distionary: Create and Evaluate Probability Distributions
Description
Create and evaluate probability distribution objects from a variety of families or define custom distributions. Automatically compute distributional properties, even when they have not been specified. This package supports statistical modeling and simulations, and forms the core of the probaverse suite of R packages.
Overview
The distionary package provides a comprehensive framework for working with probability distributions in R. With distionary, you can:
-
Specify probability distributions from common families or create custom distributions.
-
Evaluate distributional properties and representations.
-
Access distributional calculations even when they're not directly specified.
The main purpose of distionary is to implement a distribution object that powers the wider probaverse ecosystem for making probability distributions that are representative of your data.
Creating Distributions
Use the dst_*() family of functions to create distributions from
common families:
-
dst_norm(),dst_exp(),dst_unif(), etc. are some continuous distributions. -
dst_pois(),dst_binom(),dst_geom(), etc. are some discrete distributions. -
dst_empirical()is useful for creating a non-parametric distribution from data.
You can also make your own distribution using the
distribution() function, which allows you to specify
any combination of distributional representations and properties. For this
version of distionary, the CDF and density/PMF are required
in order to access all functionality.
Evaluating Distributions
A distribution's representations are functions that fully describe the
distribution. They can be accessed with the eval_*() functions.
For example, eval_cdf() and eval_quantile()
invoke the distribution's cumulative distribution function (CDF)
and quantile function.
Other properties of the distribution can be calculated by functions of the
property's name, such as mean() and range().
Random Samples
Generate random samples from a distribution using
realise().
Getting Started
New users should start with the package vignettes:
-
vignette("specify", package = "distionary")- Learn how to specify distributions. -
vignette("evaluate", package = "distionary")- Learn how to evaluate distributions.
Author(s)
Maintainer: Vincenzo Coia [email protected] (ORCID) [copyright holder]
Other contributors:
-
Amogh Joshi [contributor]
-
Shuyi Tan [contributor]
-
Zhipeng Zhu [contributor]
-
olivroy (GitHub contributor) [contributor]
See Also
Useful links:
-
Report bugs at https://github.com/probaverse/distionary/issues
Examples
# Create a Poisson distribution. poisson <- dst_pois(lambda = 1.5) poisson # Evaluate the probability mass function. eval_pmf(poisson, at = 0:4) plot(poisson) # Get distribution properties. mean(poisson) variance(poisson) # Create a continuous distribution (Normal). normal <- dst_norm(mean = 0, sd = 1) # Evaluate quantiles. eval_quantile(normal, at = c(0.025, 0.5, 0.975)) # Create a custom distribution. my_dist <- distribution( density = function(x) ifelse(x >= 0 & x <= 1, 2 * (1 - x), 0), cdf = function(x) ifelse(x >= 0 & x <= 1, 1 - (1 - x)^2, 0), .support = continuous(c(0, 1)), .name = "Linear" ) plot(my_dist) plot(my_dist, "cdf") # Even without specifying all properties, they can still be computed. mean(my_dist)# Create a Poisson distribution. poisson <- dst_pois(lambda = 1.5) poisson # Evaluate the probability mass function. eval_pmf(poisson, at = 0:4) plot(poisson) # Get distribution properties. mean(poisson) variance(poisson) # Create a continuous distribution (Normal). normal <- dst_norm(mean = 0, sd = 1) # Evaluate quantiles. eval_quantile(normal, at = c(0.025, 0.5, 0.975)) # Create a custom distribution. my_dist <- distribution( density = function(x) ifelse(x >= 0 & x <= 1, 2 * (1 - x), 0), cdf = function(x) ifelse(x >= 0 & x <= 1, 1 - (1 - x)^2, 0), .support = continuous(c(0, 1)), .name = "Linear" ) plot(my_dist) plot(my_dist, "cdf") # Even without specifying all properties, they can still be computed. mean(my_dist)
What a Support Is Made Of
Description
Take a support apart: atoms() gives the points it places mass on, and
regions() gives the intervals it spreads mass across. Each accepts a
support object or a distribution.
Usage
atoms(x) regions(x)atoms(x) regions(x)
Arguments
x |
A support object or a distribution. |
Details
These are the inverses of the constructors. discrete() builds a support
out of atoms and atoms() gives them back; continuous() builds one out of
regions and regions() gives those back. So the discrete part of a support
is discrete(atoms(x)), and its continuous part is continuous(regions(x)).
A support with no atoms gives an empty discretes object rather than
nothing, and one with no regions gives a matrix of no rows, so neither has
to be guarded against before being used. That is a definite answer: the
support says there is no part of that kind.
dst_null() is different. It has no support at all, so there is nothing to
take apart and nothing is known — both give NULL, as support() does
for it, rather than claiming it has no atoms. This mirrors range(), which
answers c(NA, NA) for it instead of refusing.
Value
For atoms(), a discretes object. For regions(), a two-column
numeric matrix of intervals (lower, upper), one row each.
See Also
Other Support:
empty_support(),
is_support(),
support(),
support-construction
Examples
atoms(mixed(discrete = 0, continuous = c(0, Inf))) regions(continuous(c(0, 1), c(3, 4))) # Either part can be put back together into a support of its own. s <- mixed(discrete = c(0, 5), continuous = c(0, 10)) discrete(atoms(s)) continuous(regions(s))atoms(mixed(discrete = 0, continuous = c(0, Inf))) regions(continuous(c(0, 1), c(3, 4))) # Either part can be put back together into a support of its own. s <- mixed(discrete = c(0, 5), continuous = c(0, 10)) discrete(atoms(s)) continuous(regions(s))
Build a Distribution Object
Description
Make a distribution object by specifying properties (e.g., cdf, density, mean, etc.). Some properties, if not included, will be calculated based on other properties that are included (e.g., quantile from cdf; variance from standard deviation). A list of these representations can be found in the details.
Usage
distribution( ..., .support = NULL, .vtype = NULL, .name = NULL, .parameters = list() ) is_distribution(object) is.distribution(object)distribution( ..., .support = NULL, .vtype = NULL, .name = NULL, .parameters = list() ) is_distribution(object) is.distribution(object)
Arguments
... |
Name-value pairs for defining the distribution. |
.support |
Required. Where the distribution places probability,
built with |
.vtype |
|
.name |
A name to give to the distribution. Can be any character vector of length 1. |
.parameters |
A named list with one entry per distribution parameter, each of which can be any data type. In this version of distionary, these parameters are only stored for the benefit of the user to know what distribution they are working with; the code never looks at these parameters to inform its calculations. This is anticipated to change in a future version of distionary. |
object |
Object to be tested |
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Details
The support
Every distribution has to say where it places probability, and .support
is how. It is the one thing distionary asks for rather than working out: a
CDF does hold the answer, its jumps being the atoms and its flattening out
marking where the distribution ends, but recovering that numerically means
hunting for discontinuities in a function that can only be sampled. The
estimate would be worst for small atoms and long tails, which are the cases
where it matters most.
Declared instead, it is exact, and the difference shows: quantiles at probability 0 and 1 are read off rather than searched for in the numerical tail, atoms are located exactly, and moments can be decomposed. It is the same bargain as declaring atoms — a little more to say up front, in exchange for exact answers rather than approximate ones. The "The Support of a Distribution" vignette covers what a support is and how to build one.
The variable type (vtype()) and the range() follow from the support, so
neither can be given here; see the property list below.
.vtype used to take a string such as "continuous" and is now defunct. A
variable type cannot stand in for a support: "discrete" does not say
which points carry mass, and "continuous" does not say over what
region, so there is no translating one into the other, and a guess would be
quietly wrong rather than an error. The argument is kept only so that old
code gets a message saying what to do, rather than unused argument.
Properties
Currently, the CDF (cdf) is required to be specified, along with the PMF
(pmf) for discrete distributions and density (density) for continuous
distributions. Otherwise, the full extent of distribution properties will
not be accessible.
