Title:	Algebra over Probability Distributions
Version:	0.9.1
Description:	Provides an algebra over probability distributions enabling composition, sampling, and automatic simplification to closed forms. Supports normal, exponential, gamma, Weibull, chi-squared, uniform, beta, log-normal, Poisson, multivariate normal, empirical, and mixture distributions with algebraic operators (addition, subtraction, multiplication, division, power, exp, log, min, max) that automatically simplify when mathematical identities apply. Includes closed-form MVN conditioning (Schur complement), affine transformations, mixture marginals/conditionals (Bayes rule), and limiting distribution builders (CLT, LLN, delta method). Uses S3 classes for distributions and R6 for support objects.
License:	GPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.3
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0)
Config/testthat/edition:	3
Imports:	stats, mvtnorm, R6
Depends:	R (≥ 3.5.0)
VignetteBuilder:	knitr
URL:	https://github.com/queelius/algebraic.dist, https://queelius.github.io/algebraic.dist/
BugReports:	https://github.com/queelius/algebraic.dist/issues
NeedsCompilation:	no
Packaged:	2026-02-27 11:41:13 UTC; spinoza
Author:	Alexander Towell [aut, cre]
Maintainer:	Alexander Towell <lex@metafunctor.com>
Repository:	CRAN
Date/Publication:	2026-02-27 13:30:02 UTC

Multiplication of distribution objects.

Description

Handles scalar * dist, dist * scalar, and dist * dist.

Usage

## S3 method for class 'dist'
x * y

Arguments

Value

A simplified distribution or edist

Examples

# Scalar multiplication simplifies for normal
z <- 2 * normal(0, 1)
z  # Normal(mu = 0, var = 4)

# Product of two distributions yields an edist
w <- normal(0, 1) * exponential(1)
is_edist(w)  # TRUE

Method for adding dist objects, or shifting a distribution by a scalar.

Description

Creates an expression distribution and automatically simplifies to closed form when possible (e.g., normal + normal = normal, normal + scalar = normal with shifted mean).

Usage

## S3 method for class 'dist'
x + y

Arguments

Value

A simplified distribution or edist if no closed form exists

Examples

# Sum of two normals simplifies to a normal
z <- normal(0, 1) + normal(2, 3)
z  # Normal(mu = 2, var = 4)

# Shift a distribution by a constant
normal(0, 1) + 5  # Normal(mu = 5, var = 1)

Method for negation or subtraction of dist objects.

Description

Unary: returns negated distribution (e.g., -N(mu, var) = N(-mu, var)) Binary: creates expression distribution and simplifies to closed form when possible (e.g., normal - normal = normal, normal - scalar = normal).

Usage

## S3 method for class 'dist'
x - y

Arguments

Value

A simplified distribution or edist if no closed form exists

Examples

# Difference of normals simplifies to a normal
z <- normal(5, 2) - normal(1, 3)
z  # Normal(mu = 4, var = 5)

# Unary negation
-normal(3, 1)  # Normal(mu = -3, var = 1)

Division of distribution objects.

Description

Handles dist / scalar (delegates to dist * (1/scalar)), scalar / dist, and dist / dist.

Usage

## S3 method for class 'dist'
x / y

Arguments

Value

A simplified distribution or edist

Examples

# Division by scalar reuses multiplication rule
z <- normal(0, 4) / 2
z  # Normal(mu = 0, var = 1)

Math group generic for distribution objects.

Description

Handles exp(), log(), sqrt(), abs(), cos(), sin(), etc.

Usage

## S3 method for class 'dist'
Math(x, ...)

Arguments

Value

A simplified distribution or edist

Examples

# exp(Normal) simplifies to LogNormal
z <- exp(normal(0, 1))
z

# sqrt of a distribution (no closed-form rule, remains edist)
w <- sqrt(exponential(1))
is_edist(w)  # TRUE

Summary group generic for distribution objects.

Description

Handles sum(), prod(), min(), max() of distributions.

Usage

## S3 method for class 'dist'
Summary(..., na.rm = FALSE)

Arguments

Value

A simplified distribution or edist

Examples

# sum() reduces via + operator
z <- sum(normal(0, 1), normal(2, 3))
z  # Normal(mu = 2, var = 4)

# min() of exponentials simplifies
w <- min(exponential(1), exponential(2))
w  # Exponential(rate = 3)

Power operator for distribution objects.

Description

Power operator for distribution objects.

Usage

## S3 method for class 'dist'
x ^ y

Arguments

Value

A simplified distribution or edist

Examples

# Standard normal squared yields chi-squared(1)
z <- normal(0, 1)^2
z

Affine transformation of a normal or multivariate normal distribution.

Description

Computes the distribution of AX + b where X \sim MVN(\mu, \Sigma). The result is MVN(A\mu + b, A \Sigma A^T).

Usage

affine_transform(x, A, b = NULL)

Arguments

Details

For a univariate normal, scalars A and b are promoted to 1x1 matrices and scalar internally. Returns a normal if the result is 1-dimensional.

Value

A normal or mvn distribution.

Examples

X <- mvn(c(0, 0), diag(2))
# Project to first component via 1x2 matrix
Y <- affine_transform(X, A = matrix(c(1, 0), 1, 2), b = 5)
mean(Y)

# Scale a univariate normal
Z <- affine_transform(normal(0, 1), A = 3, b = 2)
mean(Z)
vcov(Z)

Convert an object to a probability distribution.

Description

Generic method for converting objects (such as fitted models) into distribution objects from the algebraic.dist package.

Usage

as_dist(x, ...)

## S3 method for class 'dist'
as_dist(x, ...)

Arguments

Value

A dist object.

Examples

# Identity for existing distributions
d <- normal(0, 1)
identical(as_dist(d), d)

Construct a beta distribution object.

Description

Creates an S3 object representing a beta distribution with shape parameters shape1 and shape2. The PDF on (0, 1) is

f(x) = \frac{x^{a-1}(1-x)^{b-1}}{B(a,b)}

where a = shape1, b = shape2, and B(a,b) is the beta function.

Usage

beta_dist(shape1, shape2)

Arguments

Value

A beta_dist object with classes c("beta_dist", "univariate_dist", "continuous_dist", "dist").

Examples

x <- beta_dist(shape1 = 2, shape2 = 5)
mean(x)
vcov(x)
format(x)

Generic method for obtaining the cdf of an object.

Description

Generic method for obtaining the cdf of an object.

Usage

cdf(x, ...)

Arguments

Value

A function computing the cumulative distribution function.

Examples

x <- normal(0, 1)
F <- cdf(x)
F(0)    # 0.5 (median of standard normal)
F(1.96) # approximately 0.975

Cumulative distribution function for a beta distribution.

Description

Returns a function that evaluates the beta CDF at given points.

Usage

## S3 method for class 'beta_dist'
cdf(x, ...)

Arguments

Value

A function function(q, log.p = FALSE, ...) returning the CDF (or log-CDF) at q.

Examples

x <- beta_dist(2, 5)
F <- cdf(x)
F(0.3)
F(0.5)

Method for obtaining the cdf of a chi_squared object.

Description

Method for obtaining the cdf of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
cdf(x, ...)

Arguments

Value

A function that computes the cdf at point(s) t

Examples

x <- chi_squared(5)
F <- cdf(x)
F(5)
F(10)

CDF for expression distributions.

Description

Falls back to realize to materialize the distribution as an empirical_dist, then delegates to cdf.empirical_dist.

Usage

## S3 method for class 'edist'
cdf(x, ...)

Arguments

Value

A function computing the empirical CDF.

Examples

set.seed(1)
z <- normal(0, 1) * exponential(1)
Fz <- cdf(z)
Fz(0)

Method for obtaining the cdf of empirical_dist object x.

Description

If x is a multivariate empirical distribution, this function will throw an error. It's only defined for univariate empirical distributions.

Usage

## S3 method for class 'empirical_dist'
cdf(x, ...)

Arguments

Value

A function that takes a numeric vector t and returns the empirical cdf of x evaluated at t.

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
Fx <- cdf(ed)
Fx(3) # 0.6
Fx(c(1, 5)) # 0.2, 1.0

Method to obtain the cdf of an exponential object.

Description

Method to obtain the cdf of an exponential object.

Usage

## S3 method for class 'exponential'
cdf(x, ...)

Arguments

Value

A function that computes the cdf of the exponential. Accepts as input a vector t at which to compute the cdf, an input rate denoting the failure rate of the exponential distribution, and a logical log indicating whether to compute the log of the cdf. By default, rate is the failure rate of object x.

Examples

x <- exponential(rate = 1)
F <- cdf(x)
F(1)
F(2)

Method for obtaining the cdf of a gamma_dist object.

Description

Method for obtaining the cdf of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
cdf(x, ...)

Arguments

Value

A function that computes the cdf at point(s) t

Examples

x <- gamma_dist(shape = 2, rate = 1)
F <- cdf(x)
F(1)
F(2)

Cumulative distribution function for a log-normal distribution.

Description

Returns a function that evaluates the log-normal CDF at given points.

Usage

## S3 method for class 'lognormal'
cdf(x, ...)

Arguments

Value

A function function(q, log.p = FALSE, ...) returning the CDF (or log-CDF) at q.

Examples

x <- lognormal(0, 1)
F <- cdf(x)
F(1)
F(2)

Cumulative distribution function for a mixture distribution.

Description

Returns a function that evaluates the mixture CDF at given points. The mixture CDF is F(x) = \sum_k w_k F_k(x).

Usage

## S3 method for class 'mixture'
cdf(x, ...)

Arguments

Value

A function function(q, ...) returning the CDF at q.

Examples

m <- mixture(list(normal(0, 1), normal(5, 1)), c(0.5, 0.5))
F <- cdf(m)
F(0)
F(5)

Method for obtaining the CDF of a mvn object.

Description

Method for obtaining the CDF of a mvn object.

Usage

## S3 method for class 'mvn'
cdf(x, ...)

Arguments

Value

A function computing the multivariate normal CDF.

Examples

X <- mvn(c(0, 0), diag(2))
F <- cdf(X)
F(c(0, 0))

Method for obtaining the cdf of an normal object.

Description

Method for obtaining the cdf of an normal object.

Usage

## S3 method for class 'normal'
cdf(x, ...)

Arguments

Value

A function that computes the cdf of the normal distribution. It accepts as input a parameter vector q, a mean vector mu, a variance var, and a log argument determining whether to compute the log of the cdf. By default, mu and var are the mean and variance of object x and log is FALSE. Finally, it accepts additional arguments ... to pass to the pnorm function.

Examples

x <- normal(0, 1)
F <- cdf(x)
F(0)
F(1.96)

Cumulative distribution function for a Poisson distribution.

Description

Returns a function that evaluates the Poisson CDF at given points.

Usage

## S3 method for class 'poisson_dist'
cdf(x, ...)

Arguments

Value

A function function(q, log.p = FALSE, ...) returning the CDF (or log-CDF) at q.

Examples

x <- poisson_dist(5)
F <- cdf(x)
F(5)
F(10)

Cumulative distribution function for a uniform distribution.

Description

Returns a function that evaluates the uniform CDF at given points.

Usage

## S3 method for class 'uniform_dist'
cdf(x, ...)

Arguments

Value

A function function(q, log.p = FALSE, ...) returning the CDF (or log-CDF) at q.

Examples

x <- uniform_dist(0, 10)
F <- cdf(x)
F(5)
F(10)

Cumulative distribution function for a Weibull distribution.

Description

Returns a function that evaluates the Weibull CDF at given points.

