, Cheng Ding [aut],
Juan Estrada [aut] ,
Zhilang Xia [aut], Shanjie Zhang [ctb], Jianming Jin [ctb]| Title: | Generalized Inverse Normal Distribution Density and Generation |
|---|---|
| Description: | Density function and generation of random variables from the Generalized Inverse Normal (GIN) distribution from Robert (1991) <doi:10.1016/0167-7152(91)90174-P>. Also provides density functions and generation from the GIN distribution truncated to positive or negative reals. Theoretical guarantees supporting the sampling algorithms and an application to Bayesian estimation of network formation models can be found in the working paper Ding, Estrada and Montoya-Blandón (2023) <https://www.smontoyablandon.com/publication/networks/network_externalities.pdf>. |
| Authors: | Santiago Montoya-Blandón [cre, aut]
, Cheng Ding [aut],
Juan Estrada [aut] ,
Zhilang Xia [aut], Shanjie Zhang [ctb], Jianming Jin [ctb] |
| Maintainer: | Santiago Montoya-Blandón <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 0.0.2 |
| Built: | 2025-03-11 03:29:07 UTC |
| Source: | https://github.com/cran/ginormal |
Density for the generalized inverse normal distribution
dgin(z, alpha, mu, tau, log = TRUE, quasi = FALSE)dgin(z, alpha, mu, tau, log = TRUE, quasi = FALSE)
z |
quantile. |
alpha |
degrees-of-freedom parameter. |
mu |
similar to location parameter, controls asymmetry of the distribution. |
tau |
similar to scale parameter, controls spread of the distribution. |
log |
logical; should the log of the density be returned? Defaults to TRUE. |
quasi |
logical; should the quasi-density value be returned? Defaults to FALSE. |
Currently, only scalars are supported for the quantile and parameter values.
Density is supported on the entire real line, z and mu can take any value
in . Density is only defined for parameter values
alpha > 1 or tau > 0, so it is set to 0 outside of these values.
The quasi-density or kernel is the density without the normalization constant,
use quasi = TRUE for this behavior.
Numeric scalar with density.
# Computing (log) density dgin(z = 1, alpha = 3, mu = 1, tau = 1, log = TRUE, quasi = FALSE) # Generalized inverse normal density with alpha = 5, mu = 0, tau = 1 z_vals <- seq(-5, 5, length.out = 200) fz_unc <- sapply(z_vals, function(z) dgin(z, 5, 0, 1, FALSE)) plot(z_vals, fz_unc, type = "l", xlab = 'Values', ylab = 'Density')# Computing (log) density dgin(z = 1, alpha = 3, mu = 1, tau = 1, log = TRUE, quasi = FALSE) # Generalized inverse normal density with alpha = 5, mu = 0, tau = 1 z_vals <- seq(-5, 5, length.out = 200) fz_unc <- sapply(z_vals, function(z) dgin(z, 5, 0, 1, FALSE)) plot(z_vals, fz_unc, type = "l", xlab = 'Values', ylab = 'Density')
Density for the generalized inverse normal distribution truncated to the positive or negative reals
dtgin( z, alpha, mu, tau, sign = TRUE, log = TRUE, quasi = FALSE, method = "Fortran" )dtgin( z, alpha, mu, tau, sign = TRUE, log = TRUE, quasi = FALSE, method = "Fortran" )
z |
quantile. |
alpha |
degrees-of-freedom parameter. |
mu |
similar to location parameter, controls asymmetry of the distribution. |
tau |
similar to scale parameter, controls spread of the distribution. |
sign |
logical. |
log |
logical; should the log of the density be returned? Defaults to TRUE. |
quasi |
logical; should the quasi-density value be returned? Defaults to FALSE. |
method |
string with the method used to compute the parabolic cylinder function
in the normalization constant. |
Currently, only scalars are supported for the quantile and parameter values.
Density is supported on the positive reals (z > 0) when sign = TRUE and to
negative reals (z < 0) when sign = FALSE. mu can take any value
in . Density is only defined for parameter values
alpha > 1 or tau > 0, so it is set to 0 outside of these values.
The quasi-density or kernel is the density without the normalization constant,
use quasi = TRUE for this behavior.
Numeric scalar with density.