A distributional representation is a function that fully describes the
distribution. Besides cdf, density, and pmf, other options
understood by distionary include:
-
survival: the survival function, or one minus the cdf. -
hazard: the hazard function, for continuous variables only. -
chf: the cumulative hazard function, for continuous variables only. -
quantile: the quantile function, or left-inverse of the cdf. -
realiseorrealize: a function that takes an integer and generates a vector of that many random draws from the distribution. -
odds: for discrete variables, the probability odds function (pmf / (1 - pmf)) -
return: the quantiles associated with the provided return periods, where events are exceedances.
All functions should be vectorized.
Other properties that are understood by distionary include:
-
mean,stdev,variance,skewness,medianare self-explanatory. -
kurtosis_excandkurtosisare the distribution's excess kurtosis and regular kurtosis.
range and vtype are properties too, and eval_property() reads them
like any other, but they cannot be given here: the support determines both,
and a stated one could disagree with it. A name distionary does not know is
simply kept, retrievable with eval_property() and otherwise unused.
Value
A distribution object.
Examples
linear <- distribution( density = function(x) { d <- 2 * (1 - x) d[x < 0 | x > 1] <- 0 d }, cdf = function(x) { p <- 2 * x * (1 - x / 2) p[x < 0] <- 0 p[x > 1] <- 1 p }, .support = continuous(c(0, 1)), .name = "My Linear", .parameters = list(could = "include", anything = data.frame(x = 1:10)) ) # Inspect linear # Plot plot(linear)linear <- distribution( density = function(x) { d <- 2 * (1 - x) d[x < 0 | x > 1] <- 0 d }, cdf = function(x) { p <- 2 * x * (1 - x / 2) p[x < 0] <- 0 p[x > 1] <- 1 p }, .support = continuous(c(0, 1)), .name = "My Linear", .parameters = list(could = "include", anything = data.frame(x = 1:10)) ) # Inspect linear # Plot plot(linear)
Bernoulli Distribution
Description
Makes a Bernoulli distribution, representing the outcome of a single trial with a given success probability.
Usage
dst_bern(prob)dst_bern(prob)
Arguments
prob |
Probability of success; single numeric between 0 and 1. |
Value
A Bernoulli distribution.
Examples
dst_bern(0.3)dst_bern(0.3)
Beta Distribution
Description
Makes a Beta distribution.
Usage
dst_beta(shape1, shape2)dst_beta(shape1, shape2)
Arguments
shape1, shape2
|
Shape parameters of the distribution; single positive numerics. |
Value
A Beta distribution.
Examples
dst_beta(2, 3)dst_beta(2, 3)
Binomial Distribution
Description
Makes a Binomial distribution, representing the number of successes in a fixed number of independent trials.
Usage
dst_binom(size, prob)dst_binom(size, prob)
Arguments
size |
Number of trials; single positive integer. |
prob |
Success probability of each trial; single numeric between 0 and 1. |
Value
A binomial distribution.
Examples
dst_binom(10, 0.6)dst_binom(10, 0.6)
Cauchy Distribution
Description
Makes a Cauchy distribution.
Usage
dst_cauchy(location, scale)dst_cauchy(location, scale)
Arguments
location |
Location parameter; single numeric. |
scale |
Scale parameter; single positive numeric. |
Value
A Cauchy distribution.
Examples
d <- dst_cauchy(0, 1) # Moments do not exist for the Cauchy distribution. mean(d) variance(d)d <- dst_cauchy(0, 1) # Moments do not exist for the Cauchy distribution. mean(d) variance(d)
Chi-Squared Distribution
Description
Makes a Chi-Squared distribution.
Usage
dst_chisq(df)dst_chisq(df)
Arguments
df |
degrees of freedom parameter; single positive numeric. |
Value
A Chi-Squared distribution
Examples
dst_chisq(3)dst_chisq(3)
Degenerate Distribution
Description
A degenerate distribution assigns a 100% probability to one outcome.
Usage
dst_degenerate(location)dst_degenerate(location)
Arguments
location |
Outcome of the distribution; single positive numeric. |
Value
A degenerate distribution
Examples
d <- dst_degenerate(5) realise(d) variance(d)d <- dst_degenerate(5) realise(d) variance(d)
Empirical Distribution
Description
An empirical distribution is a non-parametric way to
estimate a distribution using data. By default,
it assigns equal probability to all observations
(this can be overridden with the weights argument).
Identical to dst_finite() with NA handling and with weights not needing
to add to 1.
Usage
dst_empirical( y, weights = 1, data = NULL, na_action_y = c("null", "drop", "fail"), na_action_w = c("null", "drop", "fail") )dst_empirical( y, weights = 1, data = NULL, na_action_y = c("null", "drop", "fail"), na_action_w = c("null", "drop", "fail") )
Arguments
y |
< |
weights |
< |
data |
Optionally, a data frame to compute |
na_action_y, na_action_w
|
What should be done with |
Details
y and weights are recycled to have the same length, but only
if one of them has length 1 (via vctrs::vec_recycle_common()).
na_action_y and na_action_w specify the NA action for y and weights.
Options are, in order of precedence:
-
"fail": Throw an error in the presence ofNA. -
"null": Return a Null distribution (dst_null()) in the presence ofNA. -
"drop"(the default forna_action_w): Remove outcome-weight pairs having anNAvalue in the specified vector.
Value
A finite distribution. If only one outcome, returns a degenerate
distribution. Returns a Null distribution if NA values are present
and "null" is specified as an NA action.
See Also
Examples
t <- -2:7 dst_empirical(t) # Using a data frame df <- data.frame(time = c(NA, NA, t)) dst_empirical(time * 60, data = df) # Null, since `NA` in `time`. # Drop NA `time` values. dst_empirical(time * 60, data = df, na_action_y = "drop") # Weights explicit. Zero-weight outcomes ("-120") are gone. df$w <- c(1, 1, 0:9) dst_empirical(time * 60, w, data = df, na_action_y = "drop") # "Null" takes precedence over "drop". df$w <- c(NA, NA, 0:9) df$time[1] <- -3 df$time[12] <- NA dst_empirical(time, w, data = df, na_action_w = "null", na_action_y = "drop") dst_empirical(time, w, data = df, na_action_w = "drop", na_action_y = "null") dst_empirical(time, w, data = df, na_action_w = "drop", na_action_y = "drop")t <- -2:7 dst_empirical(t) # Using a data frame df <- data.frame(time = c(NA, NA, t)) dst_empirical(time * 60, data = df) # Null, since `NA` in `time`. # Drop NA `time` values. dst_empirical(time * 60, data = df, na_action_y = "drop") # Weights explicit. Zero-weight outcomes ("-120") are gone. df$w <- c(1, 1, 0:9) dst_empirical(time * 60, w, data = df, na_action_y = "drop") # "Null" takes precedence over "drop". df$w <- c(NA, NA, 0:9) df$time[1] <- -3 df$time[12] <- NA dst_empirical(time, w, data = df, na_action_w = "null", na_action_y = "drop") dst_empirical(time, w, data = df, na_action_w = "drop", na_action_y = "null") dst_empirical(time, w, data = df, na_action_w = "drop", na_action_y = "drop")
Exponential Distribution
Description
Makes an Exponential distribution.
Usage
dst_exp(rate)dst_exp(rate)
Arguments
rate |
Rate parameter; single positive numeric. |
Value
An Exponential distribution.
Examples
dst_exp(1)dst_exp(1)
F Distribution
Description
Makes an F distribution.
Usage
dst_f(df1, df2)dst_f(df1, df2)
Arguments
df1, df2
|
Degrees of freedom of the numerator and denominator, both single positive numerics. |
Value
An F distribution.
Examples
dst_f(2, 3)dst_f(2, 3)
Finite Distribution
Description
Makes a finite distribution, which is a distribution with a finite number of possible outcomes.
Usage
dst_finite(outcomes, probs)dst_finite(outcomes, probs)
Arguments
outcomes |
Numeric vector representing the potential outcomes of the distribution. |
probs |
Numeric vector of probabilities corresponding to the outcomes
in |
Value
A distribution with finite outcomes.
See Also
Examples
dst_finite(2:5, probs = 1:4 / 10)dst_finite(2:5, probs = 1:4 / 10)
Gamma Distribution
Description
Makes a Gamma distribution.
Usage
dst_gamma(shape, rate)dst_gamma(shape, rate)
Arguments
shape |
Shape parameter; single positive numeric. |
rate |
Rate parameter; single positive numeric. |
Value
A Gamma distribution.
