Cauchy

Cauchy base and mixtures

A Cauchy distribution with location (x_0) and scale (>0) has density [ f(yx_0,)=, y. ]

A finite Cauchy mixture is [ f(y)={j=1}^J w_j,(yx{0j},_j). ]

Parameter mapping (math () code): (x_0) location, () scale; in mixtures (x_{0j}) location[j], (_j) scale[j], (w_j) w[j].

Without GPD (base and mixture)

Code

grid <- seq(-8, 8, length.out = 400)
cauchy_sets <- list(
  list(label = "Base", location = 0, scale = 1),
  list(label = "Mix A", w = c(0.6, 0.3, 0.1), location = c(-1, 0, 1), scale = c(1.0, 1.2, 2)),
  list(label = "Mix B", w = c(0.5, 0.3, 0.2), location = c(-1.5, 0.5, 1.5), scale = c(1.1, 1.0, 0.9))
)

base_par <- cauchy_sets[[1]]
mix_par1 <- cauchy_sets[[2]]

Code

dCauchy(0, location = base_par$location, scale = base_par$scale)

[1] 0.3183099

Code

dCauchy(0, location = base_par$location, scale = base_par$scale, log = TRUE)

[1] -1.14473

Code

pCauchy(0, location = base_par$location, scale = base_par$scale)

[1] 0.5

Code

pCauchy(0, location = base_par$location, scale = base_par$scale, lower.tail = FALSE)

[1] 0.5

Code

pCauchy(0, location = base_par$location, scale = base_par$scale, log.p = TRUE)

[1] -0.6931472

Code

q_vec(qCauchy, c(0.25, 0.5, 0.75), location = base_par$location,
      scale = base_par$scale)

[1] -1  0  1

Code

q_vec(qCauchy, c(0.25, 0.5, 0.75), location = base_par$location,
      scale = base_par$scale, lower.tail = FALSE)

[1]  1  0 -1

Code

q_vec(qCauchy, c(log(0.25), log(0.5), log(0.75)), location = base_par$location,
      scale = base_par$scale, log.p = TRUE)

[1] -1  0  1

Code

draw_many(rCauchy, list(location = base_par$location, scale = base_par$scale))

[1] -0.9070131 -0.4248394  0.2329576  3.3710605 -1.3611982

Code

dCauchyMix(0, w = mix_par1$w, location = mix_par1$location, scale = mix_par1$scale)

[1] 0.1878028

Code

dCauchyMix(0, w = mix_par1$w, location = mix_par1$location, scale = mix_par1$scale, log = TRUE)

[1] -1.672363

Code

draw_many(rCauchyMix, list(w = mix_par1$w, location = mix_par1$location, scale = mix_par1$scale))

[1]  6.8345157  0.5153257 -2.3234165 -0.3339875  0.1331388

Code

df_cauchy <- rbind(
  data.frame(x = grid, density = density_curve(grid, dCauchy, list(location = base_par$location, scale = base_par$scale)), label = "Base"),
  data.frame(x = grid, density = density_curve(grid, dCauchyMix, list(w = mix_par1$w, location = mix_par1$location, scale = mix_par1$scale)), label = "Mix A"),
  data.frame(x = grid, density = density_curve(grid, dCauchyMix, list(w = cauchy_sets[[3]]$w, location = cauchy_sets[[3]]$location, scale = cauchy_sets[[3]]$scale)), label = "Mix B")
)

ggplot(df_cauchy, aes(x = x, y = density, color = label)) +
  geom_line(linewidth = 1) +
  labs(title = "Cauchy base and mixtures", 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.

Cauchy

Cauchy base and mixtures

Without GPD (base and mixture)

Prereqs

Outputs

Interpretation

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