Create pdqr-functions with
new_*()
Package ‘pdqr’ supports two types of distributions:
- Type “discrete”: random variable has finite number
of output values. It is explicitly defined by the collection of its
values with their corresponding probability.
- Type “continuous”: there are infinite number of
output values in the form of continuous random variable. It is
explicitly defined by piecewise-linear density function.
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:
- Sample input is processed based on type. For
“discrete” type it gets tabulated with frequency of unique values
serving as their probability. For “continuous” type distribution density
is estimated using
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).
- Data frame input should completely define
distribution. For “discrete” type it should have “x” and “prob” columns
for output values and their probabilities. For “continuous” type - “x”
and “y” columns for points, which define piecewise-linear continuous
density function. Columns “prob” and “y” will be automatically
normalized to represent proper distribution: sum of “prob” will be 1 and
total square under graph of piecewise-linear function will be 1.
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:
- Four sections about how to create p-, d-, q-, and r-functions (both
from sample and data frame).
- Section “Special cases”, which describes two special cases of
pdqr-functions: dirac-like and boolean.
- Section “Using
density() arguments” describes how to
use density() arguments to tweak smoothing during creation
of “continuous” pdqr-functions.
- “Metadata of pdqr-functions” describes the concept of metadata of
pdqr-functions.
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")
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")
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")
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")
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")
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")
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")
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")
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")