Examples
dst_gamma(2, 1)dst_gamma(2, 1)
Geometric Distribution
Description
Makes a Geometric distribution, corresponding to the number of failures in a sequence of independent trials before observing a success.
Usage
dst_geom(prob)dst_geom(prob)
Arguments
prob |
Probability of success in each trial; single numeric between 0 and 1. |
Value
A Geometric distribution.
Examples
d <- dst_geom(0.4) # This version of the Geometric distribution does not count the success. range(d)d <- dst_geom(0.4) # This version of the Geometric distribution does not count the success. range(d)
Generalised Extreme Value Distribution
Description
Makes a Generalised Extreme Value (GEV) distribution, which is the limiting distribution of the maximum.
Usage
dst_gev(location, scale, shape)dst_gev(location, scale, shape)
Arguments
location |
Location parameter; single numeric. |
scale |
Scale parameter; single positive numeric. |
shape |
Shape parameter; single numeric.
This is also the extreme value index,
so that |
Value
A GEV distribution.
Examples
# Short-tailed example short <- dst_gev(0, 1, -1) range(short) mean(short) # Heavy-tailed example heavy <- dst_gev(0, 1, 1) range(heavy) mean(heavy) # Light-tailed example (a Gumbel distribution) light <- dst_gev(0, 1, 0) range(light) mean(light)# Short-tailed example short <- dst_gev(0, 1, -1) range(short) mean(short) # Heavy-tailed example heavy <- dst_gev(0, 1, 1) range(heavy) mean(heavy) # Light-tailed example (a Gumbel distribution) light <- dst_gev(0, 1, 0) range(light) mean(light)
Generalised Pareto Distribution
Description
Makes a Generalized Pareto (GP) distribution, corresponding to the limiting distribution of excesses over a threshold.
Usage
dst_gp(scale, shape)dst_gp(scale, shape)
Arguments
scale |
Scale parameter; single positive numeric. |
shape |
Shape parameter; single positive numeric.
This is also the extreme value index, so that |
Value
A Generalised Pareto Distribution.
Examples
# Short-tailed example short <- dst_gp(1, -1) range(short) mean(short) # Heavy-tailed example heavy <- dst_gp(1, 1) range(heavy) mean(heavy) # Light-tailed example (a Gumbel distribution) light <- dst_gp(1, 0) range(light) mean(light)# Short-tailed example short <- dst_gp(1, -1) range(short) mean(short) # Heavy-tailed example heavy <- dst_gp(1, 1) range(heavy) mean(heavy) # Light-tailed example (a Gumbel distribution) light <- dst_gp(1, 0) range(light) mean(light)
Gumbel Distribution
Description
Makes a Gumbel distribution, which is a special case of the Generalised Extreme Value (GEV) distribution when the shape parameter is 0.
Usage
dst_gumbel(location, scale)dst_gumbel(location, scale)
Arguments
location |
Location parameter; single numeric. |
scale |
Scale parameter; single positive numeric. |
Value
A Gumbel distribution.
Examples
dst_gumbel(0, 1)dst_gumbel(0, 1)
Hypergeometric Distribution
Description
Creates a Hypergeometric distribution. The parameterization used here
is the same as for stats::phyper(), where the outcome can be thought
of as the number of red balls drawn from an urn of coloured balls,
using a scoop that holds a fixed number of balls.
Usage
dst_hyper(m, n, k)dst_hyper(m, n, k)
Arguments
m |
The number of red balls in the urn; single positive integer. |
n |
The number of non-red balls in the urn; single positive integer. |
k |
the number of balls drawn from the urn (between 0 and |
Value
A Hypergeometric distribution.
Examples
dst_hyper(15, 50, 10)dst_hyper(15, 50, 10)
Log Normal Distribution
Description
Makes a Log Normal distribution, which is the distribution of the exponential of a Normally distributed random variable.
Usage
dst_lnorm(meanlog, sdlog)dst_lnorm(meanlog, sdlog)
Arguments
meanlog |
Mean of the log of the random variable; single numeric. |
sdlog |
Standard deviation of the log of the random variable; single positive numeric. |
Value
A Log Normal distribution.
Examples
dst_lnorm(0, 1)dst_lnorm(0, 1)
Log Pearson Type III distribution
Description
Makes a Log Pearson Type III distribution, which is the distribution of the exponential of a random variable following a Pearson Type III distribution.
Usage
dst_lp3(meanlog, sdlog, skew)dst_lp3(meanlog, sdlog, skew)
Arguments
meanlog |
Mean of the log of the random variable; single numeric. |
sdlog |
Standard deviation of the log of the random variable; single positive numeric. |
skew |
Skewness of the log of the random variable; single numeric. |
Value
A Log Pearson Type III distribution.
Examples
dst_lp3(0, 1, 1)dst_lp3(0, 1, 1)
Negative binomial Distribution
Description
Makes a Negative Binomial distribution, corresponding to the number of failures in a sequence of independent trials until a given number of successes are observed.
Usage
dst_nbinom(size, prob)dst_nbinom(size, prob)
Arguments
size |
Number of successful trials; single positive numeric. |
prob |
Probability of a successful trial; single numeric between 0 and 1. |
Value
A Negative Binomial distribution.
Examples
d <- dst_nbinom(10, 0.5) # This version of the Negative Binomial distribution does not count # the successes. range(d)d <- dst_nbinom(10, 0.5) # This version of the Negative Binomial distribution does not count # the successes. range(d)
Normal (Gaussian) Distribution
Description
Makes a Normal (Gaussian) distribution.
Usage
dst_norm(mean, sd)dst_norm(mean, sd)
Arguments
mean |
Mean of the distribution. Single numeric. |
sd |
Standard deviation of the distribution. Single positive numeric. |
Value
A Normal distribution.
Examples
dst_norm(0, 1)dst_norm(0, 1)
Null Distribution
Description
Sometimes it's convenient to work with a distribution object that is
akin to a missing value. This is especially true when programmatically
outputting distributions, such as when a distribution fails to fit to
data. This function makes such a distribution object. It always evaluates
to NA.
Usage
dst_null()dst_null()
Details
The Null distribution is the missing value of the distribution world, and
every query about it answers NA in whatever type that query returns:
NA_real_ from mean() and the eval_*() functions, NA_character_ from
vtype(), c(NA, NA) from range(), and no support at all — support()
returns NULL, R's absent-object value. It is also the one distribution
that is.na() finds; see length.dst().
Because of that it is assembled with the package's low-level constructor
rather than through distribution(). A Null distribution cannot satisfy
what distribution() asks of a real one, since it has nothing to declare;
building it here keeps that bypass internal, so a distribution with no
support cannot be made through the front door.
Value
A Null distribution.
Examples
x <- dst_null() mean(x) eval_pmf(x, at = 1:10) # It is the distribution that `is.na()` finds. is.na(x) is.na(dst_norm(0, 1)) # Everything about it is missing, including its support. vtype(x) range(x) support(x)x <- dst_null() mean(x) eval_pmf(x, at = 1:10) # It is the distribution that `is.na()` finds. is.na(x) is.na(dst_norm(0, 1)) # Everything about it is missing, including its support. vtype(x) range(x) support(x)
Pearson Type III distribution
Description
Makes a Pearson Type III distribution, which is a Gamma distribution, but shifted.
Usage
dst_pearson3(location, scale, shape)dst_pearson3(location, scale, shape)
Arguments
location |
Location parameter, specifying the boundary of the
distribution; single numeric. It is the left endpoint when |
scale |
Scale parameter of the Gamma distribution; single positive numeric. |
shape |
Shape parameter of the Gamma distribution; single numeric.
A negative value gives the distribution reflected about |
Value
A Pearson Type III distribution.
Examples
dst_pearson3(1, 1, 1) # A negative shape reflects the distribution about `location`: dst_pearson3(1, 1, -1)dst_pearson3(1, 1, 1) # A negative shape reflects the distribution about `location`: dst_pearson3(1, 1, -1)
Poisson Distribution
Description
Makes a Poisson distribution.
