_scale, tail_shape = example$tail_shape)

[1] 0.4545372

Code

pLognormalMixGpd(2.5, w = example$w, meanlog = example$meanlog, sdlog = example$sdlog, threshold = example$threshold, tail_scale = example$tail_scale, tail_shape = example$tail_shape)

[1] 0.7727314

Code

q_vec(qLognormalMixGpd, c(0.5, 0.9), w = example$w, meanlog = example$meanlog, sdlog = example$sdlog, threshold = example$threshold, tail_scale = example$tail_scale, tail_shape = example$tail_shape)

[1] 1.273895 2.946103

Code

draw_many(rLognormalMixGpd, example)

[1] 9.0460364 4.3148364 1.5313834 0.9141464 1.0786034

Code

df_logn_gpd <- do.call(rbind, lapply(logn_gpd_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dLognormalMixGpd, list(w = ps$w, meanlog = ps$meanlog, sdlog = ps$sdlog, threshold = ps$threshold, tail_scale = ps$tail_scale, tail_shape = ps$tail_shape)), label = ps$label)
}))

ggplot(df_logn_gpd, aes(x = x, y = density, color = label)) +
  geom_line(linewidth = 1) +
  labs(title = "Lognormal 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.

Lognormal

Lognormal mixture kernel

Lognormal mixture with GPD tail

Without GPD (mixture kernel)

With GPD tail

Prereqs

Outputs

Interpretation

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