Usage

## S3 method for class 'weibull_dist'
cdf(x, ...)

Arguments

Value

A function function(q, log.p = FALSE, ...) returning the CDF (or log-CDF) at q.

Examples

x <- weibull_dist(shape = 2, scale = 3)
F <- cdf(x)
F(1)
F(3)

Construct a chi-squared distribution object.

Description

Construct a chi-squared distribution object.

Usage

chi_squared(df)

Arguments

Value

A chi_squared object

Examples

x <- chi_squared(df = 5)
mean(x)
vcov(x)
format(x)

Central Limit Theorem Limiting Distribution

Description

Returns the limiting distribution of the standardized sample mean \sqrt{n}(\bar{X}_n - \mu) under the Central Limit Theorem. For a univariate distribution with variance \sigma^2, this is N(0, \sigma^2). For a multivariate distribution with covariance matrix \Sigma, this is MVN(0, \Sigma).

Usage

clt(base_dist)

Arguments

Value

A normal or mvn distribution representing the CLT limiting distribution.

Examples

# CLT for Exp(2): sqrt(n)(Xbar - 1/2) -> N(0, 1/4)
x <- exponential(rate = 2)
z <- clt(x)
mean(z)
vcov(z)

Generic method for obtaining the conditional distribution of a distribution object x given condition P.

Description

Generic method for obtaining the conditional distribution of a distribution object x given condition P.

Usage

conditional(x, P, ...)

Arguments

Value

A distribution object for the conditional distribution.

Examples

d <- empirical_dist(1:100)
# condition on values greater than 50
d_gt50 <- conditional(d, function(x) x > 50)
mean(d_gt50)

Method for obtaining the condition distribution, x | P(x), of dist object x.

Description

Falls back to MC: materializes x via ensure_realized() and then conditions on the resulting empirical distribution.

Usage

## S3 method for class 'dist'
conditional(x, P, n = 10000L, ...)

Arguments

Value

An empirical_dist approximating the conditional distribution.

Examples

set.seed(1)
x <- exponential(1)
# Condition on X > 2
x_gt2 <- conditional(x, function(t) t > 2)
mean(x_gt2)

Conditional distribution for expression distributions.

Description

Falls back to realize and delegates to conditional.empirical_dist.

Usage

## S3 method for class 'edist'
conditional(x, P, ...)

Arguments

Value

A conditional empirical_dist.

Examples

set.seed(1)
z <- normal(0, 1) + exponential(1)
z_pos <- conditional(z, function(t) t > 2)
mean(z_pos)

Method for obtaining the condition distribution, x | P(x), of empirical_dist object x.

Description

In other words, we condition the data on the predicate function. In order to do so, we simply remove all rows from the data that do not satisfy the predicate P. For instance, if we have a 2-dimensional distribution, and we want to condition on the first dimension being greater than the second dimension, we would do the following:

Usage

## S3 method for class 'empirical_dist'
conditional(x, P, ...)

Arguments

Details

x_cond <- conditional(x, function(d) d[1] > d[2])

This would return a new empirical distribution object with the same dimensionality as x, but with all rows where the first dimension is less than or equal to the second dimension removed.

Value

An empirical_dist containing only rows satisfying P.

Examples

mat <- matrix(c(1, 5, 2, 3, 4, 1, 6, 2), ncol = 2)
ed <- empirical_dist(mat)
# Condition on first column being greater than second
ed_cond <- conditional(ed, function(d) d[1] > d[2])
nobs(ed_cond)

Conditional distribution of a mixture.

Description

For a mixture of distributions that support closed-form conditioning (e.g. MVN), uses Bayes' rule to update the mixing weights:

w_k' \propto w_k f_k(x_{given})

where f_k is the marginal density of component k at the observed values. The component conditionals are computed via conditional(component_k, given_indices = ..., given_values = ...).

Usage

## S3 method for class 'mixture'
conditional(x, P = NULL, ..., given_indices = NULL, given_values = NULL)

Arguments

Details

Falls back to MC realization if P is provided or if any component does not support given_indices/given_values.

Value

A mixture or empirical_dist object.

Examples

# Closed-form conditioning on MVN mixture
m <- mixture(
  list(mvn(c(0, 0), diag(2)), mvn(c(3, 3), diag(2))),
  c(0.5, 0.5)
)
# Condition on X2 = 1
mc <- conditional(m, given_indices = 2, given_values = 1)
mean(mc)

Conditional distribution for multivariate normal.

Description

Supports two calling patterns:

1. Closed-form (via given_indices and given_values): Uses the exact Schur complement formula. Returns a normal (1D result) or mvn.

2. Predicate-based (via P): Falls back to MC realization via ensure_realized.

Usage

## S3 method for class 'mvn'
conditional(x, P = NULL, ..., given_indices = NULL, given_values = NULL)

Arguments

Value

A normal, mvn, or empirical_dist object.

Examples

# Closed-form conditioning: X2 | X1 = 1
sigma <- matrix(c(1, 0.5, 0.5, 1), 2, 2)
X <- mvn(c(0, 0), sigma)
X2_given <- conditional(X, given_indices = 1, given_values = 1)
mean(X2_given)
vcov(X2_given)

# Predicate-based MC fallback (slower)

set.seed(42)
X2_mc <- conditional(X, P = function(x) x[1] > 0)

Countable Set

Description

A countably infinite support set, such as the non-negative integers. It satisfies the concept of a support (see has, infimum, supremum, dim).

Public fields

Methods

Public methods

• countable_set$new()

• countable_set$clone()

Method new()

Initialize a countable set.

Usage

countable_set$new(lower = 0L)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

countable_set$clone(deep = FALSE)

Arguments

Delta Method CLT Limiting Distribution

Description

Returns the limiting distribution of \sqrt{n}(g(\bar{X}_n) - g(\mu)) under the Delta Method. For a univariate distribution, this is N(0, g'(\mu)^2 \sigma^2). For a multivariate distribution with Jacobian J = Dg(\mu), this is MVN(0, J \Sigma J^T).

Usage

delta_clt(base_dist, g, dg)

Arguments

Value

A normal or mvn distribution representing the Delta Method limiting distribution.

Examples

# Delta method: g = exp, dg = exp
x <- exponential(rate = 1)
z <- delta_clt(x, g = exp, dg = exp)
mean(z)
vcov(z)

Probability density function for a beta distribution.

Description

Returns a function that evaluates the beta PDF at given points.

Usage

## S3 method for class 'beta_dist'
density(x, ...)

Arguments

Value

A function function(t, log = FALSE, ...) returning the density (or log-density) at t.

Examples

x <- beta_dist(2, 5)
f <- density(x)
f(0.3)
f(0.5)

Method for obtaining the density (pdf) of a chi_squared object.

Description

Method for obtaining the density (pdf) of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
density(x, ...)

Arguments

Value

A function that computes the pdf at point(s) t

Examples

x <- chi_squared(5)
f <- density(x)
f(5)
f(10)

Density for expression distributions.

Description

Falls back to realize and delegates to density.empirical_dist.

Usage

## S3 method for class 'edist'
density(x, ...)

Arguments

Value

A function computing the empirical density (PMF).

Examples

set.seed(1)
z <- normal(0, 1) * exponential(1)
fz <- density(z)

Method for obtaining the pdf of a empirical_dist object.

Description

Method for obtaining the pdf of a empirical_dist object.

Usage

## S3 method for class 'empirical_dist'
density(x, ...)

Arguments

Value

A function computing the empirical PMF at given points.

Note

sort tibble lexicographically and do a binary search to find upper and lower bound in log(nobs(x)) time.

Examples

ed <- empirical_dist(c(1, 2, 2, 3, 3, 3))
f <- density(ed)
f(2)          # 2/6
f(3, log = TRUE) # log(3/6)

Method to obtain the pdf of an exponential object.

Description

Method to obtain the pdf of an exponential object.

Usage

## S3 method for class 'exponential'
density(x, ...)

Arguments

Value

A function that computes the pdf of the exponential distribution at a given point t. Also accepts a rate argument that determines the failure rate of the exponential distribution (defaults to the failure rate of object x) and a log argument that determines whether to compute the log of the pdf.

Examples

x <- exponential(rate = 2)
f <- density(x)
f(0)
f(1)

Method for obtaining the density (pdf) of a gamma_dist object.

Description

Method for obtaining the density (pdf) of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
density(x, ...)

Arguments

Value

A function that computes the pdf at point(s) t

Examples

x <- gamma_dist(shape = 2, rate = 1)
f <- density(x)
f(1)
f(2)

Probability density function for a log-normal distribution.

Description

Returns a function that evaluates the log-normal PDF at given points.

Usage

## S3 method for class 'lognormal'
density(x, ...)

Arguments

Value

A function function(t, log = FALSE, ...) returning the density (or log-density) at t.

Examples

x <- lognormal(0, 1)
f <- density(x)
f(1)
f(2)

Probability density function for a mixture distribution.

Description

Returns a function that evaluates the mixture density at given points. The mixture density is f(x) = \sum_k w_k f_k(x).

Usage

## S3 method for class 'mixture'
density(x, ...)

Arguments

Value

A function function(t, log = FALSE, ...) returning the density (or log-density) at t.

Examples

m <- mixture(list(normal(0, 1), normal(5, 1)), c(0.5, 0.5))
f <- density(m)
f(0)
f(2.5)

Function generator for obtaining the pdf of an mvn object (multivariate normal).

Description

Function generator for obtaining the pdf of an mvn object (multivariate normal).

Usage

## S3 method for class 'mvn'
density(x, ...)

Arguments

Value

A function that computes the pdf of the mvn distribution. It accepts as input: - obs: vector or matrix of quantiles. when x is a matrix, each row is taken to be a quantile and columns correspond to the number of dimensions, p. - mu: a a vector denoting the population mean. Defaults to the mean of x (an mvn object) - sigma: a matrix denoting the variance-covariance of observations. Defaults to the variance-covariance of x. - log: logical, determines whether to compute the log of the pdf. Defaults to FALSE. - ...: any additional parameters to pass to dmvnorm.

Examples

X <- mvn(c(0, 0), diag(2))
f <- density(X)
f(c(0, 0))
f(c(1, 1))

Method for obtaining the pdf of an normal object.

Description

Method for obtaining the pdf of an normal object.

Usage

## S3 method for class 'normal'
density(x, ...)

Arguments

Value

A function that computes the pdf of the normal distribution. It accepts as input a parameter vector x, a mean vector mu, a variance-covariance matrix var, and a log argument determining whether to compute the log of the pdf. By default, mu and var are the mean and variance of object x.

Examples

x <- normal(0, 1)
f <- density(x)
f(0)
f(1)

Probability mass function for a Poisson distribution.

Description

Returns a function that evaluates the Poisson PMF at given points.

Usage

## S3 method for class 'poisson_dist'
density(x, ...)

Arguments

Value

A function function(k, log = FALSE, ...) returning the probability mass (or log-probability) at k.

Examples

x <- poisson_dist(5)
f <- density(x)
f(5)
f(0)

Probability density function for a uniform distribution.

Description

Returns a function that evaluates the uniform PDF at given points.

Usage

## S3 method for class 'uniform_dist'
density(x, ...)

Arguments

Value

A function function(t, log = FALSE, ...) returning the density (or log-density) at t.

Examples

x <- uniform_dist(0, 10)
f <- density(x)
f(5)
f(15)

Probability density function for a Weibull distribution.

Description

Returns a function that evaluates the Weibull PDF at given points.

