Generating pseudo-random observations from a probability distribution is a common task in statistics. Being able to generate pseudo-random observations from a probability distribution is useful for simulating scenarios, in Monte-Carlo methods, which are useful for evaluating various statistical models.
The acceptance-rejection method was developed by John von Neumann in 1951 and is a well-known technique present in various computational statistics books. The original reference can be found at the link below:
The inversion method is a common way to do this, but it is not always
possible to find a closed-form formula for the inverse function of the
cumulative distribution function of a random variable
Whenever possible, it is preferable to use the inversion method to generate pseudo-random observations from a probability distribution. However, when it is not possible to find a closed-form formula for the inverse function of the cumulative distribution function of a random variable, it is necessary to resort to other methods. One way to do this is through the acceptance-rejection method, which is a Monte-Carlo procedure. This package aims to provide a function that implements the Acceptance and Rejection method for generating pseudo-random observations from probability distributions that are difficult to sample directly.
The package AcceptReject
provides the
accept_reject()
function, in addition to other functions, that implements the
acceptance-rejection method in an optimized manner to generate
pseudo-random observations for discrete or continuous random variables.
The
accept_reject()
function operates in parallel on Unix-based operating systems such as
Linux and MacOS and operates sequentially on Windows-based operating
systems; however, it still exhibits good performance. By default, on
Unix-based systems, observations are generated sequentially, but it is
possible to generate observations in parallel if desired, by using the
parallel = TRUE argument.
The
accept_reject()
function, by default, attempts to maximize the probability of acceptance
of the pseudo-random observations generated. Suppose
By default, the accept_reject function attempts to find the value of accept_reject()
function through the argument c, where accept_reject()
function, it is not necessary to specify the probability function or
probability density function of
Since the probability of acceptance is accept_reject()
function attempts to find the minimum value of c argument of the
accept_reject()
function, it is recommended to use c = NULL (default), allowing a
value of
The package is being versioned on GitHub. You can install the development version of AcceptReject, and to do this, you must first install the remotes package and then run the following command:
# install.packages("remotes")
# or remotes::install_github("prdm0/AcceptReject", force = TRUE)
library(AcceptReject)The force = TRUE argument is not necessary. It is only needed in
situations where you have already installed the package and want to
reinstall it to have a new version.
You can also install the package version available on Comprehensive R Archive Network - CRAN:
install.packages("AcceptReject")Please note the examples below on how to use the
accept_reject()
function to generate pseudo-random observations of discrete and
continuous random variables. For further details, refer to the
function’s documentation
Reference and the
Vignette.
In the examples, we use well-known distributions, but it’s important to
note that the accept-reject method can be used to generate observations
from probability distributions for which we don’t know how to generate
observations directly. Thus, if you need to generate observations from a
random variable with a distribution for which there are no built-in
functions in R that you can use to generate observations (such as
functions like rnorm(), rgamma(), rweibull(), rchisq(), among
others), the AcceptReject
package might be what you need. 🎉
As an example, let accept_reject() function. Note that it is necessary to
provide the xlim argument. Try to set an upper limit value for which
the probability of xlim = c(0, 6), where
dpois(x = 6, lambda = 0.7) is very close to zero (8.1142728^{-5}).
library(AcceptReject)
#>
#> Anexando pacote: 'AcceptReject'
#> O seguinte objeto é mascarado por 'package:stats':
#>
#> qqplot
library(cowplot) # install.packages("cowplot")
# Ensuring Reproducibility
set.seed(0)
simulation <- function(n){
AcceptReject::accept_reject(
n = n,
f = dpois,
continuous = FALSE,
args_f = list(lambda = 0.7),
xlim = c(0, 6),
parallel = TRUE
)
}
a <- simulation(25L)
b <- simulation(250L)
c <- simulation(2500L)
d <- simulation(25000L)
# Plots
p1 <- plot(a)
p2 <- plot(b)
p3 <- plot(c)
p4 <- plot(d)
plot_grid(p1, p2, p3, p4, nrow = 2L, labels = c("a", "b", "c", "d"))
# QQ-Plots
q1 <- qqplot(a, size_points = 2)
q2 <- qqplot(b, size_points = 2)
q3 <- qqplot(c, size_points = 2)
q4 <- qqplot(d, size_points = 2)
plot_grid(q1, q2, q3, q4, nrow = 2L, labels = c("a", "b", "c", "d"))
To expand beyond examples of generating pseudo-random observations of
discrete random variables, consider now that we want to generate
observations from a random variable
library(AcceptReject)
library(cowplot) # install.packages("cowplot")
# Ensuring reproducibility
set.seed(0)
simulation <- function(n){
AcceptReject::accept_reject(
n = n,
f = dnorm,
continuous = TRUE,
args_f = list(mean = 0, sd = 1),
xlim = c(-4, 4),
parallel = TRUE
)
}
a <- simulation(n = 100L)
b <- simulation(n = 150L)
c <- simulation(n = 250L)
d <- simulation(n = 2500L)
# Plots
p1 <- plot(a)
p2 <- plot(b)
p3 <- plot(c)
p4 <- plot(d)
plot_grid(p1, p2, p3, p4, nrow = 2L, labels = c("a", "b", "c", "d"))
# QQ-plots
q1 <- qqplot(a)
q2 <- qqplot(b)
q3 <- qqplot(c)
q4 <- qqplot(d)
plot_grid(q1, q2, q3, q4, nrow = 2L, labels = c("a", "b", "c", "d"))
The
accept_reject()
function supports, for the continuous case, specifying a base
probability density function if you don’t want to use the continuous
uniform distribution as the default base.