Usage
dst_pois(lambda)dst_pois(lambda)
Arguments
lambda |
Mean of the Poisson distribution; single positive numeric. |
Value
A Poisson distribution.
Examples
dst_pois(1)dst_pois(1)
Student t Distribution
Description
Makes a Student t distribution.
Usage
dst_t(df)dst_t(df)
Arguments
df |
Degrees of freedom; single positive numeric. |
Value
A Student t distribution.
Examples
dst_t(3)dst_t(3)
Uniform Distribution
Description
Makes a Uniform distribution.
Usage
dst_unif(min, max)dst_unif(min, max)
Arguments
min, max
|
Minimum and maximum of the distribution. Single numerics. |
Value
A Uniform distribution.
Examples
dst_unif(0, 1)dst_unif(0, 1)
Weibull Distribution
Description
Makes a Weibull distribution.
Usage
dst_weibull(shape, scale)dst_weibull(shape, scale)
Arguments
shape |
Shape parameter; single positive numeric. |
scale |
Scale parameter; single positive numeric. |
Value
A Weibull distribution.
Examples
dst_weibull(1, 1)dst_weibull(1, 1)
The Empty Support
Description
A support containing nothing: no atoms and no continuous part.
Usage
empty_support()empty_support()
Details
No distribution has an empty support — probability has to go somewhere —
and distribution() rejects one. It exists so that operations on supports
are closed: restricting a support to a region it does not reach has to
return something, and that something is the empty support. It is also the
identity for taking unions.
Its variable type is "empty", which is a different claim from
"unknown". Empty says there is nowhere to place probability; unknown says
nobody specified where.
Value
A support object (class "support") with both parts empty.
See Also
is_empty_support() to test for it, discrete(),
continuous(), and mixed() for supports a distribution can actually have.
Other Support:
atoms(),
is_support(),
support(),
support-construction
Examples
empty_support() is_empty_support(empty_support()) # It is also what any of the constructors gives when handed nothing. continuous(numeric(0)) discrete(numeric(0)) mixed()empty_support() is_empty_support(empty_support()) # It is also what any of the constructors gives when handed nothing. continuous(numeric(0)) discrete(numeric(0)) mixed()
Cumulative Distribution Function
Description
Access a distribution's cumulative distribution function (cdf).
Usage
eval_cdf(distribution, at) enframe_cdf(..., at, arg_name = ".arg", fn_prefix = "cdf", sep = "_")eval_cdf(distribution, at) enframe_cdf(..., at, arg_name = ".arg", fn_prefix = "cdf", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_chf(),
eval_density(),
eval_hazard(),
eval_odds(),
eval_pmf(),
eval_quantile(),
eval_return(),
eval_survival()
Examples
d1 <- dst_unif(0, 4) d2 <- dst_pois(1.1) eval_cdf(d1, at = 0:4) enframe_cdf(d1, at = 0:4) enframe_cdf(d1, d2, at = 0:4) enframe_cdf(model1 = d1, model2 = d2, at = 0:4) enframe_cdf( model1 = d1, model2 = d2, at = 0:4, arg_name = "value" )d1 <- dst_unif(0, 4) d2 <- dst_pois(1.1) eval_cdf(d1, at = 0:4) enframe_cdf(d1, at = 0:4) enframe_cdf(d1, d2, at = 0:4) enframe_cdf(model1 = d1, model2 = d2, at = 0:4) enframe_cdf( model1 = d1, model2 = d2, at = 0:4, arg_name = "value" )
Cumulative Hazard Function
Description
Access a distribution's cumulative hazard function (chf).
Usage
eval_chf(distribution, at) enframe_chf(..., at, arg_name = ".arg", fn_prefix = "chf", sep = "_")eval_chf(distribution, at) enframe_chf(..., at, arg_name = ".arg", fn_prefix = "chf", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_density(),
eval_hazard(),
eval_odds(),
eval_pmf(),
eval_quantile(),
eval_return(),
eval_survival()
Examples
d <- dst_unif(0, 4) eval_chf(d, at = 0:4) enframe_chf(d, at = 0:4)d <- dst_unif(0, 4) eval_chf(d, at = 0:4) enframe_chf(d, at = 0:4)
Probability Density Function
Description
Access a distribution's probability density function (pdf).
Usage
eval_density(distribution, at) enframe_density(..., at, arg_name = ".arg", fn_prefix = "density", sep = "_")eval_density(distribution, at) enframe_density(..., at, arg_name = ".arg", fn_prefix = "density", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_hazard(),
eval_odds(),
eval_pmf(),
eval_quantile(),
eval_return(),
eval_survival()
Examples
d <- dst_unif(0, 4) eval_density(d, at = 0:4) enframe_density(d, at = 0:4)d <- dst_unif(0, 4) eval_density(d, at = 0:4) enframe_density(d, at = 0:4)
Hazard Function
Description
Access a distribution's hazard function.
Usage
eval_hazard(distribution, at) enframe_hazard(..., at, arg_name = ".arg", fn_prefix = "hazard", sep = "_")eval_hazard(distribution, at) enframe_hazard(..., at, arg_name = ".arg", fn_prefix = "hazard", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_density(),
eval_odds(),
eval_pmf(),
eval_quantile(),
eval_return(),
eval_survival()
Examples
d <- dst_unif(0, 4) eval_hazard(d, at = 0:4) enframe_hazard(d, at = 0:4)d <- dst_unif(0, 4) eval_hazard(d, at = 0:4) enframe_hazard(d, at = 0:4)
Odds Function
Description
Access a distribution's odds function. The odds of an event having
probability p is p / (1 - p).
Usage
eval_odds(distribution, at) enframe_odds(..., at, arg_name = ".arg", fn_prefix = "odds", sep = "_")eval_odds(distribution, at) enframe_odds(..., at, arg_name = ".arg", fn_prefix = "odds", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_density(),
eval_hazard(),
eval_pmf(),
eval_quantile(),
eval_return(),
eval_survival()
Examples
d <- dst_pois(1) eval_pmf(d, at = c(1, 2, 2.5)) eval_odds(d, at = c(1, 2, 2.5)) enframe_odds(d, at = 0:4)d <- dst_pois(1) eval_pmf(d, at = c(1, 2, 2.5)) eval_odds(d, at = c(1, 2, 2.5)) enframe_odds(d, at = 0:4)
Probability Mass Function
Description
Access a distribution's probability mass function (pmf).
Usage
eval_pmf(distribution, at) enframe_pmf(..., at, arg_name = ".arg", fn_prefix = "pmf", sep = "_")eval_pmf(distribution, at) enframe_pmf(..., at, arg_name = ".arg", fn_prefix = "pmf", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_density(),
eval_hazard(),
eval_odds(),
eval_quantile(),
eval_return(),
eval_survival()
Examples
d <- dst_pois(5) eval_pmf(d, at = c(1, 2, 2.5)) enframe_pmf(d, at = 0:4) eval_pmf(dst_norm(0, 1), at = -3:3)d <- dst_pois(5) eval_pmf(d, at = c(1, 2, 2.5)) enframe_pmf(d, at = 0:4) eval_pmf(dst_norm(0, 1), at = -3:3)
Evaluate a distribution
Description
Evaluate a distribution property. The distribution itself is first searched for the property, and if it can't be found, will attempt to calculate the property from other entries.
Usage
eval_property(distribution, entry, ...)eval_property(distribution, entry, ...)
Arguments
distribution |
Distribution object. |
entry |
Name of the property, such as "cdf" or "mean". Length 1 character vector. |
... |
If the property is a function, arguments to the function go here. Need not be named; inserted in the order they appear. |
Value
The distribution's property, evaluated. If cannot be
evaluated, returns NULL.
Examples
d <- distribution( cdf = function(x) { (x > 0) * pmin(x^2, 1) }, g = 9.81, .support = continuous(c(0, 1)) ) eval_property(d, "g") eval_property(d, "quantile", 1:9 / 10) eval_property(d, "mean") eval_property(d, "realise", 10) eval_property(d, "foofy") eval_property(d, "foofy", 1:10)d <- distribution( cdf = function(x) { (x > 0) * pmin(x^2, 1) }, g = 9.81, .support = continuous(c(0, 1)) ) eval_property(d, "g") eval_property(d, "quantile", 1:9 / 10) eval_property(d, "mean") eval_property(d, "realise", 10) eval_property(d, "foofy") eval_property(d, "foofy", 1:10)
Distribution Quantiles
Description
Access a distribution's quantiles.
