Amoroso (bulk kernel and mixture)

Amoroso mixture kernel

An Amoroso component with location (a), scale (), and shapes (>0), (>0) has density (for (y>a)) [ f(ya,,,) = || ()^{} !. ]

A finite Amoroso mixture is [ f(y)=_{j=1}^J w_j,f_j(ya_j,_j,_j,_j). ]

Parameter mapping (math () code): (a_j) loc[j], (_j) scale[j], (_j) shape1[j], (_j) shape2[j], (w_j) w[j].

Amoroso mixture with GPD tail

The spliced kernel attaches a GPD tail above (u).

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

Without GPD

Code

grid <- seq(0.01, 6, length.out = 400)
amor_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.3, 0.1), loc = c(0.5, 0.5, 0.5), scale = c(1.0, 1.3, 1.6), shape1 = c(2.5, 3.0, 4.0), shape2 = c(1.2, 1.2, 1.2)),
  list(label = "Mix B", w = c(0.5, 0.3, 0.2), loc = c(0.4, 0.6, 0.6), scale = c(1.1, 1.2, 1.5), shape1 = c(2.2, 2.8, 3.8), shape2 = c(1.1, 1.2, 1.3)),
  list(label = "Mix C", w = c(0.4, 0.35, 0.25), loc = c(0.6, 0.6, 0.6), scale = c(0.9, 1.1, 1.4), shape1 = c(2.0, 2.6, 3.5), shape2 = c(1.0, 1.1, 1.2))
)

example <- amor_sets[[1]]

Code

dAmorosoMix(2, w = example$w, loc = example$loc, scale = example$scale,
            shape1 = example$shape1, shape2 = example$shape2)

[1] 0.3047048

Code

dAmorosoMix(2, w = example$w, loc = example$loc, scale = example$scale,
            shape1 = example$shape1, shape2 = example$shape2, log = TRUE)

[1] -1.188412

Code

pAmorosoMix(2, w = example$w, loc = example$loc, scale = example$scale,
            shape1 = example$shape1, shape2 = example$shape2)

[1] 0.2402211

Code

pAmorosoMix(2, w = example$w, loc = example$loc, scale = example$scale,
            shape1 = example$shape1, shape2 = example$shape2, lower.tail = FALSE)

[1] 0.7597789

Code

pAmorosoMix(2, w = example$w, loc = example$loc, scale = example$scale,
            shape1 = example$shape1, shape2 = example$shape2, log.p = TRUE)

[1] -1.426196

Code

q_vec(qAmorosoMix, c(0.25, 0.5, 0.75), w = example$w, loc = example$loc,
      scale = example$scale, shape1 = example$shape1, shape2 = example$shape2)

[1] 2.031993 2.858225 3.993931

Code

q_vec(qAmorosoMix, c(0.25, 0.5, 0.75), w = example$w, loc = example$loc,
      scale = example$scale, shape1 = example$shape1, shape2 = example$shape2,
      lower.tail = FALSE)

[1] 3.993931 2.858225 2.031993

Code

q_vec(qAmorosoMix, c(log(0.25), log(0.5), log(0.75)), w = example$w,
      loc = example$loc, scale = example$scale, shape1 = example$shape1,
      shape2 = example$shape2, log.p = TRUE)

[1] 2.031993 2.858225 3.993931

Code

draw_many(rAmorosoMix, list(w = example$w, loc = example$loc,
                            scale = example$scale, shape1 = example$shape1,
                            shape2 = example$shape2))

[1] 2.082904 4.153333 4.064689 6.122636 1.683525

Code

df_amor <- do.call(rbind, lapply(amor_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dAmorosoMix, list(w = ps$w, loc = ps$loc, scale = ps$scale, shape1 = ps$shape1, shape2 = ps$shape2)), label = ps$label)
}))