When choosing to specify another probability density function different from the uniform one, it’s necessary to specify the following arguments:
f_base: base probability density function;random_base: sampling from the base probability density function;args_f_base: list with the parameters of the base density.
By default, all of them are NULL, and the continuous uniform
distribution in xlim is used as the base. If at least one of these
arguments is not specified, no error will occur, and the continuous
uniform distribution in xlim will still be used as the base.
For the discrete case, if the user mistakenly specifies any of these
arguments, i.e., when continuous = FALSE, the
accept_reject()
function will ignore these arguments and use the discrete uniform
distribution as the base.
If you choose to specify a base density, it’s convenient to inspect it
by comparing the base density function with the theoretical probability
density function. The
inspect()
function facilitates this task. The
inspect()
function will plot the base probability density function and the
theoretical probability density function, find the intersection between
the densities, and display the value of the intersection area on the
plot. These are important pieces of information to decide if the base
probability density function specified in the args_f_base argument and
the value of c (default is 1) are appropriate.
Another example, considering
library(AcceptReject)
library(cowplot)
# Ensuring reproducibility
set.seed(0)
x <- accept_reject(
n = 100000L,
f = dbeta,
continuous = TRUE,
args_f = list(shape1 = 2, shape2 = 2),
xlim = c(0, 1)
)
print(x)
#>
#> ── Accept-Reject Samples ───────────────────────────────────────────────────────
#> ℹ It's not necessary, but if you want to extract the observations, use as.vector().
#> ✔ Case: continuous
#> ✔ Number of observations: 100000
#> ✔ c: 1.5
#> ✔ Probability of acceptance (1/c): 0.6667
#> ✔ Observations: 0.8967 0.2655 0.3721 0.5729 0.6608 0.6291 0.206 0.1766 0.687 0.3841...
#> ✔ xlim = 0 1
#>
#> ────────────────────────────────────────────────────────────────────────────────
a <- plot(x)
b <- qqplot(x)
plot_grid(a, b, nrow = 2L, labels = c("a", "b"))
# Inspecting the estimated value of c by accept_reject()
# Note that c = 1.5 indeed causes the base density,
# which in this case was the default (uniform), to overlap the # density of the beta:
inspect(
f = dbeta,
args_f = list(shape1 = 2, shape2 = 2),
f_base = dunif,
args_f_base = c(min = 0, max = 1),
xlim = c(0, 1),
c = attr(x, "c")
)
Note that c is an attribute of x. To access all attributes, you can
use attributes(x). Most likely, you won’t need to access these
attributes. They are useful for plotting methods.
library(AcceptReject)
library(cowplot) # install.packages("cowplot")
# Ensuring reproducibility
set.seed(0)
# Inspecting
# Case a
a <- inspect(
f = dweibull,
args_f = list(shape = 2.1, scale = 2.2),
f_base = dgamma,
args_f_base = list(shape = 2.8, rate = 1.2),
xlim = c(0, 6),
c = 1.2
)
# Inspecting
# Case b
b <- inspect(
f = dweibull,
args_f = list(shape = 2.1, scale = 2.2),
f_base = dgamma,
args_f_base = list(shape = 2.9, rate = 2.5),
xlim = c(0, 6),
c = 1.4
)
plot_grid(a, b, nrow = 2L, labels = c("a", "b"))
Notice that considering the distribution in scenario “a” in the code
above is more convenient. Note that the area is approximately 1, the
base probability density function with parameters shape = 2.8 and
rate = 1.2 provides a shape close to the theoretical distribution, and
c = 1.2 ensures that the base density function upper bounds the
theoretical probability density function. Therefore, considering
f_base with c = 1.2 is a
reasonable choice for a base distribution.
Therefore, passing arguments to f_base = dgamma,
args_f_base = list(shape = 2.8, rate = 1.2), and c = 1.2 to the
accept_reject()
function will lead us to an even more efficient code.
library(AcceptReject)
library(tictoc) # install.packages("tictoc")
# Ensuring reproducibility
set.seed(0)
# Not specifying the base probability density function
tic()
case_1 <- accept_reject(
n = 2000,
continuous = TRUE,
f = dweibull,
args_f = list(shape = 2.1, scale = 2.2),
xlim = c(0, 6)
)
toc()
#> 0.005 sec elapsed
# Specifying the base probability density function
tic()
case_2 <- accept_reject(
n = 2000,
continuous = TRUE,
f = dweibull,
args_f = list(shape = 2.1, scale = 2.2),
f_base = dgamma,
random_base = rgamma,
args_f_base = list(shape = 2.8, rate = 1.2),
xlim = c(0, 6),
c = 1.2
)
toc()
#> 0.005 sec elapsed
# Visualizing the results
p1 <- plot(case_1)
p2 <- plot(case_2)
plot_grid(p1, p2, nrow = 2L)
# QQ-plot
q1 <- qqplot(case_1)
q2 <- qqplot(case_2)
plot_grid(q1, q2, nrow = 1L)
Notice that the results were very close in a graphical analysis. However, the execution time specifying a convenient base density was lower for a very large sample.