Usage
eval_quantile(distribution, at) enframe_quantile(..., at, arg_name = ".arg", fn_prefix = "quantile", sep = "_")eval_quantile(distribution, at) enframe_quantile(..., at, arg_name = ".arg", fn_prefix = "quantile", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Details
The 0- and 1-quantiles are the ends of the distribution's support: the
0-quantile is its lower end and the 1-quantile its upper end. They are
read from the support (see support()) rather than computed, so an
unbounded distribution gives -Inf and Inf instead of a large finite
number found by searching into the tail.
When a quantile function does not exist, the remaining probabilities are found by inverting the CDF by bisection: an interval known to contain the solution is progressively cut in half, moving into whichever half still contains it. The whole vector is solved together — one vectorized CDF evaluation per step rather than one per probability — so evaluating many quantiles at once is considerably faster than one at a time. Because the support says where the atoms (discrete mass points) are, a probability landing inside an atom's jump in the CDF is returned as that atom exactly, rather than approximately. Tolerance is roughly 1e-9 in the quantile value, unless the maximum number of iterations (200) is reached.
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_density(),
eval_hazard(),
eval_odds(),
eval_pmf(),
eval_return(),
eval_survival()
Examples
d <- dst_unif(0, 4) eval_quantile(d, at = 1:9 / 10) enframe_quantile(d, at = 1:9 / 10)d <- dst_unif(0, 4) eval_quantile(d, at = 1:9 / 10) enframe_quantile(d, at = 1:9 / 10)
Return Level Function
Description
Compute return levels (quantiles) from a distribution by inputting return periods. The return periods correspond to events that are exceedances of a quantile, not non-exceedances.
Usage
eval_return(distribution, at) enframe_return(..., at, arg_name = ".arg", fn_prefix = "return", sep = "_")eval_return(distribution, at) enframe_return(..., at, arg_name = ".arg", fn_prefix = "return", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of return periods >=1. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Details
This function is simply the quantile
function evaluated at 1 - 1 / at.
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_density(),
eval_hazard(),
eval_odds(),
eval_pmf(),
eval_quantile(),
eval_survival()
Examples
d <- dst_gp(24, 0.3) eval_return(d, at = c(2, 25, 100, 200))d <- dst_gp(24, 0.3) eval_return(d, at = c(2, 25, 100, 200))
Survival Function
Description
Access a distribution's survival function.
Usage
eval_survival(distribution, at) enframe_survival(..., at, arg_name = ".arg", fn_prefix = "survival", sep = "_")eval_survival(distribution, at) enframe_survival(..., at, arg_name = ".arg", fn_prefix = "survival", sep = "_")
Arguments
distribution, ...
|
A distribution, or possibly multiple
distributions in the case of |
at |
Vector of values to evaluate the representation at. |
arg_name |
For |
fn_prefix |
For |
sep |
When |
Value
The evaluated representation in vector form (for eval_)
with length matching the length of at, and data frame
or tibble form (for enframe_) with number of rows matching the
length of at. The at input occupies the first column,
named .arg by default, or the specification in arg_name;
the evaluated representations for each distribution in ...
go in the subsequent columns (one column per distribution). For a
single distribution, this column is named according to the
representation by default (cdf, survival, quantile, etc.),
or the value in fn_prefix. For multiple distributions, unnamed
distributions are auto-named, and columns are named
<fn_prefix><sep><distribution_name> (e.g., cdf_distribution1).
See Also
Other distributional representations:
eval_cdf(),
eval_chf(),
eval_density(),
eval_hazard(),
eval_odds(),
eval_pmf(),
eval_quantile(),
eval_return()
Examples
d <- dst_unif(0, 4) eval_survival(d, at = 0:4) enframe_survival(d, at = 0:4)d <- dst_unif(0, 4) eval_survival(d, at = 0:4) enframe_survival(d, at = 0:4)
Test for a Support Object
Description
is_empty_support() tests whether a support is the empty one: no atoms and
no continuous part. It is FALSE for anything that is not a support.
Usage
is_support(x) is_empty_support(x)is_support(x) is_empty_support(x)
Arguments
x |
Object to test. |
Value
TRUE if x is a support object, otherwise FALSE.
See Also
Other Support:
atoms(),
empty_support(),
support(),
support-construction
Moments of a Distribution
Description
Get common moment-related quantities of a
distribution: mean, variance, standard deviation (stdev),
skewness, and kurtosis or excess kurtosis (kurtosis_exc).
If these quantities are not supplied in the
distribution's definition, a numerical algorithm may be used.
Usage
kurtosis(distribution) kurtosis_exc(distribution) ## S3 method for class 'dst' mean(x, ...) skewness(distribution) stdev(distribution) variance(distribution)kurtosis(distribution) kurtosis_exc(distribution) ## S3 method for class 'dst' mean(x, ...) skewness(distribution) stdev(distribution) variance(distribution)
Arguments
x, distribution
|
Distribution to evaluate. |
... |
When calculating the mean via integration of the quantile
function, arguments passed to |
Details
If a moment is not supplied in the distribution's definition, it is computed numerically over the distribution's support: a sum over the atoms (the discrete part) plus integration of the density over the continuous part. An infinite atomic support (such as a Poisson's) is summed by walking outward through its atoms until the tail contribution is negligible.
Value
A single numeric.
Note
When a moment is computed numerically and the underlying sum or
integral does not converge — for example, a heavy-tailed distribution
whose moment is not finite — the result is NaN.
Examples
a <- dst_gp(1, 0.5) b <- dst_unif(0, 1) c <- dst_norm(3, 4) mean(a) variance(b) kurtosis(c) kurtosis_exc(c)a <- dst_gp(1, 0.5) b <- dst_unif(0, 1) c <- dst_norm(3, 4) mean(a) variance(b) kurtosis(c) kurtosis_exc(c)
Median of a Distribution
Description
Finds the median of a distribution.
Usage
## S3 method for class 'dst' median(x, ...)## S3 method for class 'dst' median(x, ...)
Arguments
x |
Distribution to calculate median from. |
... |
Not used. |
Details
Median is calculated as the 0.5-quantile when not found in the distribution. So, when the median is non-unique, the smallest of the possibilities is taken.
Value
Median of a distribution; single numeric.
Examples
d <- dst_gamma(3, 3) median(d)d <- dst_gamma(3, 3) median(d)
Parameters of a Distribution
Description
Get or set the parameters of a distribution, if applicable. See details.
Usage
parameters(distribution) parameters(distribution) <- valueparameters(distribution) parameters(distribution) <- value
Arguments
distribution |
Distribution. |
value |
A list of named parameter values, or |
Details
If a distribution is made by specifying parameter values
(e.g., mean and variance for a Normal distribution; shape parameters
for a Beta distribution), it is useful to keep track of what
these parameters are. This is done by adding parameters
to the list of objects defining the distribution; for instance,
distribution(parameters = c(shape1 = 1.4, shape2 = 3.4)).
Note that no checks are made to ensure the parameters are valid.
It's important to note that, in this version of distionary,
manually changing the parameters after the distribution has been
created will not change the functionality of the distribution,
because the parameters are never referred to when making calculations.
Value
A list of the distribution parameters. More specifically,
returns the "parameters" entry of the list making up the
probability distribution.
Examples
a <- dst_beta(1, 2) parameters(a) b <- distribution(mean = 5, .support = continuous()) parameters(b) parameters(b) <- list(t = 7) parameters(b)a <- dst_beta(1, 2) parameters(a) b <- distribution(mean = 5, .support = continuous()) parameters(b) parameters(b) <- list(t = 7) parameters(b)
Representations of the Generalized Extreme Value Distribution
Description
Representations of the Generalized Extreme Value Distribution
Usage
pgev(q, location, scale, shape) qgev(p, location, scale, shape) dgev(x, location, scale, shape)pgev(q, location, scale, shape) qgev(p, location, scale, shape) dgev(x, location, scale, shape)
Arguments
location |
Location parameter; numeric vector. |
scale |
Scale parameter; positive numeric vector. |
shape |
Shape parameter; numeric vector.
