knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) library(pdqr) set.seed(102)
Family of as_*()
functions should be used to convert existing distribution functions into desired class ("p", "d", "q", or "r"). Roughly, this is a new_*()
family but with function as an input.
There are two main use cases:
Converting existing pdqr-function to desired type is done straightforwardly by changing function's class without touching the underlying distribution ("x_tbl" metadata is the same):
d_fin <- new_d(1:4, "discrete") meta_x_tbl(d_fin) # This is equivalent to `new_p(1:4, "discrete")` (p_fin <- as_p(d_fin)) meta_x_tbl(p_fin)
Another important use case for as_*()
functions is to convert some other distribution functions to be pdqr-functions. Except small number of special cases, output of as_*()
function will have "continuous" type. The reason is because identifying exact values of distribution in discrete case is very hard in this setup (when almost nothing is known about the input function). It is assumed that if user knows those values, some new_*()
function with data frame input can be used to create arbitrary "discrete" pdqr-function.
General conversion algorithm is as follows:
new_*()
function is created which approximates input function. Approximation precision can be tweaked with n_grid
(and n_sample
for as_r()
) argument: bigger values lead to better approximation precision, but worse memory usage and evaluation speed (direct and of summ_*()
functions).For input distribution function to be recognized as "honored", it should be supplied directly with its original name:
# "Honored" distributions as_d(dnorm) # Underlying distribution doesn't depend on class ("p", "d", "q", "r"). # Following code has the same effect as `as_r(as_d(dnorm))` as_r(rnorm) # Different picewise-linear approximation precision is achieved with different # `n_grid` argument value as_d(dnorm, n_grid = 101) # Different extra arguments for input as_d(dnorm, mean = 10, sd = 0.1) # Currently only five distributions result into "discrete" output of `as_*()` as_d(dbinom, size = 10, prob = 0.3) as_d(dgeom, prob = 0.3) as_d(dhyper, m = 10, n = 10, k = 7) as_d(dnbinom, size = 10, prob = 0.3) as_d(dpois, lambda = 1) # This isn't recognized as "honored", but output is very close to "honored" as_d(function(x) {dnorm(x)})
Support detection is implemented for more smooth user experience. For more details on algorithms behind it, see section "Support detection" in as_p()
documentation. Generally, if you know exactly what support should be, it is better to provide it.
my_d <- function(x) {ifelse(x >= -1 & x <= 1, 0.75 * (1 - x^2), 0)} # With default support detection as_d(my_d) # Providing custom, maybe only partially known, support as_d(my_d, support = c(-1, NA)) as_d(my_d, support = c(NA, 1)) as_d(my_d, support = c(-1, 1))
Here is a comparison of support detection performance. One important note here is that algorithm has random nature in as_r()
(which is reasonable because the only information available about distribution is its random generation function).
(p_norm <- as_p(function(x) {pnorm(x)})) (d_norm <- as_d(function(x) {dnorm(x)})) (q_norm <- as_q(function(x) {qnorm(x)})) (r_norm <- as_r(function(x) {rnorm(x)})) plot( as_d(p_norm), col = "black", main = "Comparison of `as_*()` functions support detection" ) lines(d_norm, col = "blue") lines(as_d(q_norm), col = "red") lines(as_d(r_norm), col = "green")
If for some point density function goes to infinity, it is imputed linearly from its neighborhood. For not "honored" distribution functions, it can be more robust to use as_p()
for initial conversion.
x_grid <- seq(0, 0.06, by = 1e-5) # "Honored" distribution plot( as_d(dchisq, df = 1), col = "black", xlim = c(0, 0.05), ylim = c(0, 20), main = "Infinity imputation for Chi-squared distribution" ) lines(x_grid, dchisq(x_grid, df = 1), col = "red") # Custom function plot( as_d(function(x) {-log(x)}, support = c(0, 1)), col = "black", xlim = c(0, 0.001), ylim = c(6, 12), main = "Infinity imputation for custom function" ) lines(x_grid, -log(x_grid), col = "red")
Note that output distribution is usually an approximation (albeit a reasonably good one) of input due to the following facts:
'pdqr' provides a diagnostic function pdqr_approx_error()
to look at the precision of approximation. It accepts a pdqr-function and original reference distribution function with its possible extra arguments. It constructs a grid that is more dense than "x" column in pdqr-function's "x_tbl" metadata (to actually test the precision of piecewise-linear nature). Output is a data frame with rows corresponding to that grid elements and columns with two kinds of errors: "error" (with direct, signed error as difference between values of reference function and pdqr-function) and "abserror" (with absolute error):
approx_err <- pdqr_approx_error(as_d(dnorm, sd = 2), dnorm, sd = 2) head(approx_err) summary(approx_err)
Here are estimation of median and maximum errors for most common "honored" distributions using default n_grid
value (tested for d-functions, but can be used also for p- and q-functions):
abserror_stat <- function(f, ref_f, ...) { approx_err <- pdqr_approx_error(f, ref_f, ...) c( median_abserror = median(approx_err[["abserror"]]), max_abserror = max(approx_err[["abserror"]]) ) } abserror_stat_fin <- function(f, ref_f, grid, ...) { abserror <- abs(f(grid) - ref_f(grid, ...)) c(median_abserror = median(abserror), max_abserror = max(abserror)) } # Normal abserror_stat(as_d(dnorm), dnorm) # Beta abserror_stat( as_d(dbeta, shape1 = 10, shape2 = 20), dbeta, shape1 = 10, shape2 = 20 ) # By default, `pdqr_approx_error()` removes infinity errors. As one can see, # when density goes to infinity, error can be quite big abserror_stat( as_d(dbeta, shape1 = 0.1, shape2 = 0.2), dbeta, shape1 = 0.1, shape2 = 0.2 ) # Exponential abserror_stat(as_d(dexp, rate = 10), dexp, rate = 10) # Student abserror_stat(as_d(dt, df = 5), dt, df = 5) # Cauchy. Heavy tails also affect approximation error abserror_stat(as_d(dcauchy), dcauchy) # Poisson. Pdqr-function isn't exact because of tail trimming. abserror_stat_fin(as_d(dpois, lambda = 10), dpois, grid = 0:30, lambda = 10) # For some distributions functions are exact # Uniform abserror_stat(as_d(dunif), dunif) # Binomial abserror_stat_fin( as_d(dbinom, size = 10, prob = 0.1), dbinom, grid = 0:10, size = 10, prob = 0.1 )
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