Package ‘pdqr’ supports two types of distributions:
Note that all distributions assume finite support (output values are bounded from below and above) and finite values of density function (density function in case of “continuous” type can’t go to infinity).
All new_*()
functions create a pdqr-function of certain type (“discrete” or “continuous”) based on sample or data frame of appropriate structure:
density()
function if input has at least 2 elements. For 1 element special “dirac-like” pdqr-function is created: an approximation single number with triangular distribution of very narrow support (1e-8 of magnitude). Basically, sample input is converted into data frame of appropriate structure that defines distribution (see next list item).We will use the following data frame inputs in examples:
# For type "discrete"
dis_df <- data.frame(x = 1:4, prob = 4:1 / 10)
# For type "continuous"
con_df <- data.frame(x = 1:4, y = c(0, 1, 1, 1))
This vignette is organized as follows:
density()
arguments” describes how to use density()
arguments to tweak smoothing during creation of “continuous” pdqr-functions.P-function (analogue of p*()
functions in base R) represents a cumulative distribution function of distribution.
# Treating input as discrete
p_mpg_dis <- new_p(mtcars$mpg, type = "discrete")
p_mpg_dis
#> Cumulative distribution function of discrete type
#> Support: [10.4, 33.9] (25 elements)
# Treating input as continuous
p_mpg_con <- new_p(mtcars$mpg, type = "continuous")
p_mpg_con
#> Cumulative distribution function of continuous type
#> Support: ~[2.96996, 41.33004] (511 intervals)
# Outputs are actually vectorized functions
p_mpg_dis(15:20)
#> [1] 0.18750 0.31250 0.34375 0.40625 0.46875 0.56250
p_mpg_con(15:20)
#> [1] 0.2185498 0.2804962 0.3465967 0.4143147 0.4818869 0.5478942
# You can plot them directly using base `plot()` and `lines()`
plot(p_mpg_con, main = "P-functions from sample")
lines(p_mpg_dis, col = "blue")
p_df_dis <- new_p(dis_df, type = "discrete")
p_df_dis
#> Cumulative distribution function of discrete type
#> Support: [1, 4] (4 elements)
p_df_con <- new_p(con_df, type = "continuous")
p_df_con
#> Cumulative distribution function of continuous type
#> Support: [1, 4] (3 intervals)
plot(p_df_con, main = "P-functions from data frame")
lines(p_df_dis, col = "blue")
D-function (analogue of d*()
functions in base R) represents a probability mass function for “discrete” type and density function for “continuous”:
# Treating input as discrete
d_mpg_dis <- new_d(mtcars$mpg, type = "discrete")
d_mpg_dis
#> Probability mass function of discrete type
#> Support: [10.4, 33.9] (25 elements)
# Treating input as continuous
d_mpg_con <- new_d(mtcars$mpg, type = "continuous")
d_mpg_con
#> Density function of continuous type
#> Support: ~[2.96996, 41.33004] (511 intervals)
# Outputs are actually vectorized functions
d_mpg_dis(15:20)
#> [1] 0.03125 0.00000 0.00000 0.00000 0.00000 0.00000
d_mpg_con(15:20)
#> [1] 0.05888171 0.06450605 0.06726441 0.06788664 0.06703401 0.06469730
# You can plot them directly using base `plot()` and `lines()`
op <- par(mfrow = c(1, 2))
plot(d_mpg_con, main = '"continuous" d-function\nfrom sample')
plot(d_mpg_dis, main = '"discrete" d-function\nfrom sample', col = "blue")
par(op)
d_df_dis <- new_d(dis_df, type = "discrete")
d_df_dis
#> Probability mass function of discrete type
#> Support: [1, 4] (4 elements)
d_df_con <- new_d(con_df, type = "continuous")
d_df_con
#> Density function of continuous type
#> Support: [1, 4] (3 intervals)
op <- par(mfrow = c(1, 2))
plot(d_df_con, main = '"continuous" d-function\nfrom data frame')
plot(d_df_dis, main = '"discrete" d-function\nfrom data frame', col = "blue")
