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:

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:

P-functions

P-function (analogue of p*() functions in base R) represents a cumulative distribution function of distribution.

From sample

# 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")

plot of chunk p-fun_sample

From data frame

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")

plot of chunk p-fun_data-frame

D-functions

D-function (analogue of d*() functions in base R) represents a probability mass function for “discrete” type and density function for “continuous”:

From sample

# 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")

plot of chunk d-fun_sample

par(op)

From data frame

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")

plot of chunk d-fun_data-frame

par(op)

Q-functions

Q-function (analogue of q*() functions in base R) represents a quantile function, an inverse of corresponding p-function:

From sample

# 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")

plot of chunk q-fun_sample

From data frame

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")

plot of chunk q-fun_data-frame

R-functions

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().

From sample

# 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")

plot of chunk r-fun_sample

par(op)

From data frame

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")

plot of chunk r-fun_data-frame

par(op)

Special cases

Dirac-like

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

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)

Using density() arguments

When 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")

plot of chunk density-args

Metadata of pdqr-functions

Every pdqr-function has metadata, information which describes underline distribution and pdqr-function. Family of meta_*() functions are implemented to extract that information:

# 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().