Usage

## S3 method for class 'weibull_dist'
density(x, ...)

Arguments

Value

A function function(t, log = FALSE, ...) returning the density (or log-density) at t.

Examples

x <- weibull_dist(shape = 2, scale = 3)
f <- density(x)
f(1)
f(3)

Dimension of a beta distribution (always 1).

Description

Dimension of a beta distribution (always 1).

Usage

## S3 method for class 'beta_dist'
dim(x)

Arguments

Value

1.

Examples

dim(beta_dist(2, 5))

Retrieve the dimension of a chi_squared object.

Description

Retrieve the dimension of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
dim(x)

Arguments

Value

1 (univariate)

Examples

dim(chi_squared(5))

Get the dimension of a countable set.

Description

Get the dimension of a countable set.

Usage

## S3 method for class 'countable_set'
dim(x)

Arguments

Value

1 (always univariate).

Examples

cs <- countable_set$new(0L)
dim(cs)  # 1

Method for obtaining the dimension of a empirical_dist object.

Description

Method for obtaining the dimension of a empirical_dist object.

Usage

## S3 method for class 'empirical_dist'
dim(x)

Arguments

Value

Integer; the number of dimensions.

Examples

ed1 <- empirical_dist(c(1, 2, 3))
dim(ed1) # 1

ed2 <- empirical_dist(matrix(1:6, ncol = 2))
dim(ed2) # 2

Method to obtain the dimension of an exponential object.

Description

Method to obtain the dimension of an exponential object.

Usage

## S3 method for class 'exponential'
dim(x)

Arguments

Value

The dimension of the exponential object

Examples

dim(exponential(rate = 1))

Return the dimension of the finite set.

Description

Return the dimension of the finite set.

Usage

## S3 method for class 'finite_set'
dim(x)

Arguments

Value

Integer; the dimension of the set.

Examples

fs <- finite_set$new(c(1, 3, 5, 7))
dim(fs) # 1

Retrieve the dimension of a gamma_dist object.

Description

Retrieve the dimension of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
dim(x)

Arguments

Value

1 (univariate)

Examples

dim(gamma_dist(2, 1))

Return the dimension of the interval.

Description

Return the dimension of the interval.

Usage

## S3 method for class 'interval'
dim(x)

Arguments

Value

Integer; the number of interval components.

Examples

iv <- interval$new(lower = 0, upper = 1)
dim(iv) # 1

Dimension of a log-normal distribution (always 1).

Description

Dimension of a log-normal distribution (always 1).

Usage

## S3 method for class 'lognormal'
dim(x)

Arguments

Value

1.

Examples

dim(lognormal(0, 1))

Dimension of a mixture distribution.

Description

Returns the dimension of the first component (all components are assumed to have the same dimension).

Usage

## S3 method for class 'mixture'
dim(x)

Arguments

Value

The dimension of the distribution.

Examples

m <- mixture(list(normal(0, 1), normal(5, 1)), c(0.5, 0.5))
dim(m)

Method for obtaining the dimension of an mvn object.

Description

Method for obtaining the dimension of an mvn object.

Usage

## S3 method for class 'mvn'
dim(x)

Arguments

Value

The dimension of the mvn object

Examples

dim(mvn(c(0, 0, 0)))

Method for obtaining the dimension of a normal object.

Description

Method for obtaining the dimension of a normal object.

Usage

## S3 method for class 'normal'
dim(x)

Arguments

Value

The dimension of the normal object

Examples

dim(normal(0, 1))

Dimension of a Poisson distribution (always 1).

Description

Dimension of a Poisson distribution (always 1).

Usage

## S3 method for class 'poisson_dist'
dim(x)

Arguments

Value

1.

Examples

dim(poisson_dist(5))

Dimension of a uniform distribution (always 1).

Description

Dimension of a uniform distribution (always 1).

Usage

## S3 method for class 'uniform_dist'
dim(x)

Arguments

Value

1.

Examples

dim(uniform_dist(0, 1))

Dimension of a Weibull distribution (always 1).

Description

Dimension of a Weibull distribution (always 1).

Usage

## S3 method for class 'weibull_dist'
dim(x)

Arguments

Value

1.

Examples

dim(weibull_dist(2, 3))

Takes an expression e and a list vars and returns a lazy edist (expression distribution object), that is a subclass of dist that can be used in place of a dist object.

Description

Takes an expression e and a list vars and returns a lazy edist (expression distribution object), that is a subclass of dist that can be used in place of a dist object.

Usage

edist(e, vars)

Arguments

Value

An edist object.

Examples

x <- normal(0, 1)
y <- normal(2, 3)
e <- edist(quote(x + y), list(x = x, y = y))
e

Construct empirical distribution object.

Description

Construct empirical distribution object.

Usage

empirical_dist(data)

Arguments

Value

An empirical_dist object.

Examples

# Univariate empirical distribution from a vector
ed <- empirical_dist(c(1, 2, 3, 4, 5))
mean(ed)

# Multivariate empirical distribution from a matrix
mat <- matrix(c(1, 2, 3, 4, 5, 6), ncol = 2)
ed_mv <- empirical_dist(mat)
dim(ed_mv)

Memoized MC fallback materialization.

Description

Single internal entry point for all Monte Carlo fallback paths. If x is already an empirical_dist, returns it unchanged. If x has a .cache environment (e.g. edist objects), caches the realization so that multiple method calls (e.g. cdf + density) share the same samples. Sample-size-aware: if the cached realization has fewer than n samples, re-realizes.

Usage

ensure_realized(x, n = 10000L)

Arguments

Value

An empirical_dist (or realized_dist).

Generic method for obtaining the expectation of f with respect to x.

Description

Generic method for obtaining the expectation of f with respect to x.

Usage

expectation(x, g, ...)

Arguments

Value

The expected value of g(x).

Examples

x <- exponential(1)
# E[X] for Exp(1) is 1
expectation(x, function(t) t)

Expectation of a Function Applied to a dist Object

Description

Expectation operator applied to x of type dist with respect to a function g. Optionally, constructs a confidence interval for the expectation estimate using the Central Limit Theorem.

Usage

## S3 method for class 'dist'
expectation(x, g = function(t) t, ..., control = list())

Arguments

Value

If compute_stats is FALSE, then the estimate of the expectation, otherwise a list with the following components: value - The estimate of the expectation ci - The confidence intervals for each component of the expectation n - The number of samples

Examples

# MC expectation of X^2 where X ~ Exp(1)
set.seed(1)
ex <- exponential(1)
expectation(ex, g = function(t) t^2)

Method for obtaining the expectation of empirical_dist object x under function g.

Description

Method for obtaining the expectation of empirical_dist object x under function g.

Usage

## S3 method for class 'empirical_dist'
expectation(x, g = function(t) t, ..., control = list())

Arguments

Value

If compute_stats is FALSE, then the estimate of the expectation, otherwise a list with the following components: value - The estimate of the expectation ci - The confidence intervals for each component of the expectation n - The number of samples

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
expectation(ed)                     # E[X] = 3
expectation(ed, function(x) x^2)    # E[X^2] = 11

Exact expectation for a Poisson distribution.

Description

Computes E[g(X)] using truncated summation over the support. The summation is truncated at the 1 - 10^{-12} quantile to ensure negligible truncation error.

Usage

## S3 method for class 'poisson_dist'
expectation(x, g, ...)

Arguments

Value

The expected value E[g(X)].

Examples

x <- poisson_dist(5)
expectation(x, identity)
expectation(x, function(k) k^2)

Method for obtaining the expectation of f with respect to a univariate_dist object x.

Description

Assumes the support is a contiguous interval that has operations for retrieving the lower and upper bounds.

Usage

## S3 method for class 'univariate_dist'
expectation(x, g, ..., control = list())

Arguments

Value

The expected value (numeric scalar), or the full integrate() result if compute_stats = TRUE.

Examples

x <- normal(3, 4)
# E[X] for Normal(3, 4) is 3
expectation(x, function(t) t)

# E[X^2] for Exp(1) is 2
expectation(exponential(1), function(t) t^2)

Function used for computing expectations given data (e.g., from an MC simulation or bootstrap). it expects a matrix, or something that can be coerced to a matrix (e.g., a data frame). it also expects a function g to apply to each row of the data, and returns the expectation of g under the empirical distribution of the data. it also returns a confidence interval for the expectation, and the number of samples used to compute the expectation.

Description

example: expectation_data(D, function(x) (x-colMeans(D)) %*% t(x-colMeans(D))) computes the covariance of the data D, except the matrix structure is lost (it's just a vector, which can be coerced back to a matrix if needed).

Usage

expectation_data(
  data,
  g = function(x) x,
  ...,
  compute_stats = TRUE,
  alpha = 0.05
)

Arguments

Value

if compute_stats is TRUE, then a list with the following components: value - The estimate of the expectation ci - The confidence intervals for each component of the expectation n - The number of samples otherwise, just the value of the expectation.

Examples

set.seed(42)
data <- matrix(rnorm(200), ncol = 2)
# sample mean with confidence interval
expectation_data(data)

# just the point estimate, no CI
expectation_data(data, compute_stats = FALSE)

# expectation of a function of the data (row-wise)
expectation_data(data, g = function(x) sum(x^2))

Construct exponential distribution object.

Description

Construct exponential distribution object.

Usage

exponential(rate)

Arguments

Value

An exponential distribution object.

Examples

x <- exponential(rate = 2)
mean(x)
vcov(x)
format(x)

Finite set

Description

A finite set. It also satisfies the concept of a support.

Public fields

Methods

Public methods

• finite_set$new()

• finite_set$has()

• finite_set$infimum()

• finite_set$supremum()

• finite_set$dim()

• finite_set$clone()

Method new()

Initialize a finite set.

Usage

finite_set$new(values)

Arguments

Method has()

Determine if a value is contained in the finite set.

Usage

finite_set$has(x)

Arguments

Method infimum()

Get the infimum of the finite set.

Usage

finite_set$infimum()

Returns

A numeric vector of infimums.

Method supremum()

Get the supremum of the finite set.

Usage

finite_set$supremum()

Returns

A numeric vector of supremums.

Method dim()

Get the dimension of the finite set.

Usage

finite_set$dim()

Returns

The dimension of the finite set.

Method clone()

The objects of this class are cloneable with this method.

Usage

finite_set$clone(deep = FALSE)

Arguments

Format a beta_dist object as a character string.

Description

Format a beta_dist object as a character string.

Usage

## S3 method for class 'beta_dist'
format(x, ...)

Arguments

Value

A character string describing the distribution.

Examples

format(beta_dist(2, 5))

Format a chi_squared object as a character string.

Description

Format a chi_squared object as a character string.

Usage

## S3 method for class 'chi_squared'
format(x, ...)

Arguments

Value

A character string describing the distribution

Examples

format(chi_squared(5))

Format method for edist objects.

Description

Format method for edist objects.

Usage

## S3 method for class 'edist'
format(x, ...)

Arguments

Value

A character string

Examples

z <- normal(0, 1) * exponential(2)
format(z)

Format method for empirical_dist objects.

Description

Format method for empirical_dist objects.

Usage

## S3 method for class 'empirical_dist'
format(x, ...)

Arguments

Value

A character string

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
format(ed)

Format method for exponential objects.

Description

Format method for exponential objects.

Usage

## S3 method for class 'exponential'
format(x, ...)

Arguments

Value

A character string

Examples

format(exponential(rate = 2))

Format a gamma_dist object as a character string.

Description

Format a gamma_dist object as a character string.