This is also the extreme value index,
so that |
p |
Vector of probabilities. |
x, q
|
Vector of quantiles. |
Value
Vector of evaluated GEV distribution, with length
equal to the recycled lengths of q/x/p, location, scale, and
shape.
Examples
pgev(1:10, 0, 1, 1) dgev(1:10, 1:10, 2, 0) qgev(1:9 / 10, 2, 10, -2)pgev(1:10, 0, 1, 1) dgev(1:10, 1:10, 2, 0) qgev(1:9 / 10, 2, 10, -2)
Representations of the Generalized Pareto Distribution
Description
Representations of the Generalized Pareto Distribution
Usage
pgp(q, scale, shape, lower.tail = TRUE) qgp(p, scale, shape) dgp(x, scale, shape)pgp(q, scale, shape, lower.tail = TRUE) qgp(p, scale, shape) dgp(x, scale, shape)
Arguments
scale |
Vector of scale parameters; positive numeric. |
shape |
Vector of shape parameters; positive numeric. |
lower.tail |
Single logical. If |
p |
Vector of probabilities. |
x, q
|
Vector of quantiles. |
Value
Vector of evaluated GP distribution, with length
equal to the recycled lengths of q/x/p, scale, and shape.
Examples
pgp(1:10, 1, 1) dgp(1:10, 2, 0) qgp(1:9 / 10, 10, -2)pgp(1:10, 1, 1) dgp(1:10, 2, 0) qgp(1:9 / 10, 10, -2)
Plot a Distribution
Description
Plot a distribution's representation.
Usage
## S3 method for class 'dst' plot( x, what = c("density", "pmf", "cdf", "survival", "quantile", "hazard", "chf"), ... )## S3 method for class 'dst' plot( x, what = c("density", "pmf", "cdf", "survival", "quantile", "hazard", "chf"), ... )
Arguments
x |
Distribution object |
what |
Name of the representation to plot. |
... |
Other arguments to pass to the |
Value
This function is run for its graphics byproduct, and therefore returns the original distribution, invisibly.
Examples
d <- dst_norm(0, 1) plot(d, from = -4, to = 4) plot(d, "cdf", n = 1000) plot(d, "survival") plot(d, "quantile") plot(d, "hazard") plot(d, "chf") p <- dst_pois(4) plot(p)d <- dst_norm(0, 1) plot(d, from = -4, to = 4) plot(d, "cdf", n = 1000) plot(d, "survival") plot(d, "quantile") plot(d, "hazard") plot(d, "chf") p <- dst_pois(4) plot(p)
Representations of the Log Pearson Type III Distribution
Description
Representations of the Log Pearson Type III (LP3) Distribution
Usage
plp3(x, meanlog, sdlog, skew, lower.tail = TRUE) dlp3(x, meanlog, sdlog, skew) qlp3(p, meanlog, sdlog, skew) rlp3(n, meanlog, sdlog, skew)plp3(x, meanlog, sdlog, skew, lower.tail = TRUE) dlp3(x, meanlog, sdlog, skew) qlp3(p, meanlog, sdlog, skew) rlp3(n, meanlog, sdlog, skew)
Arguments
x |
Vector of quantiles. |
meanlog |
Parameter representing the mean of the random variable
in log (base e) space; numeric. Vectors are allowed except for |
sdlog |
Parameter representing the standard deviation of the
random variable in log (base e) space; positive numeric.
Vectors are allowed except for |
skew |
Parameter representing the skewness of the random variable
in log (base e) space; numeric.
Vectors are allowed except for |
lower.tail |
Logical; if |
p |
Vector of probabilities. |
n |
Single positive whole number; number of observations to draw from the distribution. |
Value
Vector of evaluated LP3 distribution, with length
equal to the recycled lengths of x/p, meanlog, sdlog, and
skew. For rlp3(), a vector of length n.
Examples
plp3(1:10, meanlog = 0, sdlog = 1, skew = 1) dlp3(1:10, meanlog = 1:10, sdlog = 2, skew = 0) qlp3(1:9 / 10, meanlog = 2, sdlog = 10, skew = 2) set.seed(1) rlp3(10, meanlog = 2, sdlog = 10, skew = 2)plp3(1:10, meanlog = 0, sdlog = 1, skew = 1) dlp3(1:10, meanlog = 1:10, sdlog = 2, skew = 0) qlp3(1:9 / 10, meanlog = 2, sdlog = 10, skew = 2) set.seed(1) rlp3(10, meanlog = 2, sdlog = 10, skew = 2)
Representations of the Pearson Type III Distribution
Description
Representations of the Pearson Type III Distribution
Usage
ppearson3(x, location, scale, shape, lower.tail = TRUE) dpearson3(x, location, scale, shape, log = FALSE) qpearson3(p, location, scale, shape) rpearson3(n, location, scale, shape)ppearson3(x, location, scale, shape, lower.tail = TRUE) dpearson3(x, location, scale, shape, log = FALSE) qpearson3(p, location, scale, shape) rpearson3(n, location, scale, shape)
Arguments
x |
Vector of quantiles. |
location |
Parameter representing the boundary (endpoint) of the
distribution; numeric. This is the left endpoint when |
scale |
Scale parameter; positive numeric.
Vectors are allowed except for |
shape |
Shape parameter; numeric. A negative value gives the
distribution reflected about |
lower.tail |
Logical; if |
log |
Logical; if |
p |
Vector of probabilities. |
n |
Single positive whole number; number of observations to draw from the distribution. |
Value
Vector of evaluated Pearson Type III distribution, with length
equal to the recycled lengths of x/p, location, scale, and
shape. For rpearson3(), a vector of length n.
Examples
ppearson3(1:10, location = 0, scale = 1, shape = 1) dpearson3(1:10, location = 1:10, scale = 2, shape = 0) qpearson3(1:9 / 10, location = -2, scale = 10, shape = 2) set.seed(1) rpearson3(10, location = 2, scale = 10, shape = 2)ppearson3(1:10, location = 0, scale = 1, shape = 1) dpearson3(1:10, location = 1:10, scale = 2, shape = 0) qpearson3(1:9 / 10, location = -2, scale = 10, shape = 2) set.seed(1) rpearson3(10, location = 2, scale = 10, shape = 2)
Distribution name
Description
Print the name of a distribution, possibly with parameters.
Usage
pretty_name(distribution, param_digits = 0)pretty_name(distribution, param_digits = 0)
Arguments
distribution |
Distribution object. |
param_digits |
How many significant digits to include when displaying
the parameters? |
Value
A character containing the distribution's name, possibly followed by parameters in brackets.
Examples
d <- dst_norm(0.3552, 1.1453) pretty_name(d) pretty_name(d, 2)d <- dst_norm(0.3552, 1.1453) pretty_name(d) pretty_name(d, 2)
Find the probability left or right of a number
Description
Probability to the left or right of a number, inclusive or not.
prob_left() is a more general cdf defined using either < or <=, and
prob_right() is a more general survival function defined using either
> or >=.
Usage
prob_left(distribution, of, inclusive) prob_right(distribution, of, inclusive)prob_left(distribution, of, inclusive) prob_right(distribution, of, inclusive)
Arguments
distribution |
Distribution to find probabilities of. |
of |
Find the probability to the left or right of this number. Could be a vector. |
inclusive |
Should |
Value
A vector of probabilities.
Examples
d <- dst_pois(5) prob_left(d, of = 3, inclusive = TRUE) prob_left(d, of = 3, inclusive = FALSE) prob_right(d, of = 0:3, inclusive = TRUE)d <- dst_pois(5) prob_left(d, of = 3, inclusive = TRUE) prob_left(d, of = 3, inclusive = FALSE) prob_right(d, of = 0:3, inclusive = TRUE)
Range of Distribution
Description
Range returns a vector of length two, with the minimum and maximum values of the (support of the) distribution.
The support method gives the smallest and largest values the support
reaches — its two outermost points, taking the atoms and the continuous
regions together. Gaps in between are not represented.