# Computing (log) truncated densities dtgin(z = 1, alpha = 3, mu = 1, tau = 1, sign = TRUE, log = TRUE, quasi = FALSE) dtgin(z = -1, alpha = 3, mu = -1, tau = 1, sign = FALSE, log = TRUE, quasi = FALSE) # Generalized inverse normal density with alpha = 5, mu = 0, tau = 1 n_values <- 200 z_vals <- seq(-5, 5, length.out = n_values) # Truncated to positive reals (z > 0) fz_p <- sapply(z_vals[z_vals > 0], function(z) dtgin(z, 5, 0, 1, TRUE, FALSE)) fz_p <- c(rep(0, n_values - sum(z_vals > 0)), fz_p) plot(z_vals, fz_p, type = "l", xlab = 'Values', ylab = 'Density') # Truncated to positive reals (z < 0) fz_n <- sapply(z_vals[z_vals < 0], function(z) dtgin(z, 5, 0, 1, FALSE, FALSE)) fz_n <- c(fz_n, rep(0, n_values - sum(z_vals < 0))) plot(z_vals, fz_n, type = "l", xlab = 'Values', ylab = 'Density') # Both truncated densities together plot(z_vals, fz_p, type = "l", xlab = 'Values', ylab = 'Density') lines(z_vals, fz_n, col = 'blue', lty = 2) legend('topright', legend = c('z > 0', 'z < 0'), col = c('black', 'blue'), lty = 1:2)# Computing (log) truncated densities dtgin(z = 1, alpha = 3, mu = 1, tau = 1, sign = TRUE, log = TRUE, quasi = FALSE) dtgin(z = -1, alpha = 3, mu = -1, tau = 1, sign = FALSE, log = TRUE, quasi = FALSE) # Generalized inverse normal density with alpha = 5, mu = 0, tau = 1 n_values <- 200 z_vals <- seq(-5, 5, length.out = n_values) # Truncated to positive reals (z > 0) fz_p <- sapply(z_vals[z_vals > 0], function(z) dtgin(z, 5, 0, 1, TRUE, FALSE)) fz_p <- c(rep(0, n_values - sum(z_vals > 0)), fz_p) plot(z_vals, fz_p, type = "l", xlab = 'Values', ylab = 'Density') # Truncated to positive reals (z < 0) fz_n <- sapply(z_vals[z_vals < 0], function(z) dtgin(z, 5, 0, 1, FALSE, FALSE)) fz_n <- c(fz_n, rep(0, n_values - sum(z_vals < 0))) plot(z_vals, fz_n, type = "l", xlab = 'Values', ylab = 'Density') # Both truncated densities together plot(z_vals, fz_p, type = "l", xlab = 'Values', ylab = 'Density') lines(z_vals, fz_n, col = 'blue', lty = 2) legend('topright', legend = c('z > 0', 'z < 0'), col = c('black', 'blue'), lty = 1:2)
Generating random numbers from the generalized inverse normal distribution
rgin(size, alpha, mu, tau, algo = "hormann", method = "Fortran")rgin(size, alpha, mu, tau, algo = "hormann", method = "Fortran")
size |
number of desired draws. Output is numpy vector of length equal to size. |
alpha |
degrees-of-freedom parameter. |
mu |
similar to location parameter, controls asymmetry of the distribution. |
tau |
similar to scale parameter, controls spread of the distribution. |
algo |
string with desired algorithm to compute minimal bounding rectangle. If "hormann", use the method from Hörmann and Leydold (2014). When "leydold", use the one from Leydold (2001). Defaults to "hormann" and returns an error for any other values. |
method |
string with the method used to compute the parabolic cylinder function
in the normalization constant. |
Currently, only values of alpha > 2 are supported. For Bayesian posterior sampling,
alpha is always larger than 2 even for non-informative priors. The algorithm requires
calculating the probability of truncation region (either z < 0 or z > 0).
It is more stable to compute a probability bounded away from 0. As mu controls asymmetry,
when mu > 0, P(truncation region) = P(z > 0) >= 50%, and this probability is computed.
If mu < 0, P(z < 0) >= 50% and this region's probability is used.
Numeric vector of length size.
# Generate 1000 values from the distribution with alpha = 5, mu = 0, tau = 1 set.seed(123456) z_unc <- rgin(1000, 5, 0, 1) # Compare histogram to true density z_vals <- seq(-5, 5, length.out = 200) fz_unc <- sapply(z_vals, function(z) dgin(z, 5, 0, 1, FALSE)) temp <- hist(z_unc, breaks = 200, plot = FALSE) plot(temp, freq = FALSE, xlim = c(-5, 5), ylim = range(c(fz_unc, temp$density)), main = '', xlab = 'Values', ylab = 'Density', col = 'blue') lines(z_vals, fz_unc, col = 'red', lwd = 2)# Generate 1000 values from the distribution with alpha = 5, mu = 0, tau = 1 set.seed(123456) z_unc <- rgin(1000, 5, 0, 1) # Compare histogram to true density z_vals <- seq(-5, 5, length.out = 200) fz_unc <- sapply(z_vals, function(z) dgin(z, 5, 0, 1, FALSE)) temp <- hist(z_unc, breaks = 200, plot = FALSE) plot(temp, freq = FALSE, xlim = c(-5, 5), ylim = range(c(fz_unc, temp$density)), main = '', xlab = 'Values', ylab = 'Density', col = 'blue') lines(z_vals, fz_unc, col = 'red', lwd = 2)
Generating random numbers from the generalized inverse normal distribution truncated to the positive or negative reals
rtgin( size, alpha, mu, tau, sign, algo = "hormann", method = "Fortran", verbose = FALSE )rtgin( size, alpha, mu, tau, sign, algo = "hormann", method = "Fortran", verbose = FALSE )
size |
number of desired draws. Output is numpy vector of length equal to size. |
alpha |
degrees-of-freedom parameter. |
mu |
similar to location parameter, controls asymmetry of the distribution. |
tau |
similar to scale parameter, controls spread of the distribution. |
sign |
logical. |
algo |
string with desired algorithm to compute minimal bounding rectangle. If "hormann", use the method from Hörmann and Leydold (2014). When "leydold", use the one from Leydold (2001). Defaults to "hormann" and returns an error for any other values. |
method |
string with the method used to compute the parabolic cylinder function
in the normalization constant. |
verbose |
logical; should the acceptance rate from the ratio-of-uniforms method be provided along with additional information? Defaults to FALSE. |
Currently, only values of alpha > 2 are supported. For Bayesian posterior sampling,
alpha is always larger than 2 even for non-informative priors.