Usage

## S3 method for class 'gamma_dist'
format(x, ...)

Arguments

Value

A character string describing the distribution

Examples

format(gamma_dist(2, 1))

Format a lognormal object as a character string.

Description

Format a lognormal object as a character string.

Usage

## S3 method for class 'lognormal'
format(x, ...)

Arguments

Value

A character string describing the distribution.

Examples

format(lognormal(0, 1))

Format a mixture object as a character string.

Description

Format a mixture object as a character string.

Usage

## S3 method for class 'mixture'
format(x, ...)

Arguments

Value

A character string describing the mixture.

Examples

m <- mixture(list(normal(0, 1), normal(5, 1)), c(0.5, 0.5))
format(m)

Format method for mvn objects.

Description

Format method for mvn objects.

Usage

## S3 method for class 'mvn'
format(x, ...)

Arguments

Value

A character string

Examples

format(mvn(c(0, 0)))

Format method for normal objects.

Description

Format method for normal objects.

Usage

## S3 method for class 'normal'
format(x, ...)

Arguments

Value

A character string

Examples

x <- normal(2, 3)
format(x)

Format a poisson_dist object as a character string.

Description

Format a poisson_dist object as a character string.

Usage

## S3 method for class 'poisson_dist'
format(x, ...)

Arguments

Value

A character string describing the distribution.

Examples

format(poisson_dist(5))

Format a realized_dist object as a character string.

Description

Shows the number of samples and a summary of the source distribution.

Usage

## S3 method for class 'realized_dist'
format(x, ...)

Arguments

Value

A character string.

Examples

rd <- realize(normal(0, 1), n = 100)
format(rd)

Format a uniform_dist object as a character string.

Description

Format a uniform_dist object as a character string.

Usage

## S3 method for class 'uniform_dist'
format(x, ...)

Arguments

Value

A character string describing the distribution.

Examples

format(uniform_dist(0, 10))

Format a weibull_dist object as a character string.

Description

Format a weibull_dist object as a character string.

Usage

## S3 method for class 'weibull_dist'
format(x, ...)

Arguments

Value

A character string describing the distribution.

Examples

format(weibull_dist(2, 3))

Construct a gamma distribution object.

Description

Construct a gamma distribution object.

Usage

gamma_dist(shape, rate)

Arguments

Value

A gamma_dist object

Examples

x <- gamma_dist(shape = 2, rate = 1)
mean(x)
vcov(x)
format(x)

Support

Description

support is a class that represents the support of a random element or distribution, i.e. the set of values that it realize.

It's a conceptual class. To satisfy the concept of a support, the following methods must be implemented:

1. has: a function that returns a logical vector indicating whether each value in a vector is contained in the support

2. infimum: a function that returns the infimum of the support

3. supremum: a function that returns the supremum of the support

4. dim: a function that returns the dimension of the support

We provide two implementations that satisfy the concept:

• interval: a support that is an infiite set of contiguous numeric values

• finite_set: a support that is a finite set of values Determine if a value is contained in the support.

Usage

has(object, x)

Arguments

Value

Logical vector indicating membership.

Examples

I <- interval$new(0, 1, lower_closed = TRUE, upper_closed = TRUE)
has(I, 0.5)  # TRUE
has(I, 2)    # FALSE

S <- finite_set$new(c(1, 2, 3))
has(S, 2)    # TRUE
has(S, 4)    # FALSE

Check membership in a countable set.

Description

Returns TRUE if all values are integers (within floating-point tolerance) that are at least as large as the lower bound.

Usage

## S3 method for class 'countable_set'
has(object, x)

Arguments

Value

Logical; TRUE if all values are valid members of the set.

Examples

cs <- countable_set$new(0L)
has(cs, c(0, 3, 5))   # TRUE
has(cs, c(-1, 2))     # FALSE (negative integer)
has(cs, 1.5)          # FALSE (not integer)

Determine if a value is contained in the finite set.

Description

Determine if a value is contained in the finite set.

Usage

## S3 method for class 'finite_set'
has(object, x)

Arguments

Value

Logical indicating membership.

Examples

fs <- finite_set$new(c(1, 3, 5, 7))
has(fs, 3) # TRUE
has(fs, 4) # FALSE

Determine if a value is contained in the interval.

Description

Determine if a value is contained in the interval.

Usage

## S3 method for class 'interval'
has(object, x)

Arguments

Value

Logical vector indicating containment.

Examples

iv <- interval$new(lower = 0, upper = 1)
has(iv, 0.5) # TRUE
has(iv, 2.0) # FALSE

Generic method for obtaining the hazard function of an object.

Description

Generic method for obtaining the hazard function of an object.

Usage

hazard(x, ...)

Arguments

Value

A function computing the hazard rate at given points.

Examples

x <- exponential(2)
h <- hazard(x)
h(1)  # hazard rate at t = 1 (constant for exponential)

Method for obtaining the hazard function of a chi_squared object.

Description

Method for obtaining the hazard function of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
hazard(x, ...)

Arguments

Value

A function that computes h(t) = f(t) / S(t)

Examples

x <- chi_squared(5)
h <- hazard(x)
h(5)

Method to obtain the hazard function of an exponential object.

Description

Method to obtain the hazard function of an exponential object.

Usage

## S3 method for class 'exponential'
hazard(x, ...)

Arguments

Value

A function that computes the hazard function of the exponential distribution at a given point t and rate rate. By default, rate is the failure rate of object x Also accepts a log argument that determines whether to compute the log of the hazard function.

Examples

x <- exponential(rate = 2)
h <- hazard(x)
h(1)
h(5)

Method for obtaining the hazard function of a gamma_dist object.

Description

Method for obtaining the hazard function of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
hazard(x, ...)

Arguments

Value

A function that computes h(t) = f(t) / S(t)

Examples

x <- gamma_dist(shape = 2, rate = 1)
h <- hazard(x)
h(1)

Hazard function for a log-normal distribution.

Description

Returns a function that evaluates the log-normal hazard rate h(t) = f(t) / S(t) for t > 0.

Usage

## S3 method for class 'lognormal'
hazard(x, ...)

Arguments

Value

A function function(t, log = FALSE) returning the hazard (or log-hazard) at t.

Examples

x <- lognormal(0, 1)
h <- hazard(x)
h(1)
h(2)

Hazard function for a Weibull distribution.

Description

Returns a function that evaluates the Weibull hazard rate h(t) = (shape/scale)(t/scale)^{shape-1} for t > 0.

Usage

## S3 method for class 'weibull_dist'
hazard(x, ...)

Arguments

Value

A function function(t, log = FALSE) returning the hazard (or log-hazard) at t.

Examples

x <- weibull_dist(shape = 2, scale = 3)
h <- hazard(x)
h(1)
h(3)

Get the infimum of the support.

Description

Get the infimum of the support.

Usage

infimum(object)

Arguments

Value

The infimum (greatest lower bound) of the support.

Examples

I <- interval$new(0, 10)
infimum(I)  # 0

S <- finite_set$new(c(3, 7, 11))
infimum(S)  # 3

Get the infimum of a countable set.

Description

Get the infimum of a countable set.

Usage

## S3 method for class 'countable_set'
infimum(object)

Arguments

Value

The lower bound (integer).

Examples

cs <- countable_set$new(0L)
infimum(cs)  # 0

Return the infimum of the finite set.

Description

Return the infimum of the finite set.

Usage

## S3 method for class 'finite_set'
infimum(object)

Arguments

Value

Numeric; the minimum value(s).

Examples

fs <- finite_set$new(c(1, 3, 5, 7))
infimum(fs) # 1

Return the (vector of) infimum of the interval.

Description

Return the (vector of) infimum of the interval.

Usage

## S3 method for class 'interval'
infimum(object)

Arguments

Value

Numeric vector of lower bounds.

Examples

iv <- interval$new(lower = 0, upper = 1)
infimum(iv) # 0

Interval

Description

An interval is a support that is a finite union of intervals.

Public fields

Methods

Public methods

• interval$new()

• interval$is_empty()

• interval$has()

• interval$infimum()

• interval$supremum()

• interval$dim()

• interval$clone()

Method new()

Initialize an interval.

Usage

interval$new(
  lower = -Inf,
  upper = Inf,
  lower_closed = FALSE,
  upper_closed = FALSE
)

Arguments

Method is_empty()

Determine if the interval is empty

Usage

interval$is_empty()

Returns

A logical vector indicating whether the interval is empty.

Method has()

Determine if a value is contained in the interval.

Usage

interval$has(x)

Arguments

Returns

A logical vector indicating whether each value is contained

Method infimum()

Get the infimum of the interval.

Usage

interval$infimum()

Returns

A numeric vector of infimums.

Method supremum()

Get the supremum of the interval.

Usage

interval$supremum()

Returns

A numeric vector of supremums.

Method dim()

Get the dimension of the interval.

Usage

interval$dim()

Returns

The dimension of the interval.

Method clone()

The objects of this class are cloneable with this method.

Usage

interval$clone(deep = FALSE)

Arguments

Generic method for obtaining the quantile (inverse cdf) of an object.

Description

Generic method for obtaining the quantile (inverse cdf) of an object.

Usage

inv_cdf(x, ...)

Arguments

Value

A function computing the quantile (inverse CDF).

Examples

x <- normal(0, 1)
Q <- inv_cdf(x)
Q(0.5)   # 0 (median of standard normal)
Q(0.975) # approximately 1.96

Inverse CDF (quantile function) for a beta distribution.

Description

Returns a function that computes quantiles of the beta distribution.

Usage

## S3 method for class 'beta_dist'
inv_cdf(x, ...)

Arguments

Value

A function function(p, lower.tail = TRUE, log.p = FALSE, ...) returning the quantile at probability p.

Examples

x <- beta_dist(2, 5)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Method for obtaining the inverse cdf (quantile function) of a chi_squared object.

Description

Method for obtaining the inverse cdf (quantile function) of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
inv_cdf(x, ...)

Arguments

Value

A function that computes the quantile at probability p

Examples

x <- chi_squared(5)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Inverse CDF (quantile function) for expression distributions.

Description

Falls back to realize and delegates to inv_cdf.empirical_dist.

Usage

## S3 method for class 'edist'
inv_cdf(x, ...)

Arguments

Value

A function computing the empirical quantile function.

Examples

set.seed(1)
z <- normal(0, 1) * exponential(1)
qz <- inv_cdf(z)
qz(0.5)

Method for obtaining the inverse CDF (quantile function) of a univariate empirical_dist object.

Description

Uses the empirical quantile function from the observed data.

Usage

## S3 method for class 'empirical_dist'
inv_cdf(x, ...)

Arguments

Value

A function that accepts a vector of probabilities p and returns the corresponding quantiles.

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
qf <- inv_cdf(ed)
qf(0.5)       # median
qf(c(0.25, 0.75)) # quartiles

Method to obtain the inverse cdf of an exponential object.

Description

Method to obtain the inverse cdf of an exponential object.

Usage

## S3 method for class 'exponential'
inv_cdf(x, ...)

Arguments

Value

A function that computes the inverse cdf of the exponential distribution. Accepts as input a vector p probabilities to compute the inverse cdf, a rate value denoting the failure rate of the exponential distribution, and a logical log.p indicating whether input p denotes probability or log-probability. By default, rate is the failure rate of object x.

Examples

x <- exponential(rate = 1)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Method for obtaining the inverse cdf (quantile function) of a gamma_dist object.

Description

Method for obtaining the inverse cdf (quantile function) of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
inv_cdf(x, ...)