An empty support reaches nothing, and its range is c(Inf, -Inf) — what
R gives for the range of nothing, and reversed on purpose, being the
identity for combining ranges.
Usage
## S3 method for class 'dst' range(distribution, ...) ## S3 method for class 'support' range(support, ...)## S3 method for class 'dst' range(distribution, ...) ## S3 method for class 'support' range(support, ...)
Arguments
distribution |
Distribution to compute range from. |
... |
Not used; vestige of the |
support |
A support object. |
Details
The range is read from the distribution's support (see support()), which
is where a distribution says what values it reaches. In this it behaves like
vtype(): derived, not declared.
It is still a property, and eval_property() reaches it like any other, so
code walking a list of property names need not know which are stored and
which are worked out. What it cannot be is stated: distribution()
refuses a range entry, since a stated one would be consulted ahead of the
derived value and could disagree with it.
The Null distribution is a different case: it has no support at all, so
neither end is known, and its range is c(NA, NA) — still a vector of
length two, rather than a single NA. An empty support says there is
nothing to reach; the Null distribution says nothing at all.
Value
Vector of length two, containing the minimum and maximum values of a distribution.
Examples
a <- dst_gp(1, 0.5) b <- dst_unif(0, 1) c <- dst_norm(3, 4) range(a) range(b) range(c) range(continuous(c(0, 1), c(3, 4))) range(mixed(discrete = -1, continuous = c(0, Inf))) range(empty_support())a <- dst_gp(1, 0.5) b <- dst_unif(0, 1) c <- dst_norm(3, 4) range(a) range(b) range(c) range(continuous(c(0, 1), c(3, 4))) range(mixed(discrete = -1, continuous = c(0, Inf))) range(empty_support())
Generate a Sample from a Distribution
Description
Draw n independent observations from a distribution.
Usage
realise(distribution, n = 1) realize(distribution, n = 1)realise(distribution, n = 1) realize(distribution, n = 1)
Arguments
distribution |
Distribution object. |
n |
Number of observations to generate. |
Value
Vector of independent values drawn from the inputted distribution.
Note
realise() and realize() are aliases and do the same thing.
Examples
d <- dst_pois(5) set.seed(2) realise(d, n = 10)d <- dst_pois(5) set.seed(2) realise(d, n = 10)
A Distribution has Length 1
Description
A distribution object is one distribution, so length() gives 1 and
is.na() gives a single logical. as.list() wraps the distribution
in a list of one.
Usage
## S3 method for class 'dst' length(x) ## S3 method for class 'dst' is.na(x) ## S3 method for class 'dst' as.list(x, ...)## S3 method for class 'dst' length(x) ## S3 method for class 'dst' is.na(x) ## S3 method for class 'dst' as.list(x, ...)
Arguments
x |
A distribution object. |
... |
Not used. |
Details
A distribution is built out of a list of its properties — a CDF, a
density, a mean — and without these methods base R reports on that list
rather than on the distribution. length() counted the properties and
is.na() tested each one, so dst_norm(0, 1) answered with eleven
FALSEs. Neither answer was about the distribution.
is.na() is TRUE for the Null distribution (dst_null()) and FALSE
for every other. The Null distribution is the missing value of the
distribution world, so it is the one that is.na() finds.
Note that the properties are still reachable, and are still what the
object is made of: x[["cdf"]] and names(x) are unchanged, and
eval_property() is the supported way to get at them. Only the questions
asked of the distribution as a whole now answer about the whole.
To hold several distributions, put them in a list; in a data frame, that is a list-column. A distribution does not have length beyond one.
Value
For length(), the number 1. For is.na(), a single logical.
For as.list(), a list containing the one distribution.
Examples
d <- dst_norm(0, 1) length(d) is.na(d) # The Null distribution is the missing one. is.na(dst_null()) # Several distributions go in a list. ds <- list(dst_norm(0, 1), dst_null(), dst_pois(3)) vapply(ds, is.na, logical(1))d <- dst_norm(0, 1) length(d) is.na(d) # The Null distribution is the missing one. is.na(dst_null()) # Several distributions go in a list. ds <- list(dst_norm(0, 1), dst_null(), dst_pois(3)) vapply(ds, is.na, logical(1))
Retrieve the Support of a Distribution
Description
Returns the structured support of a distribution: its atomic part and its
continuous part. Every distribution has one, because distribution()
requires it — the single exception being dst_null(), which has nothing to
place anywhere and returns NULL.
Usage
support(distribution)support(distribution)
Arguments
distribution |
Distribution object. |
Value
A support object, or NULL if the distribution has no structured
support.
See Also
discrete(), continuous(), mixed() to build supports;
vtype() for the derived variable type.
Other Support:
atoms(),
empty_support(),
is_support(),
support-construction
Examples
support(distribution(.support = continuous(c(0, Inf))))support(distribution(.support = continuous(c(0, Inf))))
Add or Remove Atoms
Description
Add atoms to a support, or take them away.
Usage
support_add_atoms(support, atoms) support_drop_atoms(support, atoms)support_add_atoms(support, atoms) support_drop_atoms(support, atoms)
Arguments
support |
A support object, or a distribution. |
atoms |
Atoms to add or remove: a numeric vector, or a |
Details
Adding an atom that is already there changes nothing. Removing one that is not there changes nothing either. Removing an atom does not disturb the continuous part, so removing an atom sitting on an interval leaves the interval whole.
Value
A support object.
See Also
Other Support algebra:
support_contains(),
support_restrict(),
support_transform(),
support_union()
Examples
support_add_atoms(continuous(c(0, Inf)), 0) support_drop_atoms(discrete(c(1, 2, 3)), 2) # Removing an atom leaves the continuous part alone. support_drop_atoms(mixed(discrete = 0, continuous = c(0, 1)), 0)support_add_atoms(continuous(c(0, Inf)), 0) support_drop_atoms(discrete(c(1, 2, 3)), 2) # Removing an atom leaves the continuous part alone. support_drop_atoms(mixed(discrete = 0, continuous = c(0, 1)), 0)
Test Membership of a Support
Description
Is a value in the support at all, or an atom of it specifically?
Usage
support_contains(support, at) support_has_atom(support, at)support_contains(support, at) support_has_atom(support, at)
Arguments
support |
A support object, or a distribution. |
at |
Values to test. Vectorised. |
Details
support_contains() is TRUE for a value that is either an atom or inside
one of the continuous intervals. support_has_atom() is TRUE only for the
atoms, and so is the one to reach for when what matters is whether a point
carries positive probability.
Continuous intervals count their endpoints as contained. Those endpoints carry no probability, so a value can be contained in a support without being a point of positive mass — which is exactly the distinction between these two functions.
Value
A logical vector the same length as at.
See Also
Other Support algebra:
support_add_atoms(),
support_restrict(),
support_transform(),
support_union()
Examples
s <- mixed(discrete = 0, continuous = c(2, 5)) support_contains(s, at = c(0, 1, 3, 9)) support_has_atom(s, at = c(0, 1, 3, 9)) support_has_atom(dst_pois(3), at = c(-1, 0, 2.5, 4))s <- mixed(discrete = 0, continuous = c(2, 5)) support_contains(s, at = c(0, 1, 3, 9)) support_has_atom(s, at = c(0, 1, 3, 9)) support_has_atom(dst_pois(3), at = c(-1, 0, 2.5, 4))
Restrict a Support to an Interval
Description
Cut a support down to the part of it lying within [from, to].
Usage
support_restrict( support, ..., from = -Inf, to = Inf, include_from = TRUE, include_to = TRUE )support_restrict( support, ..., from = -Inf, to = Inf, include_from = TRUE, include_to = TRUE )
Arguments
support |
A support object, or a distribution. |
... |
Not used; must be empty. Present so that the arguments below are matched by name. |
from, to
|
Endpoints of the interval to restrict to. |
include_from, include_to
|
Whether the endpoints themselves are kept. |
Details
The include_* flags apply to the atoms only. An endpoint of a continuous
interval carries no probability either way, so including or excluding it
makes no difference to the continuous part.
Restricting to a region the support does not reach gives
empty_support(), which is the reason that object exists.
Value
A support object.
See Also
support_union() to combine supports instead.