par(op)
Q-function (analogue of q*()
functions in base R) represents a quantile function, an inverse of corresponding p-function:
# Treating input as discrete
q_mpg_dis <- new_q(mtcars$mpg, type = "discrete")
q_mpg_dis
#> Quantile function of discrete type
#> Support: [10.4, 33.9] (25 elements)
# Treating input as continuous
q_mpg_con <- new_q(mtcars$mpg, type = "continuous")
q_mpg_con
#> Quantile function of continuous type
#> Support: ~[2.96996, 41.33004] (511 intervals)
# Outputs are actually vectorized functions
q_mpg_dis(c(0.1, 0.3, 0.7, 1.5))
#> [1] 14.3 15.8 21.5 NaN
q_mpg_con(c(0.1, 0.3, 0.7, 1.5))
#> [1] 12.53278 16.29969 22.62140 NaN
# You can plot them directly using base `plot()` and `lines()`
plot(q_mpg_con, main = "Q-functions from sample")
lines(q_mpg_dis, col = "blue")
q_df_dis <- new_q(dis_df, type = "discrete")
q_df_dis
#> Quantile function of discrete type
#> Support: [1, 4] (4 elements)
q_df_con <- new_q(con_df, type = "continuous")
q_df_con
#> Quantile function of continuous type
#> Support: [1, 4] (3 intervals)
plot(q_df_con, main = "Q-functions from data frame")
lines(q_df_dis, col = "blue")
R-function (analogue of r*()
functions in base R) represents a random generation function. For “discrete” type it will generate only values present in input. For “continuous” function it will generate values from distribution corresponding to one estimated with density()
.
# Treating input as discrete
r_mpg_dis <- new_r(mtcars$mpg, type = "discrete")
r_mpg_dis
#> Random generation function of discrete type
#> Support: [10.4, 33.9] (25 elements)
# Treating input as continuous
r_mpg_con <- new_r(mtcars$mpg, type = "continuous")
r_mpg_con
#> Random generation function of continuous type
#> Support: ~[2.96996, 41.33004] (511 intervals)
# Outputs are actually functions
r_mpg_dis(5)
#> [1] 17.3 10.4 21.5 21.4 15.2
r_mpg_con(5)
#> [1] 16.30053 20.58094 16.80433 21.19017 19.96810
# You can plot them directly using base `plot()` and `lines()`
op <- par(mfrow = c(1, 2))
plot(r_mpg_con, main = '"continuous" r-function\nfrom sample')
plot(r_mpg_dis, main = '"discrete" r-function\nfrom sample', col = "blue")
par(op)
r_df_dis <- new_r(dis_df, type = "discrete")
r_df_dis
#> Random generation function of discrete type
#> Support: [1, 4] (4 elements)
r_df_con <- new_r(con_df, type = "continuous")
r_df_con
#> Random generation function of continuous type
#> Support: [1, 4] (3 intervals)
op <- par(mfrow = c(1, 2))
plot(r_df_con, main = '"continuous" r-function\nfrom data frame')
plot(r_df_dis, main = '"discrete" r-function\nfrom data frame', col = "blue")
par(op)
When creating “continuous” pdqr-function with new_*()
from single number, a special “dirac-like” pdqr-function is created. It is an approximation of single number with triangular distribution of very narrow support (1e-8 of magnitude):
r_dirac <- new_r(3.14, type = "continuous")
r_dirac
#> Random generation function of continuous type
#> Support: ~[3.14, 3.14] (2 intervals)
r_dirac(4)
#> [1] 3.14 3.14 3.14 3.14
# Outputs aren't exactly but approximately equal
dput(r_dirac(4))
#> c(3.13999999621714, 3.14000000258556, 3.13999999778332, 3.13999999147402
#> )
Boolean pdqr-function is a special case of “discrete” function, which values are exactly 0 and 1. Those functions are usually created after transformations involving logical operators (see vignette on transformation for more details). It is assumed that 0 represents that some expression is false, and 1 is for being true. Corresponding probabilities describe distribution of expression’s logical values. The only difference from other “discrete” pdqr-functions is in more detailed printing.