Arguments

Value

A function that computes the quantile at probability p

Examples

x <- gamma_dist(shape = 2, rate = 1)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Inverse CDF (quantile function) for a log-normal distribution.

Description

Returns a function that computes quantiles of the log-normal distribution.

Usage

## S3 method for class 'lognormal'
inv_cdf(x, ...)

Arguments

Value

A function function(p, lower.tail = TRUE, log.p = FALSE, ...) returning the quantile at probability p.

Examples

x <- lognormal(0, 1)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Method for obtaining the inverse cdf of an normal object.

Description

Method for obtaining the inverse cdf of an normal object.

Usage

## S3 method for class 'normal'
inv_cdf(x, ...)

Arguments

Value

A function that computes the inverse cdf of the normal distribution.

Examples

x <- normal(0, 1)
q <- inv_cdf(x)
q(0.5)
q(0.975)

Inverse CDF (quantile function) for a Poisson distribution.

Description

Returns a function that computes quantiles of the Poisson distribution.

Usage

## S3 method for class 'poisson_dist'
inv_cdf(x, ...)

Arguments

Value

A function function(p, lower.tail = TRUE, log.p = FALSE, ...) returning the quantile at probability p.

Examples

x <- poisson_dist(5)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Inverse CDF (quantile function) for a uniform distribution.

Description

Returns a function that computes quantiles of the uniform distribution.

Usage

## S3 method for class 'uniform_dist'
inv_cdf(x, ...)

Arguments

Value

A function function(p, lower.tail = TRUE, log.p = FALSE, ...) returning the quantile at probability p.

Examples

x <- uniform_dist(0, 10)
q <- inv_cdf(x)
q(0.5)
q(0.9)

Inverse CDF (quantile function) for a Weibull distribution.

Description

Returns a function that computes quantiles of the Weibull distribution.

Usage

## S3 method for class 'weibull_dist'
inv_cdf(x, ...)

Arguments

Value

A function function(p, lower.tail = TRUE, log.p = FALSE, ...) returning the quantile at probability p.

Examples

x <- weibull_dist(shape = 2, scale = 3)
q <- inv_cdf(x)
q(0.5)
q(0.95)

Test whether an object is a beta_dist.

Description

Test whether an object is a beta_dist.

Usage

is_beta_dist(x)

Arguments

Value

TRUE if x inherits from "beta_dist", FALSE otherwise.

Examples

is_beta_dist(beta_dist(2, 5))
is_beta_dist(normal(0, 1))

Test whether an object is a chi_squared.

Description

Test whether an object is a chi_squared.

Usage

is_chi_squared(x)

Arguments

Value

Logical; TRUE if x inherits from chi_squared

Examples

is_chi_squared(chi_squared(3))
is_chi_squared(normal(0, 1))

Function to determine whether an object x is a dist object.

Description

Function to determine whether an object x is a dist object.

Usage

is_dist(x)

Arguments

Value

Logical indicating whether x is a dist object.

Examples

is_dist(normal(0, 1))   # TRUE
is_dist(42)             # FALSE

Function to determine whether an object x is an edist object.

Description

Function to determine whether an object x is an edist object.

Usage

is_edist(x)

Arguments

Value

Logical; TRUE if x is an edist.

Examples

is_edist(normal(0, 1) * exponential(1))  # TRUE
is_edist(normal(0, 1))                   # FALSE

Function to determine whether an object x is an empirical_dist object.

Description

Function to determine whether an object x is an empirical_dist object.

Usage

is_empirical_dist(x)

Arguments

Value

Logical; TRUE if x is an empirical_dist.

Examples

ed <- empirical_dist(c(1, 2, 3))
is_empirical_dist(ed)    # TRUE
is_empirical_dist("abc") # FALSE

Function to determine whether an object x is an exponential object.

Description

Function to determine whether an object x is an exponential object.

Usage

is_exponential(x)

Arguments

Value

Logical; TRUE if x is an exponential.

Examples

is_exponential(exponential(1))
is_exponential(normal(0, 1))

Test whether an object is a gamma_dist.

Description

Test whether an object is a gamma_dist.

Usage

is_gamma_dist(x)

Arguments

Value

Logical; TRUE if x inherits from gamma_dist

Examples

is_gamma_dist(gamma_dist(2, 1))
is_gamma_dist(normal(0, 1))

Test whether an object is a lognormal.

Description

Test whether an object is a lognormal.

Usage

is_lognormal(x)

Arguments

Value

TRUE if x inherits from "lognormal", FALSE otherwise.

Examples

is_lognormal(lognormal(0, 1))
is_lognormal(normal(0, 1))

Test whether an object is a mixture distribution.

Description

Test whether an object is a mixture distribution.

Usage

is_mixture(x)

Arguments

Value

TRUE if x inherits from "mixture", FALSE otherwise.

Examples

m <- mixture(list(normal(0, 1), normal(5, 2)), c(0.5, 0.5))
is_mixture(m)
is_mixture(normal(0, 1))

Function to determine whether an object x is an mvn object.

Description

Function to determine whether an object x is an mvn object.

Usage

is_mvn(x)

Arguments

Value

Logical; TRUE if x is an mvn.

Examples

is_mvn(mvn(c(0, 0)))
is_mvn(normal(0, 1))

Function to determine whether an object x is an normal object.

Description

Function to determine whether an object x is an normal object.

Usage

is_normal(x)

Arguments

Value

Logical; TRUE if x is a normal.

Examples

is_normal(normal(0, 1))
is_normal(exponential(1))

Test whether an object is a poisson_dist.

Description

Test whether an object is a poisson_dist.

Usage

is_poisson_dist(x)

Arguments

Value

TRUE if x inherits from "poisson_dist", FALSE otherwise.

Examples

is_poisson_dist(poisson_dist(5))
is_poisson_dist(normal(0, 1))

Test whether an object is a realized_dist.

Description

Test whether an object is a realized_dist.

Usage

is_realized_dist(x)

Arguments

Value

TRUE if x inherits from "realized_dist", FALSE otherwise.

Examples

rd <- realize(normal(0, 1), n = 100)
is_realized_dist(rd)  # TRUE

is_realized_dist(normal(0, 1))  # FALSE

Test whether an object is a uniform_dist.

Description

Test whether an object is a uniform_dist.

Usage

is_uniform_dist(x)

Arguments

Value

TRUE if x inherits from "uniform_dist", FALSE otherwise.

Examples

is_uniform_dist(uniform_dist(0, 1))
is_uniform_dist(normal(0, 1))

Test whether an object is a weibull_dist.

Description

Test whether an object is a weibull_dist.

Usage

is_weibull_dist(x)

Arguments

Value

TRUE if x inherits from "weibull_dist", FALSE otherwise.

Examples

is_weibull_dist(weibull_dist(2, 3))
is_weibull_dist(normal(0, 1))

Law of Large Numbers Limiting Distribution

Description

Returns the degenerate limiting distribution of the sample mean \bar{X}_n under the Law of Large Numbers. The limit is a point mass at the population mean (represented as a normal or mvn with zero variance).

Usage

lln(base_dist)

Arguments

Value

A normal or mvn distribution with zero variance, representing the degenerate distribution at the mean.

Examples

# LLN for Exp(2): Xbar -> 1/2 (degenerate)
x <- exponential(rate = 2)
d <- lln(x)
mean(d)
vcov(d)

Construct a log-normal distribution object.

Description

Creates an S3 object representing a log-normal distribution with the given meanlog and sdlog parameters. The log-normal PDF is

f(t) = \frac{1}{t \cdot sdlog \sqrt{2\pi}} \exp\!\left(-\frac{(\log t - meanlog)^2}{2 \cdot sdlog^2}\right)

for t > 0.

Usage

lognormal(meanlog = 0, sdlog = 1)

Arguments

Value

A lognormal object with classes c("lognormal", "univariate_dist", "continuous_dist", "dist").

Examples

x <- lognormal(meanlog = 0, sdlog = 1)
mean(x)
vcov(x)
format(x)

Generic method for obtaining the marginal distribution of a distribution object x over components indices.

Description

Generic method for obtaining the marginal distribution of a distribution object x over components indices.

Usage

marginal(x, indices)

Arguments

Value

A distribution object for the marginal over indices.

Examples

x <- mvn(c(0, 0), diag(2))
m <- marginal(x, 1)  # marginal over first component
mean(m)               # 0

Method for obtaining the marginal distribution of empirical_dist object x.

Description

Method for obtaining the marginal distribution of empirical_dist object x.

Usage

## S3 method for class 'empirical_dist'
marginal(x, indices)

Arguments

Value

An empirical_dist over the selected columns.

Examples

mat <- matrix(1:12, ncol = 3)
ed <- empirical_dist(mat)
ed_marginal <- marginal(ed, c(1, 3))
dim(ed_marginal) # 2

Marginal distribution of a mixture.

Description

The marginal of a mixture is itself a mixture of the component marginals with the same mixing weights: p(x_I) = \sum_k w_k p_k(x_I).

Usage

## S3 method for class 'mixture'
marginal(x, indices)

Arguments

Details

Requires all components to support marginal.

Value

A mixture object with marginalized components.

Examples

# Mixture of bivariate normals, extract marginal over first variable
m <- mixture(
  list(mvn(c(0, 0), diag(2)), mvn(c(3, 3), diag(2))),
  c(0.5, 0.5)
)
m1 <- marginal(m, 1)
mean(m1)

Generic method for obtaining the marginal distribution of an mvn object x over components indices.

Description

Generic method for obtaining the marginal distribution of an mvn object x over components indices.

Usage

## S3 method for class 'mvn'
marginal(x, indices)

Arguments

Value

A normal (for a single index) or mvn marginal distribution.

Examples

X <- mvn(c(1, 2, 3))
# Univariate marginal
marginal(X, 1)
# Bivariate marginal
marginal(X, c(1, 3))

Mean of a beta distribution.

Description

Computes \alpha / (\alpha + \beta) where \alpha = shape1 and \beta = shape2.

Usage

## S3 method for class 'beta_dist'
mean(x, ...)

Arguments

Value

The mean of the distribution.

Examples

mean(beta_dist(2, 5))

Retrieve the mean of a chi_squared object.

Description

Retrieve the mean of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
mean(x, ...)

Arguments

Value

The mean, equal to df

Examples

mean(chi_squared(10))

Method for obtaining the mean of an edist object.

Description

Method for obtaining the mean of an edist object.

Usage

## S3 method for class 'edist'
mean(x, n = 10000, ...)

Arguments

Value

The mean of the edist object

Examples

set.seed(1)
z <- normal(0, 1) * exponential(2)
mean(z)

Method for obtaining the mean of empirical_dist object x.

Description

Method for obtaining the mean of empirical_dist object x.

Usage

## S3 method for class 'empirical_dist'
mean(x, ...)

Arguments

Value

Numeric vector of column means.

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
mean(ed) # 3

Method to obtain the mean of an exponential object.

Description

Method to obtain the mean of an exponential object.

Usage

## S3 method for class 'exponential'
mean(x, ...)

Arguments

Value

The mean of the exponential distribution (1 / rate).

Examples

x <- exponential(rate = 0.5)
mean(x)

Retrieve the mean of a gamma_dist object.

Description

Retrieve the mean of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
mean(x, ...)

Arguments

Value

The mean, shape / rate

Examples

mean(gamma_dist(shape = 3, rate = 2))

Mean of a log-normal distribution.

Description

Computes \exp(meanlog + sdlog^2 / 2).