Other Support algebra:
support_add_atoms(),
support_contains(),
support_transform(),
support_union()
Examples
support_restrict(continuous(c(0, 10)), from = 3, to = 6) support_restrict(discrete(natural0()), to = 4) # Excluding an endpoint drops the atom sitting on it. support_restrict(discrete(natural0()), to = 4, include_to = FALSE) # Restricting out of reach gives the empty support. support_restrict(continuous(c(0, 1)), from = 5, to = 6)support_restrict(continuous(c(0, 10)), from = 3, to = 6) support_restrict(discrete(natural0()), to = 4) # Excluding an endpoint drops the atom sitting on it. support_restrict(discrete(natural0()), to = 4, include_to = FALSE) # Restricting out of reach gives the empty support. support_restrict(continuous(c(0, 1)), from = 5, to = 6)
Transform a Support
Description
Push a support through a strictly monotonic map, giving the support of the transformed variable.
Usage
support_transform( support, fun, inv, ..., increasing = TRUE, domain = c(-Inf, Inf), range = c(-Inf, Inf) ) support_shift(support, by) support_scale(support, by) support_reciprocal(support)support_transform( support, fun, inv, ..., increasing = TRUE, domain = c(-Inf, Inf), range = c(-Inf, Inf) ) support_shift(support, by) support_scale(support, by) support_reciprocal(support)
Arguments
support |
A support object, or a distribution. |
fun, inv
|
The map and its inverse. Both must be vectorised, and |
... |
Not used; must be empty. Present so that the arguments below are matched by name. |
increasing |
Whether |
domain, range
|
The domain and range of |
by |
For |
Details
support_shift(), support_scale(), and support_reciprocal() are the
common cases, and avoid having to supply an inverse, a domain, and a range
by hand.
support_reciprocal() maps each side of zero separately, since 1 / x is
monotonic on each side but not across the two. A support with an atom at
zero has no reciprocal, and is an error. Zero lying inside a region is
fine: a single point carries no probability there.
Scaling by zero sends every value to 0. Density that was spread over a
region is compressed onto that single point, and density compressed onto a
point is mass. So whatever the support was, the result has a mass at 0 and
density nowhere: discrete(0). Only an empty support, having nothing to
compress, stays empty.
It only works in that direction. A mass sits on one point and lands on one point, so mass stays mass.
A strictly monotonic map stretches and shifts regions but never squashes
one down to a point, so density stays density and mass stays mass. That is
why support_transform() asks for a monotonic map, and why scaling by zero
— which is not one — is handled separately.
Value
A support object.
See Also
Other Support algebra:
support_add_atoms(),
support_contains(),
support_restrict(),
support_union()
Examples
support_shift(continuous(c(0, 1)), by = 5) support_scale(discrete(natural0()), by = 2) # A decreasing map reverses the region. support_scale(continuous(c(1, 2)), by = -1) # Scaling by zero collapses everything onto a single atom. support_scale(continuous(c(1, 2)), by = 0) # Reciprocal of a support spanning zero. support_reciprocal(continuous(c(-2, 4))) # The general form. support_transform( continuous(c(0, Inf)), fun = exp, inv = log, domain = c(0, Inf), range = c(1, Inf) )support_shift(continuous(c(0, 1)), by = 5) support_scale(discrete(natural0()), by = 2) # A decreasing map reverses the region. support_scale(continuous(c(1, 2)), by = -1) # Scaling by zero collapses everything onto a single atom. support_scale(continuous(c(1, 2)), by = 0) # Reciprocal of a support spanning zero. support_reciprocal(continuous(c(-2, 4))) # The general form. support_transform( continuous(c(0, Inf)), fun = exp, inv = log, domain = c(0, Inf), range = c(1, Inf) )
Combine Supports
Description
The union of two or more supports: everything covered by any of them.
Usage
support_union(...)support_union(...)
Arguments
... |
Supports to combine, or a single list of them. Distributions are
accepted in place of supports. With no arguments, the result is
If any of them has no support — a |
Details
The atomic parts are unioned as series, and the continuous parts are pooled and merged back into canonical form, so touching or overlapping intervals come out as one.
An atom that falls inside another support's continuous part stays an atom. The two carry different kinds of probability, and one does not absorb the other.
Value
A support object.
See Also
support_restrict() to cut a support down instead.
Other Support algebra:
support_add_atoms(),
support_contains(),
support_restrict(),
support_transform()
Examples
support_union(continuous(c(0, 1)), continuous(c(0.5, 3))) support_union(discrete(c(1, 2)), continuous(c(5, 6))) # The identity. support_union()support_union(continuous(c(0, 1)), continuous(c(0.5, 3))) support_union(discrete(c(1, 2)), continuous(c(5, 6))) # The identity. support_union()
Specify the Support of a Distribution
Description
A support says where a distribution's probability lives, and in what form.
Probability comes in two forms: mass, which sits on single points, and
density, which is spread over regions. A support records both — the
points carrying mass, its atoms, and the regions carrying density —
and discrete(), continuous() and mixed() build one from those pieces.
Usage
discrete(atoms = numeric(0)) continuous(...) mixed(discrete = numeric(0), continuous = numeric(0))discrete(atoms = numeric(0)) continuous(...) mixed(discrete = numeric(0), continuous = numeric(0))
Arguments
atoms |
For |
... |
For |
discrete, continuous
|
For |
Details
The variable type (vtype()) is derived from the support: a support with
only atoms is "discrete", only a continuous part is "continuous", both
is "mixed", and neither is "empty".
Because the type is derived, none of the three insists on being handed
something non-empty. Each builds whatever the parts describe, and describing
nothing gives empty_support(). So mixed(continuous = continuous()) is
the whole real line, discrete(numeric(0)) is empty, and mixed() is empty
too. This is what makes them usable when the parts are computed rather than
typed and may come out empty; mixed() is then the general constructor,
with discrete() and continuous() the direct way to say one kind on its
own.
A region is written as a closed interval, but its endpoints carry no probability either way, a single point having no width, so open against closed makes no difference there. An atom that happens to sit on a region's boundary is simply tracked as an atom.
Recording where the mass is and where the density is are two pieces of
information, not one. Knowing which values are possible is not enough:
continuous(c(0, 1)) and mixed(discrete = 0, continuous = c(0, 1)) cover
the same values, but they are different supports and the distributions over
them differ: one has P(X = 0) = 0, the other does not. This is why an
atom lying inside a region is kept rather than absorbed into it.
The third kind
Strictly, a measure on the real line splits into three parts, not two: mass on points, density over regions, and a third kind with neither — all of its probability on a set of zero total length, none of it sitting on any point. The Cantor distribution is the usual example. This is the Lebesgue decomposition, and the third part is called singular continuous. A support here has no way to describe one, so such distributions are out of reach.
Value
A support object (class "support").
See Also
support() to retrieve a distribution's support, vtype() for the
derived variable type.
Other Support:
atoms(),
empty_support(),
is_support(),
support()
Examples
discrete(discretes::natural0()) # e.g. the support of a Poisson discrete(c(3.5, 1.2, 6.7)) # finitely many atoms continuous(c(0, Inf)) # e.g. the support of a Gamma continuous(c(0, 1), c(3, 4)) # a union of regions mixed(discrete = 0, continuous = c(0, Inf)) # an atom, plus a taildiscrete(discretes::natural0()) # e.g. the support of a Poisson discrete(c(3.5, 1.2, 6.7)) # finitely many atoms continuous(c(0, Inf)) # e.g. the support of a Gamma continuous(c(0, 1), c(3, 4)) # a union of regions mixed(discrete = 0, continuous = c(0, Inf)) # an atom, plus a tail
Variable Type of a Distribution
Description
Retrieve the variable type of a distribution, such as "continuous" or "discrete".
Usage
vtype(distribution)vtype(distribution)
Arguments
distribution |
Distribution object. |
Value
Single character with the variable type.
Examples
vtype(dst_beta(1, 2)) vtype(dst_bern(0.4)) vtype(distribution( cdf = pnorm, density = dnorm, .support = continuous() ))vtype(dst_beta(1, 2)) vtype(dst_bern(0.4)) vtype(distribution( cdf = pnorm, density = dnorm, .support = continuous() ))