Generate from positive region (z > 0) hen sign = TRUE, and from
negative region (z < 0) when sign = FALSE. When verbose = TRUE,
a list is returned containing the actual draw in value, as well as average
acceptance rate avg_arate and total number of acceptance-rejection steps ARiters.
If verbose = FALSE (default), a numeric vector of length size.
Otherwise, a list with components value, avg_arate, and ARiters
# Generate 1000 values from the truncated distributions with alpha = 5, mu = 0, tau = 1 set.seed(123456) n_draws <- 1000 z_p <- rtgin(n_draws, 5, 0, 1, TRUE) z_n <- rtgin(n_draws, 5, 0, 1, FALSE) # Compare generation from truncation to positive reals with true density n_values <- 200 z_vals <- seq(-5, 5, length.out = n_values) fz_p <- sapply(z_vals[z_vals > 0], function(z) dtgin(z, 5, 0, 1, TRUE, FALSE)) fz_p <- c(rep(0, n_values - sum(z_vals > 0)), fz_p) temp <- hist(z_p, breaks = 100, plot = FALSE) plot(temp, freq = FALSE, xlim = c(-5, 5), ylim = range(c(fz_p, temp$density)), main = '', xlab = 'Values', ylab = 'Density', col = 'blue') lines(z_vals, fz_p, col = 'red', lwd = 2) # Compare generation from truncation to negative reals with true density fz_n <- sapply(z_vals[z_vals < 0], function(z) dtgin(z, 5, 0, 1, FALSE, FALSE)) fz_n <- c(fz_n, rep(0, n_values - sum(z_vals < 0))) temp <- hist(z_n, breaks = 100, plot = FALSE) plot(temp, freq = FALSE, xlim = c(-5, 5), ylim = range(c(fz_n, temp$density)), main = '', xlab = 'Values', ylab = 'Density', col = 'blue') lines(z_vals, fz_n, col = 'red', lwd = 2) # verbose = TRUE provides info on the acceptance rate of the # ratio-of-uniforms acceptance-rejection method for sampling the variables draw_list <- rtgin(50, 5, 0, 1, sign = TRUE, verbose = TRUE) draw_list$ARiters # Acceptance-Rejection iterations draw_list$avg_arate # Average of 1/ARiters# Generate 1000 values from the truncated distributions with alpha = 5, mu = 0, tau = 1 set.seed(123456) n_draws <- 1000 z_p <- rtgin(n_draws, 5, 0, 1, TRUE) z_n <- rtgin(n_draws, 5, 0, 1, FALSE) # Compare generation from truncation to positive reals with true density n_values <- 200 z_vals <- seq(-5, 5, length.out = n_values) fz_p <- sapply(z_vals[z_vals > 0], function(z) dtgin(z, 5, 0, 1, TRUE, FALSE)) fz_p <- c(rep(0, n_values - sum(z_vals > 0)), fz_p) temp <- hist(z_p, breaks = 100, plot = FALSE) plot(temp, freq = FALSE, xlim = c(-5, 5), ylim = range(c(fz_p, temp$density)), main = '', xlab = 'Values', ylab = 'Density', col = 'blue') lines(z_vals, fz_p, col = 'red', lwd = 2) # Compare generation from truncation to negative reals with true density fz_n <- sapply(z_vals[z_vals < 0], function(z) dtgin(z, 5, 0, 1, FALSE, FALSE)) fz_n <- c(fz_n, rep(0, n_values - sum(z_vals < 0))) temp <- hist(z_n, breaks = 100, plot = FALSE) plot(temp, freq = FALSE, xlim = c(-5, 5), ylim = range(c(fz_n, temp$density)), main = '', xlab = 'Values', ylab = 'Density', col = 'blue') lines(z_vals, fz_n, col = 'red', lwd = 2) # verbose = TRUE provides info on the acceptance rate of the # ratio-of-uniforms acceptance-rejection method for sampling the variables draw_list <- rtgin(50, 5, 0, 1, sign = TRUE, verbose = TRUE) draw_list$ARiters # Acceptance-Rejection iterations draw_list$avg_arate # Average of 1/ARiters