Usage

## S3 method for class 'lognormal'
mean(x, ...)

Arguments

Value

The mean of the distribution.

Examples

mean(lognormal(0, 1))

Mean of a mixture distribution.

Description

The mean of a mixture is the weighted sum of the component means: E[X] = \sum_k w_k \mu_k.

Usage

## S3 method for class 'mixture'
mean(x, ...)

Arguments

Value

The mean of the mixture distribution.

Examples

m <- mixture(list(normal(0, 1), normal(10, 1)), c(0.5, 0.5))
mean(m)

Retrieve the mean of a mvn object.

Description

Retrieve the mean of a mvn object.

Usage

## S3 method for class 'mvn'
mean(x, ...)

Arguments

Value

The mean of the mvn object

Examples

X <- mvn(c(1, 2, 3))
mean(X)

Retrieve the mean of a normal object.

Description

Retrieve the mean of a normal object.

Usage

## S3 method for class 'normal'
mean(x, ...)

Arguments

Value

The mean of the normal object

Examples

x <- normal(5, 2)
mean(x)

Mean of a Poisson distribution.

Description

Mean of a Poisson distribution.

Usage

## S3 method for class 'poisson_dist'
mean(x, ...)

Arguments

Value

The mean, equal to lambda.

Examples

mean(poisson_dist(5))

Mean of a uniform distribution.

Description

Computes (min + max) / 2.

Usage

## S3 method for class 'uniform_dist'
mean(x, ...)

Arguments

Value

The mean of the distribution.

Examples

mean(uniform_dist(0, 10))

Method for obtaining the mean of univariate_dist object x.

Description

Method for obtaining the mean of univariate_dist object x.

Usage

## S3 method for class 'univariate_dist'
mean(x, ...)

Arguments

Value

Numeric scalar; the mean of the distribution.

Examples

mean(normal(5, 2))    # 5
mean(exponential(2))  # 0.5

Mean of a Weibull distribution.

Description

Computes scale \cdot \Gamma(1 + 1/shape).

Usage

## S3 method for class 'weibull_dist'
mean(x, ...)

Arguments

Value

The mean of the distribution.

Examples

mean(weibull_dist(shape = 2, scale = 3))

Construct a mixture distribution.

Description

Creates an S3 object representing a finite mixture distribution. The density is f(x) = \sum_{k=1}^{K} w_k f_k(x) where f_k are the component densities and w_k are the mixing weights.

Usage

mixture(components, weights)

Arguments

Details

The class hierarchy is determined by the components: if all components are univariate (or multivariate, continuous, discrete), the mixture inherits those classes as well.

Value

A mixture object with appropriate class hierarchy.

Examples

m <- mixture(
  components = list(normal(0, 1), normal(5, 2)),
  weights = c(0.3, 0.7)
)
mean(m)
vcov(m)
format(m)

Construct a multivariate or univariate normal distribution object.

Description

This function constructs an object representing a normal distribution. If the length of the mean vector mu is 1, it creates a univariate normal distribution. Otherwise, it creates a multivariate normal distribution.

Usage

mvn(mu, sigma = diag(length(mu)))

Arguments

Value

If mu has length 1, it returns a normal object. If mu has length > 1, it returns an mvn object. Both types of objects contain mu and sigma as their properties.

Examples

# Bivariate normal with identity covariance
X <- mvn(mu = c(0, 0))
mean(X)
vcov(X)

# 1D case returns a normal object
is_normal(mvn(mu = 1, sigma = matrix(4)))

Method for obtaining the number of observations used to construct a empirical_dist object.

Description

Method for obtaining the number of observations used to construct a empirical_dist object.

Usage

## S3 method for class 'empirical_dist'
nobs(object, ...)

Arguments

Value

Integer; number of observations.

Examples

ed <- empirical_dist(c(10, 20, 30, 40))
nobs(ed) # 4

Construct univariate normal distribution object.

Description

Construct univariate normal distribution object.

Usage

normal(mu = 0, var = 1)

Arguments

Value

A normal distribution object.

Examples

x <- normal(mu = 0, var = 1)
mean(x)
vcov(x)
format(x)

Moment-Matching Normal Approximation

Description

Constructs a normal (or multivariate normal) distribution that matches the mean and variance-covariance of the input distribution. This is useful as a quick Gaussian approximation for any distribution whose first two moments are available.

Usage

normal_approx(x)

Arguments

Value

A normal distribution (for univariate inputs) or an mvn distribution (for multivariate inputs) with the same mean and variance-covariance as x.

Examples

# Approximate a Gamma(5, 2) with a normal
g <- gamma_dist(shape = 5, rate = 2)
n <- normal_approx(g)
mean(n)
vcov(n)

Generic method for obtaining the number of parameters of distribution-like object x.

Description

Generic method for obtaining the number of parameters of distribution-like object x.

Usage

nparams(x)

Arguments

Value

Integer; the number of parameters.

Examples

d <- empirical_dist(matrix(rnorm(30), ncol = 3))
nparams(d)  # 0 (non-parametric)

Method for obtaining the name of a empirical_dist object. Since the empirical distribution is parameter-free, this function returns 0.

Description

Method for obtaining the name of a empirical_dist object. Since the empirical distribution is parameter-free, this function returns 0.

Usage

## S3 method for class 'empirical_dist'
nparams(x)

Arguments

Value

0 (empirical distributions are non-parametric).

Examples

ed <- empirical_dist(c(1, 2, 3))
nparams(ed) # 0

Number of parameters for a mixture distribution.

Description

The total number of parameters is the sum of component parameters plus the number of mixing weights.

Usage

## S3 method for class 'mixture'
nparams(x)

Arguments

Value

An integer count of parameters.

Examples

m <- mixture(list(normal(0, 1), normal(5, 2)), c(0.3, 0.7))
nparams(m)

Retrieve the observations used to construct a distribution-like object. This is useful for obtaining the data used to construct an empirical distribution, but it is also useful for, say, retrieving the sample that was used by a fitted object, like an maximum likelihood estimate.

Description

Retrieve the observations used to construct a distribution-like object. This is useful for obtaining the data used to construct an empirical distribution, but it is also useful for, say, retrieving the sample that was used by a fitted object, like an maximum likelihood estimate.

Usage

obs(x)

Arguments

Value

The data (matrix or vector) used to construct x.

Examples

d <- empirical_dist(1:10)
obs(d)  # returns the vector 1:10

Method for obtaining the observations used to construct a empirical_dist object.

Description

Method for obtaining the observations used to construct a empirical_dist object.

Usage

## S3 method for class 'empirical_dist'
obs(x)

Arguments

Value

A matrix of observations (rows = observations, columns = dimensions).

Examples

ed <- empirical_dist(c(5, 10, 15))
obs(ed)

Generic method for obtaining the parameters of an object.

Description

Generic method for obtaining the parameters of an object.

Usage

params(x)

Arguments

Value

A named vector (or list) of distribution parameters.

Examples

x <- normal(5, 2)
params(x)  # mu = 5, var = 2

y <- exponential(3)
params(y)  # rate = 3

Retrieve the parameters of a beta_dist object.

Description

Retrieve the parameters of a beta_dist object.

Usage

## S3 method for class 'beta_dist'
params(x)

Arguments

Value

A named numeric vector with elements shape1 and shape2.

Examples

params(beta_dist(2, 5))

Method for obtaining the parameters of a chi_squared object.

Description

Method for obtaining the parameters of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
params(x)

Arguments

Value

A named numeric vector of parameters

Examples

params(chi_squared(5))

Method for obtaining the parameters of an edist object.

Description

Method for obtaining the parameters of an edist object.

Usage

## S3 method for class 'edist'
params(x)

Arguments

Value

A named vector of parameters

Examples

z <- normal(0, 1) * exponential(2)
params(z)

empirical_dist objects have no parameters, so this function returns NULL.

Description

empirical_dist objects have no parameters, so this function returns NULL.

Usage

## S3 method for class 'empirical_dist'
params(x)

Arguments

Value

NULL (empirical distributions have no parameters).

Examples

ed <- empirical_dist(c(1, 2, 3))
params(ed) # NULL

Method for obtaining the parameters of an exponential object.

Description

Method for obtaining the parameters of an exponential object.

Usage

## S3 method for class 'exponential'
params(x)

Arguments

Value

A named vector of parameters

Examples

x <- exponential(rate = 0.5)
params(x)

Method for obtaining the parameters of a gamma_dist object.

Description

Method for obtaining the parameters of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
params(x)

Arguments

Value

A named numeric vector of parameters

Examples

params(gamma_dist(2, 1))

Retrieve the parameters of a lognormal object.

Description

Retrieve the parameters of a lognormal object.

Usage

## S3 method for class 'lognormal'
params(x)

Arguments

Value

A named numeric vector with elements meanlog and sdlog.

Examples

params(lognormal(0, 1))

Retrieve the parameters of a mixture object.

Description

Returns a named numeric vector containing all component parameters (flattened) followed by the mixing weights.

Usage

## S3 method for class 'mixture'
params(x)

Arguments

Value

A named numeric vector.

Examples

m <- mixture(list(normal(0, 1), normal(5, 2)), c(0.3, 0.7))
params(m)

Method for obtaining the parameters of a mvn object.

Description

Method for obtaining the parameters of a mvn object.

Usage

## S3 method for class 'mvn'
params(x)

Arguments

Value

A named vector of parameters

Examples

X <- mvn(c(0, 0), diag(2))
params(X)

Method for obtaining the parameters of a normal object.

Description

Method for obtaining the parameters of a normal object.

Usage

## S3 method for class 'normal'
params(x)

Arguments

Value

A named vector of parameters

Examples

x <- normal(3, 2)
params(x)

Retrieve the parameters of a poisson_dist object.

Description

Retrieve the parameters of a poisson_dist object.

Usage

## S3 method for class 'poisson_dist'
params(x)

Arguments

Value

A named numeric vector with element lambda.

Examples

params(poisson_dist(5))

Retrieve the parameters of a uniform_dist object.

Description

Retrieve the parameters of a uniform_dist object.

Usage

## S3 method for class 'uniform_dist'
params(x)

Arguments

Value

A named numeric vector with elements min and max.

Examples

params(uniform_dist(0, 10))

Retrieve the parameters of a weibull_dist object.

Description

Retrieve the parameters of a weibull_dist object.

Usage

## S3 method for class 'weibull_dist'
params(x)

Arguments

Value

A named numeric vector with elements shape and scale.

Examples

params(weibull_dist(2, 3))

Construct a Poisson distribution object.

Description

Creates an S3 object representing a Poisson distribution with rate parameter \lambda. The PMF is P(X = k) = \lambda^k e^{-\lambda} / k! for k = 0, 1, 2, \ldots.

Usage

poisson_dist(lambda)

Arguments

Value

A poisson_dist object with classes c("poisson_dist", "univariate_dist", "discrete_dist", "dist").

Examples

x <- poisson_dist(lambda = 5)
mean(x)
vcov(x)
format(x)

Print a beta_dist object.

Description

Print a beta_dist object.

Usage

## S3 method for class 'beta_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(beta_dist(2, 5))

Print method for chi_squared objects.

Description

Print method for chi_squared objects.

Usage

## S3 method for class 'chi_squared'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(chi_squared(5))

Print method for edist objects.

Description

Print method for edist objects.

Usage

## S3 method for class 'edist'
print(x, ...)

Arguments

Examples

z <- normal(0, 1) * exponential(2)
print(z)

Print method for empirical_dist objects.