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

With GPD tail

Code

amor_gpd_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.3, 0.1), loc = c(0.5, 0.5, 0.5), scale = c(1.0, 1.3, 1.6), shape1 = c(2.5, 3.0, 4.0), shape2 = c(1.2, 1.2, 1.2), threshold = 2.8, tail_scale = 0.4, tail_shape = 0.2),
  list(label = "Mix B", w = c(0.5, 0.3, 0.2), loc = c(0.4, 0.6, 0.6), scale = c(1.1, 1.2, 1.5), shape1 = c(2.2, 2.8, 3.8), shape2 = c(1.1, 1.2, 1.3), threshold = 3.0, tail_scale = 0.35, tail_shape = 0.18),
  list(label = "Mix C", w = c(0.4, 0.35, 0.25), loc = c(0.6, 0.6, 0.6), scale = c(0.9, 1.1, 1.4), shape1 = c(2.0, 2.6, 3.5), shape2 = c(1.0, 1.1, 1.2), threshold = 2.5, tail_scale = 0.45, tail_shape = 0.22)
)
example <- amor_gpd_sets[[1]]

Code

dAmorosoMixGpd(3, w = example$w, loc = example$loc, scale = example$scale,
               shape1 = example$shape1, shape2 = example$shape2,
               threshold = example$threshold, tail_scale = example$tail_scale,
               tail_shape = example$tail_shape)

[1] 0.7287094

Code

dAmorosoMixGpd(3, w = example$w, loc = example$loc, scale = example$scale,
               shape1 = example$shape1, shape2 = example$shape2,
               threshold = example$threshold, tail_scale = example$tail_scale,
               tail_shape = example$tail_shape, log = TRUE)

[1] -0.3164802

Code

pAmorosoMixGpd(3, w = example$w, loc = example$loc, scale = example$scale,
               shape1 = example$shape1, shape2 = example$shape2,
               threshold = example$threshold, tail_scale = example$tail_scale,
               tail_shape = example$tail_shape)

[1] 0.6793679

Code

pAmorosoMixGpd(3, w = example$w, loc = example$loc, scale = example$scale,
               shape1 = example$shape1, shape2 = example$shape2,
               threshold = example$threshold, tail_scale = example$tail_scale,
               tail_shape = example$tail_shape, lower.tail = FALSE)

[1] 0.3206321

Code

pAmorosoMixGpd(3, w = example$w, loc = example$loc, scale = example$scale,
               shape1 = example$shape1, shape2 = example$shape2,
               threshold = example$threshold, tail_scale = example$tail_scale,
               tail_shape = example$tail_shape, log.p = TRUE)

[1] -0.3865925

Code

q_vec(qAmorosoMixGpd, c(0.25, 0.5, 0.75), w = example$w, loc = example$loc,
      scale = example$scale, shape1 = example$shape1, shape2 = example$shape2,
      threshold = example$threshold, tail_scale = example$tail_scale,
      tail_shape = example$tail_shape)

[1] 2.031993 2.812937 3.112257

Code

q_vec(qAmorosoMixGpd, c(0.25, 0.5, 0.75), w = example$w, loc = example$loc,
      scale = example$scale, shape1 = example$shape1, shape2 = example$shape2,
      threshold = example$threshold, tail_scale = example$tail_scale,
      tail_shape = example$tail_shape, lower.tail = FALSE)

[1] 3.112257 2.812937 2.031993

Code

q_vec(qAmorosoMixGpd, c(log(0.25), log(0.5), log(0.75)), w = example$w,
      loc = example$loc, scale = example$scale, shape1 = example$shape1,
      shape2 = example$shape2, threshold = example$threshold,
      tail_scale = example$tail_scale, tail_shape = example$tail_shape,
      log.p = TRUE)

[1] 2.031993 2.812937 3.112257

Code

draw_many(rAmorosoMixGpd, example)

[1] 2.974135 3.442089 6.040809 6.227795 2.037521

Code

df_amor_gpd <- do.call(rbind, lapply(amor_gpd_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dAmorosoMixGpd, list(w = ps$w, loc = ps$loc, scale = ps$scale, shape1 = ps$shape1, shape2 = ps$shape2, threshold = ps$threshold, tail_scale = ps$tail_scale, tail_shape = ps$tail_shape)), label = ps$label)
}))

ggplot(df_amor_gpd, aes(x = x, y = density, color = label)) +
  geom_line(linewidth = 1) +
  labs(title = "Amoroso mixtures with GPD tail", 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
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