new_d(data.frame(x = c(0, 1), prob = c(0.25, 0.75)), type = "discrete")
#> Probability mass function of discrete type
#> Support: [0, 1] (2 elements, probability of 1: 0.75)
density()
argumentsWhen creating pdqr-function of “continuous” type, density()
is used to estimate density. To tweak its performance, supply its extra arguments directly to new_*()
functions. Here are some examples:
plot(
new_d(mtcars$mpg, "continuous"), lwd = 3,
main = "Examples of `density()` options"
)
# Argument `adjust` of `density()` helps to define smoothing bandwidth
lines(new_d(mtcars$mpg, "continuous", adj = 0.3), col = "blue")
# Argument `n` defines number of points to be used in piecewise-linear
# approximation
lines(new_d(mtcars$mpg, "continuous", n = 5), col = "green")
# Argument `cut` defines the "extending" property of density estimation.
# Using `cut = 0` assumes that density can't go outside of input's range
lines(new_d(mtcars$mpg, "continuous", cut = 0), col = "magenta")
Every pdqr-function has metadata, information which describes underline distribution and pdqr-function. Family of meta_*()
functions are implemented to extract that information:
meta_x_tbl()
) completely defines distribution. It is a data frame with structure depending on type of pdqr-function:
meta_class()
) - class of pdqr-function. This can be one of “p”, “d”, “q”, “r”. Represents how pdqr-function describes underlying distribution.meta_type()
) - type of pdqr-function. This can be one of “discrete” or “continuous”. Represents type of underlying distribution.meta_support()
) - support of distribution. This is a range of “x” column from “x_tbl” metadata.# Type "discrete"
d_dis <- new_d(1:4, type = "discrete")
meta_x_tbl(d_dis)
#> x prob cumprob
#> 1 1 0.25 0.25
#> 2 2 0.25 0.50
#> 3 3 0.25 0.75
#> 4 4 0.25 1.00
meta_class(d_dis)
#> [1] "d"
meta_type(d_dis)
#> [1] "discrete"
meta_support(d_dis)
#> [1] 1 4
# Type "continuous"
p_con <- new_p(1:4, type = "continuous")
head(meta_x_tbl(p_con))
#> x y cumprob
#> 1 -1.290542 0.001477707 0.000000e+00
#> 2 -1.275706 0.001568091 2.259340e-05
#> 3 -1.260870 0.001660241 4.654081e-05
#> 4 -1.246034 0.001759392 7.190728e-05
#> 5 -1.231199 0.001862292 9.877253e-05
#> 6 -1.216363 0.001970915 1.272068e-04
meta_class(p_con)
#> [1] "p"
meta_type(p_con)
#> [1] "continuous"
meta_support(p_con)
#> [1] -1.290542 6.290542
# Dirac-like "continuous" function
r_dirac <- new_r(1, type = "continuous")
dput(meta_x_tbl(r_dirac))
#> structure(list(x = c(0.99999999, 1, 1.00000001), y = c(0, 100000000.052636,
#> 0), cumprob = c(0, 0.5, 1)), row.names = c(NA, -3L), class = "data.frame")
dput(meta_support(r_dirac))
#> c(0.99999999, 1.00000001)
# `meta_all()` returns all metadata in a single list
meta_all(d_dis)
#> $class
#> [1] "d"
#>
#> $type
#> [1] "discrete"
#>
#> $support
#> [1] 1 4
#>
#> $x_tbl
#> x prob cumprob
#> 1 1 0.25 0.25
#> 2 2 0.25 0.50
#> 3 3 0.25 0.75
#> 4 4 0.25 1.00
For more details go to help page of meta_all()
.