Description

Print method for empirical_dist objects.

Usage

## S3 method for class 'empirical_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
print(ed)

Print method for exponential objects.

Description

Print method for exponential objects.

Usage

## S3 method for class 'exponential'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(exponential(rate = 2))

Print method for gamma_dist objects.

Description

Print method for gamma_dist objects.

Usage

## S3 method for class 'gamma_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(gamma_dist(2, 1))

Print the interval.

Description

Print the interval.

Usage

## S3 method for class 'interval'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

iv <- interval$new(lower = 0, upper = 1, lower_closed = TRUE)
print(iv) # [0, 1)

Print a lognormal object.

Description

Print a lognormal object.

Usage

## S3 method for class 'lognormal'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(lognormal(0, 1))

Print a mixture object.

Description

Print a mixture object.

Usage

## S3 method for class 'mixture'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

m <- mixture(list(normal(0, 1), normal(5, 1)), c(0.5, 0.5))
print(m)

Method for printing an mvn object.

Description

Method for printing an mvn object.

Usage

## S3 method for class 'mvn'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(mvn(c(0, 0)))

Print method for normal objects.

Description

Print method for normal objects.

Usage

## S3 method for class 'normal'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

x <- normal(2, 3)
print(x)

Print a poisson_dist object.

Description

Print a poisson_dist object.

Usage

## S3 method for class 'poisson_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(poisson_dist(5))

Print a realized_dist object.

Description

Print a realized_dist object.

Usage

## S3 method for class 'realized_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

rd <- realize(normal(0, 1), n = 100)
print(rd)

Print method for summary_dist objects.

Description

Print method for summary_dist objects.

Usage

## S3 method for class 'summary_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

s <- summary(normal(5, 2))
print(s)

Print a uniform_dist object.

Description

Print a uniform_dist object.

Usage

## S3 method for class 'uniform_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(uniform_dist(0, 10))

Print a weibull_dist object.

Description

Print a weibull_dist object.

Usage

## S3 method for class 'weibull_dist'
print(x, ...)

Arguments

Value

x, invisibly.

Examples

print(weibull_dist(2, 3))

Materialize any distribution to empirical_dist by sampling.

Description

realize draws n samples from a distribution and wraps them in an empirical_dist. This is the universal fallback that lets any dist object be converted to a discrete approximation on which methods like cdf, density, and conditional are always available.

Usage

realize(x, n = 10000, ...)

## S3 method for class 'dist'
realize(x, n = 10000, ...)

## S3 method for class 'empirical_dist'
realize(x, ...)

## S3 method for class 'realized_dist'
realize(x, n = 10000, ...)

Arguments

Details

For non-empirical distributions, the result is a realized_dist that preserves the source distribution as provenance metadata. This enables re-sampling via realize(x$source, n = ...) and informative printing.

The empirical_dist method is a no-op: the distribution is already materialized.

The realized_dist method re-samples from the original source distribution, allowing cheap regeneration with a different sample size.

Value

An empirical_dist (or realized_dist) object.

Examples

set.seed(1)
x <- normal(0, 1)
rd <- realize(x, n = 1000)
mean(rd)

Construct a realized distribution object.

Description

A realized_dist is an empirical_dist that preserves provenance: it remembers the source distribution that generated its samples. All empirical_dist methods work unchanged via S3 inheritance.

Usage

realized_dist(data, source, n)

Arguments

Value

A realized_dist object.

Generic method for applying a map f to distribution object x.

Description

Generic method for applying a map f to distribution object x.

Usage

rmap(x, g, ...)

Arguments

Value

A distribution representing the push-forward of x through g.

Examples

d <- empirical_dist(1:20)
d_sq <- rmap(d, function(x) x^2)
mean(d_sq)  # E[X^2] for uniform 1..20

Method for obtaining g(x)) where x is a dist object.

Description

Falls back to MC: materializes x via ensure_realized() and then applies rmap with g to the resulting empirical distribution.

Usage

## S3 method for class 'dist'
rmap(x, g, n = 10000L, ...)

Arguments

Value

An empirical_dist of the transformed samples.

Examples

set.seed(1)
x <- exponential(1)
# Distribution of log(X) where X ~ Exp(1)
log_x <- rmap(x, log)
mean(log_x)

Map function over expression distribution.

Description

Falls back to realize and delegates to rmap.empirical_dist.

Usage

## S3 method for class 'edist'
rmap(x, g, ...)

Arguments

Value

A transformed empirical_dist.

Examples

set.seed(1)
z <- normal(0, 1) * exponential(1)
abs_z <- rmap(z, abs)
mean(abs_z)

Method for obtaining the empirical distribution of a function of the observations of empirical_dist object x.

Description

Method for obtaining the empirical distribution of a function of the observations of empirical_dist object x.

Usage

## S3 method for class 'empirical_dist'
rmap(x, g, ...)

Arguments

Value

An empirical_dist of the transformed observations.

Examples

ed <- empirical_dist(c(1, 2, 3, 4))
ed2 <- rmap(ed, function(x) x^2)
mean(ed2) # mean of 1, 4, 9, 16

Computes the distribution of g(x) where x is an mvn object.

Description

By the invariance property, if x is an mvn object, then under the right conditions, asymptotically, g(x) is an MVN distributed, g(x) ~ normal(g(mean(x)), sigma) where sigma is the variance-covariance of g(x)

Usage

## S3 method for class 'mvn'
rmap(x, g, n = 10000L, ...)

Arguments

Value

An mvn distribution fitted to the transformed samples.

Examples

X <- mvn(c(1, 2), diag(2))
set.seed(42)
Y <- rmap(X, function(x) x^2)
mean(Y)

Function for obtaining sample points for an mvn object that is within the p-probability region. That is, it samples from the smallest region of the distribution that contains p probability mass. This is done by first sampling from the entire distribution, then rejecting samples that are not in the probability region (using the statistical distance mahalanobis from mu).

Description

Function for obtaining sample points for an mvn object that is within the p-probability region. That is, it samples from the smallest region of the distribution that contains p probability mass. This is done by first sampling from the entire distribution, then rejecting samples that are not in the probability region (using the statistical distance mahalanobis from mu).

Usage

sample_mvn_region(n, mu, sigma, p = 0.95, ...)

Arguments

Value

An n by length(mu) matrix of samples within the probability region.

Examples

set.seed(42)
pts <- sample_mvn_region(10, mu = c(0, 0), sigma = diag(2), p = 0.95)
dim(pts)

Generic method for sampling from distribution-like objects.

Description

It creates a sampler for the x object. It returns a function that accepts a parameter n denoting the number of samples to draw from the x object and also any additional parameters ... are passed to the generated function.

Usage

sampler(x, ...)

Arguments

Value

A function that takes n and returns n samples.

Examples

x <- normal(0, 1)
samp <- sampler(x)
set.seed(42)
samp(5)  # draw 5 samples from standard normal

Sampler for a beta distribution.

Description

Returns a function that draws n independent samples from the beta distribution.

Usage

## S3 method for class 'beta_dist'
sampler(x, ...)

Arguments

Value

A function function(n = 1, ...) returning a numeric vector of length n.

Examples

x <- beta_dist(2, 5)
s <- sampler(x)
set.seed(42)
s(5)

Method for sampling from a chi_squared object.

Description

Method for sampling from a chi_squared object.

Usage

## S3 method for class 'chi_squared'
sampler(x, ...)

Arguments

Value

A function that generates n samples from the chi-squared distribution

Examples

x <- chi_squared(5)
s <- sampler(x)
set.seed(42)
s(5)

Sampler for non-dist objects (degenerate distributions).

Description

Sampler for non-dist objects (degenerate distributions).

Usage

## Default S3 method:
sampler(x, ...)

Arguments

Value

A function that takes n and returns n copies of x

Examples

s <- sampler(5)
s(3)  # returns c(5, 5, 5)

Method for obtaining the sampler of an edist object.

Description

Method for obtaining the sampler of an edist object.

Usage

## S3 method for class 'edist'
sampler(x, ...)

Arguments

Value

A function that takes a number of samples n, ... which is passed into the expression x$e and returns the result of applying the expression x$e to the sampled values.

Examples

set.seed(1)
z <- normal(0, 1) * exponential(2)
s <- sampler(z)
samples <- s(100)
head(samples)

Method for obtaining the sampler for a empirical_dist object.

Description

Method for obtaining the sampler for a empirical_dist object.

Usage

## S3 method for class 'empirical_dist'
sampler(x, ...)

Arguments

Value

A function that takes n and returns n resampled observations.

Examples

ed <- empirical_dist(c(10, 20, 30))
s <- sampler(ed)
set.seed(42)
s(5)

Method to sample from an exponential object.

Description

Method to sample from an exponential object.

Usage

## S3 method for class 'exponential'
sampler(x, ...)

Arguments

Value

A function that allows sampling from the exponential distribution. Accepts an argument n denoting sample size and rate denoting the failure rate. Defaults to the failure rate of object x.

Examples

x <- exponential(rate = 2)
s <- sampler(x)
set.seed(42)
s(5)

Method for sampling from a gamma_dist object.

Description

Method for sampling from a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
sampler(x, ...)

Arguments

Value

A function that generates n samples from the gamma distribution

Examples

x <- gamma_dist(shape = 2, rate = 1)
s <- sampler(x)
set.seed(42)
s(5)

Sampler for a log-normal distribution.

Description

Returns a function that draws n independent samples from the log-normal distribution.

Usage

## S3 method for class 'lognormal'
sampler(x, ...)

Arguments

Value

A function function(n = 1, ...) returning a numeric vector of length n.

Examples

x <- lognormal(0, 1)
s <- sampler(x)
set.seed(42)
s(5)

Sampler for a mixture distribution.

Description

Returns a function that draws samples from the mixture by first selecting a component according to the mixing weights, then sampling from the selected component.

Usage

## S3 method for class 'mixture'
sampler(x, ...)

Arguments

Value

A function function(n = 1, ...) returning a numeric vector of length n.

Examples

m <- mixture(list(normal(0, 1), normal(5, 1)), c(0.5, 0.5))
s <- sampler(m)
set.seed(42)
s(6)

Function generator for sampling from a mvn (multivariate normal) object.

Description

Function generator for sampling from a mvn (multivariate normal) object.

Usage

## S3 method for class 'mvn'
sampler(x, ...)

Arguments

Value

A function that samples from the mvn distribution. It accepts as input: - n: number of samples to generate. Defaults to 1. - mu: a vector denoting the population mean. Defaults to the mean of x (an mvn object) - sigma: a matrix denoting the covariance of observations. Defaults to the variance-covariance of x. - p: probability region to sample from. Defaults to 1, which corresponds to the entire distribution. sample_mvn_region method. It's used when p is less than 1. - ...: any additional parameters to pass to rmvnorm or sample_mvn_region which can be different during each call.

Examples

X <- mvn(c(0, 0), diag(2))
s <- sampler(X)
set.seed(42)
s(3)

Method for sampling from a normal object.

Description

Method for sampling from a normal object.

Usage

## S3 method for class 'normal'
sampler(x, ...)

Arguments

Value

A function that samples from the normal distribution. As input, it accepts a sample size n, a numeric mu, and a variance numeric var. By default, mu and var are the mean and variance of object x.

Examples

x <- normal(0, 1)
s <- sampler(x)
set.seed(42)
s(5)

Sampler for a Poisson distribution.

Description

Returns a function that draws n independent samples from the Poisson distribution.

