Gamma

Gamma mixture kernel

A gamma component with shape (>0) and scale (>0) has density [ f(y,) = ,y^{}!(-), y>0. ]

A finite gamma mixture with (J) components is [ f(y)=_{j=1}^J w_j,f_j(y_j,j), w_j, {j=1}^J w_j=1. ]

Parameter mapping (math () code): () shape, () scale, and weights (w_j) w[j].

The *MixGpd variant uses the same splicing idea: gamma mixture below (u) and a GPD tail above (u).

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

Without GPD (mixture kernel)

Code

grid <- seq(0, 10, length.out = 400)
gamma_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.3, 0.1), shape = c(2.0, 5.0, 9.0), scale = c(1.0, 0.6, 0.3)),
  list(label = "Mix B", w = c(0.5, 0.3, 0.2), shape = c(1.5, 4.0, 7.0), scale = c(1.2, 0.7, 0.35)),
  list(label = "Mix C", w = c(0.4, 0.3, 0.3), shape = c(1.2, 3.5, 6.0), scale = c(1.4, 0.75, 0.4))
)
example <- gamma_sets[[1]]

dens <- do.call(rbind, lapply(gamma_sets, function(s) {
  data.frame(
    x = grid,
    density = dGammaMix(grid, w = s$w, shape = s$shape, scale = s$scale),
    label = s$label
  )
}))

Code

dGammaMix(2, w = example$w, shape = example$shape, scale = example$scale)

[1] 0.2952082

Code

dGammaMix(2, w = example$w, shape = example$shape, scale = example$scale, log = TRUE)

[1] -1.220074

Code

pGammaMix(2, w = example$w, shape = example$shape, scale = example$scale)

[1] 0.452308

Code

pGammaMix(2, w = example$w, shape = example$shape, scale = example$scale, lower.tail = FALSE)

[1] 0.547692

Code

pGammaMix(2, w = example$w, shape = example$shape, scale = example$scale, log.p = TRUE)

[1] -0.7933919

Code

qGammaMix(0.95, w = example$w, shape = example$shape, scale = example$scale)

[1] 5.008484

Code

qGammaMix(0.95, w = example$w, shape = example$shape, scale = example$scale, lower.tail = FALSE)

[1] 0.4747765

Code

df_gamma <- do.call(rbind, lapply(gamma_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dGammaMix, list(w = ps$w, shape = ps$shape, scale = ps$scale)), label = ps$label)
}))

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

Gamma mixture with GPD tail

Code

u <- 6
tail_scale <- 1.0
tail_shape <- 0.2

Code

dGammaMixGpd(6.5, w = example$w, shape = example$shape, scale = example$scale,
             threshold = u, tail_scale = tail_scale, tail_shape = tail_shape)

[1] 0.01094814

Code

dGammaMixGpd(6.5, w = example$w, shape = example$shape, scale = example$scale,
             threshold = u, tail_scale = tail_scale, tail_shape = tail_shape, log = TRUE)

[1] -4.514586

Code

pGammaMixGpd(6.5, w = example$w, shape = example$shape, scale = example$scale,
             threshold = u, tail_scale = tail_scale, tail_shape = tail_shape)

[1] 0.987957

Code

pGammaMixGpd(6.5, w = example$w, shape = example$shape, scale = example$scale,
             threshold = u, tail_scale = tail_scale, tail_shape = tail_shape, lower.tail = FALSE)

[1] 0.01204295

Code

qGammaMixGpd(0.95, w = example$w, shape = example$shape, scale = example$scale,
             threshold = u, tail_scale = tail_scale, tail_shape = tail_shape)

[1] 5.008484

Code

gamma_gpd_sets <- list(
  list(label = "Mix A", w = c(0.6, 0.4), shape = c(2.0, 5.0), scale = c(1.0, 0.6), threshold = 6.0, tail_scale = 1.0, tail_shape = 0.2),
  list(label = "Mix B", w = c(0.5, 0.5), shape = c(1.5, 4.0), scale = c(1.2, 0.7), threshold = 5.5, tail_scale = 0.8, tail_shape = 0.15),
  list(label = "Mix C", w = c(0.7, 0.3), shape = c(1.2, 3.5), scale = c(1.4, 0.75), threshold = 6.5, tail_scale = 1.2, tail_shape = 0.25)
)



df_gamma_gpd <- do.call(rbind, lapply(gamma_gpd_sets, function(ps) {
  data.frame(x = grid, density = density_curve(grid, dGammaMixGpd, list(w = ps$w, shape = ps$shape, scale = ps$scale, threshold = ps$threshold, tail_scale = ps$tail_scale, tail_shape = ps$tail_shape)), label = ps$label)
}))

ggplot(df_gamma_gpd, aes(x = x, y = density, color = label)) +
  geom_line(linewidth = 1) +
  labs(title = "Gamma 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
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Continue to the linked canonical concept page, then return for implementation-specific details.

Gamma

Gamma mixture kernel

Without GPD (mixture kernel)

Gamma mixture with GPD tail

Prereqs

Outputs

Interpretation

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