Usage

## S3 method for class 'poisson_dist'
sampler(x, ...)

Arguments

Value

A function function(n = 1, ...) returning an integer vector of length n.

Examples

x <- poisson_dist(5)
s <- sampler(x)
set.seed(42)
s(5)

Sampler for a uniform distribution.

Description

Returns a function that draws n independent samples from the uniform distribution.

Usage

## S3 method for class 'uniform_dist'
sampler(x, ...)

Arguments

Value

A function function(n = 1, ...) returning a numeric vector of length n.

Examples

x <- uniform_dist(0, 10)
s <- sampler(x)
set.seed(42)
s(5)

Sampler for a Weibull distribution.

Description

Returns a function that draws n independent samples from the Weibull distribution.

Usage

## S3 method for class 'weibull_dist'
sampler(x, ...)

Arguments

Value

A function function(n = 1, ...) returning a numeric vector of length n.

Examples

x <- weibull_dist(shape = 2, scale = 3)
s <- sampler(x)
set.seed(42)
s(5)

Generic method for simplifying distributions.

Description

Generic method for simplifying distributions.

Usage

simplify(x, ...)

Arguments

Value

The simplified distribution

Examples

# Simplify dispatches to the appropriate method
simplify(normal(0, 1))  # unchanged (already simplified)

Default Method for simplifying a dist object. Just returns the object.

Description

Default Method for simplifying a dist object. Just returns the object.

Usage

## S3 method for class 'dist'
simplify(x, ...)

Arguments

Value

The dist object

Examples

x <- normal(0, 1)
identical(simplify(x), x)  # TRUE, returns unchanged

Method for simplifying an edist object.

Description

Attempts to reduce expression distributions to closed-form distributions when mathematical identities apply. Supported rules include:

Usage

## S3 method for class 'edist'
simplify(x, ...)

Arguments

Details

Single-variable:

• c * Normal(mu, v) -> Normal(cmu, c^2v)

• c * Gamma(a, r) -> Gamma(a, r/c) for c > 0

• c * Exponential(r) -> Gamma(1, r/c) for c > 0

• c * Uniform(a, b) -> Uniform(min(ca,cb), max(ca,cb)) for c != 0

• c * Weibull(k, lam) -> Weibull(k, c*lam) for c > 0

• c * ChiSq(df) -> Gamma(df/2, 1/(2c)) for c > 0

• c * LogNormal(ml, sl) -> LogNormal(ml + log(c), sl) for c > 0

• Normal(mu, v) + c -> Normal(mu + c, v)

• Normal(mu, v) - c -> Normal(mu - c, v)

• Uniform(a, b) + c -> Uniform(a + c, b + c)

• Uniform(a, b) - c -> Uniform(a - c, b - c)

• Normal(0, 1) ^ 2 -> ChiSquared(1)

• exp(Normal(mu, v)) -> LogNormal(mu, sqrt(v))

• log(LogNormal(ml, sl)) -> Normal(ml, sl^2)

Two-variable:

• Normal + Normal -> Normal

• Normal - Normal -> Normal

• Gamma + Gamma (same rate) -> Gamma

• Exponential + Exponential (same rate) -> Gamma(2, rate)

• Gamma + Exponential (same rate) -> Gamma(a+1, rate)

• ChiSquared + ChiSquared -> ChiSquared

• Poisson + Poisson -> Poisson

• LogNormal * LogNormal -> LogNormal

Value

The simplified distribution, or unchanged edist if no rule applies

Examples

# Normal + Normal simplifies to a Normal
z <- normal(0, 1) + normal(2, 3)
is_normal(z)  # TRUE
z             # Normal(mu = 2, var = 4)

# exp(Normal) simplifies to LogNormal
w <- exp(normal(0, 1))
is_lognormal(w)  # TRUE

Method for obtaining a summary of a dist object.

Description

Method for obtaining a summary of a dist object.

Usage

## S3 method for class 'dist'
summary(object, ..., name = NULL, nobs = NULL)

Arguments

Value

A summary_dist object

Examples

summary(normal(0, 1))

Method for constructing a summary_dist object.

Description

Method for constructing a summary_dist object.

Usage

summary_dist(name, mean, vcov, nobs = NULL)

Arguments

Value

A summary_dist object

Examples

s <- summary_dist(name = "my_dist", mean = 0, vcov = 1)
print(s)

Generic method for retrieving the support of a (dist) object x.

Description

The returned value should have the following operations:

• min: a vector, the minimum value of the support for each component.

• max: a vector, the maximum value of the support for each component.

• call: a predicate function, which returns TRUE if the value is in the support, and FALSE otherwise.

• sample: a function, which returns a sample from the support. Note that the returned value is not guaranteed to be in the support of x. You may need to call call to check.

Usage

sup(x)

Arguments

Value

A support object for x.

Examples

x <- normal(0, 1)
S <- sup(x)
infimum(S)   # -Inf
supremum(S)  # Inf

y <- exponential(1)
S2 <- sup(y)
infimum(S2)  # 0

Support of a beta distribution.

Description

The beta distribution is supported on the open interval (0, 1).

Usage

## S3 method for class 'beta_dist'
sup(x)

Arguments

Value

An interval object representing (0, 1).

Examples

sup(beta_dist(2, 5))

Support for chi-squared distribution, the positive real numbers (0, Inf).

Description

Support for chi-squared distribution, the positive real numbers (0, Inf).

Usage

## S3 method for class 'chi_squared'
sup(x)

Arguments

Value

An interval object representing (0, Inf)

Examples

sup(chi_squared(5))

Support for expression distributions.

Description

Falls back to realize and delegates to sup.empirical_dist.

Usage

## S3 method for class 'edist'
sup(x)

Arguments

Value

A finite_set support object.

Examples

set.seed(1)
z <- normal(0, 1) * exponential(1)
sup(z)

Method for obtaining the support of empirical_dist object x.

Description

Method for obtaining the support of empirical_dist object x.

Usage

## S3 method for class 'empirical_dist'
sup(x)

Arguments

Value

A finite_set object containing the support of x.

Examples

ed <- empirical_dist(c(1, 2, 2, 3))
s <- sup(ed)
s$has(2) # TRUE
s$has(4) # FALSE

Support for exponential distribution, the positive real numbers, (0, Inf).

Description

Support for exponential distribution, the positive real numbers, (0, Inf).

Usage

## S3 method for class 'exponential'
sup(x)

Arguments

Value

An interval object representing the support of the exponential

Examples

x <- exponential(rate = 1)
sup(x)

Support for gamma distribution, the positive real numbers (0, Inf).

Description

Support for gamma distribution, the positive real numbers (0, Inf).

Usage

## S3 method for class 'gamma_dist'
sup(x)

Arguments

Value

An interval object representing (0, Inf)

Examples

sup(gamma_dist(2, 1))

Support of a log-normal distribution.

Description

The log-normal distribution is supported on (0, \infty).

Usage

## S3 method for class 'lognormal'
sup(x)

Arguments

Value

An interval object representing (0, \infty).

Examples

sup(lognormal(0, 1))

Support of a mixture distribution.

Description

Returns an interval spanning the widest range of all component supports (from the smallest infimum to the largest supremum).

Usage

## S3 method for class 'mixture'
sup(x)

Arguments

Value

An interval object.

Examples

m <- mixture(list(normal(0, 1), exponential(1)), c(0.5, 0.5))
sup(m)

Method for obtaining the support of a mvn object, where the support is defined as values that have non-zero probability density.

Description

Method for obtaining the support of a mvn object, where the support is defined as values that have non-zero probability density.

Usage

## S3 method for class 'mvn'
sup(x, ...)

Arguments

Value

A support-type object (see support.R), in this case an interval object for each component.

Examples

X <- mvn(c(0, 0))
sup(X)

Method for obtaining the support of a normal object, where the support is defined as values that have non-zero probability density.

Description

Method for obtaining the support of a normal object, where the support is defined as values that have non-zero probability density.

Usage

## S3 method for class 'normal'
sup(x)

Arguments

Value

A support-type object (see support.R), in this case an interval object for each component.

Examples

x <- normal(0, 1)
sup(x)

Support of a Poisson distribution.

Description

The Poisson distribution is supported on the non-negative integers \{0, 1, 2, \ldots\}.

Usage

## S3 method for class 'poisson_dist'
sup(x)

Arguments

Value

A countable_set object with lower bound 0.

Examples

sup(poisson_dist(5))

Support of a uniform distribution.

Description

The uniform distribution is supported on [min, max].

Usage

## S3 method for class 'uniform_dist'
sup(x)

Arguments

Value

An interval object representing [min, max].

Examples

sup(uniform_dist(0, 10))

Support of a Weibull distribution.

Description

The Weibull distribution is supported on (0, \infty).

Usage

## S3 method for class 'weibull_dist'
sup(x)

Arguments

Value

An interval object representing (0, \infty).

Examples

sup(weibull_dist(2, 3))

Get the supremum of the support.

Description

Get the supremum of the support.

Usage

supremum(object)

Arguments

Value

The supremum (least upper bound) of the support.

Examples

I <- interval$new(0, 10)
supremum(I)  # 10

S <- finite_set$new(c(3, 7, 11))
supremum(S)  # 11

Get the supremum of a countable set.

Description

Get the supremum of a countable set.

Usage

## S3 method for class 'countable_set'
supremum(object)

Arguments

Value

Inf (the set is unbounded above).

Examples

cs <- countable_set$new(0L)
supremum(cs)  # Inf

Return the supremum of the finite set.

Description

Return the supremum of the finite set.

Usage

## S3 method for class 'finite_set'
supremum(object)

Arguments

Value

Numeric; the maximum value(s).

Examples

fs <- finite_set$new(c(1, 3, 5, 7))
supremum(fs) # 7

Return the (vector of) supremum of the interval.

Description

Return the (vector of) supremum of the interval.

Usage

## S3 method for class 'interval'
supremum(object)

Arguments

Value

Numeric vector of upper bounds.

Examples

iv <- interval$new(lower = 0, upper = 1)
supremum(iv) # 1

Generic method for obtaining the survival function of an object.

Description

Generic method for obtaining the survival function of an object.

Usage

surv(x, ...)

Arguments

Value

A function computing the survival function S(t) = P(X > t).

Examples

x <- exponential(1)
S <- surv(x)
S(0)  # 1 (survival at time 0)
S(1)  # exp(-1), approximately 0.368

Method for obtaining the survival function of a chi_squared object.

Description

Method for obtaining the survival function of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
surv(x, ...)

Arguments

Value

A function that computes S(t) = P(X > t)

Examples

x <- chi_squared(5)
S <- surv(x)
S(5)

Method to obtain the cdf of an exponential object.

Description

Method to obtain the cdf of an exponential object.

Usage

## S3 method for class 'exponential'
surv(x, ...)

Arguments

Value

A function that computes the cdf of the exponential. Accepts as input a vector t at which to compute the cdf, an input rate denoting the failure rate of the exponential distribution, and a logical log indicating whether to compute the log of the cdf. By default, rate is the failure rate of object x.

Examples

x <- exponential(rate = 1)
S <- surv(x)
S(1)
S(2)

Method for obtaining the survival function of a gamma_dist object.

Description

Method for obtaining the survival function of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
surv(x, ...)

Arguments

Value

A function that computes S(t) = P(X > t)

Examples

x <- gamma_dist(shape = 2, rate = 1)
S <- surv(x)
S(1)

Survival function for a log-normal distribution.

Description

Returns a function that computes S(t) = P(X > t) for the log-normal distribution.

Usage

## S3 method for class 'lognormal'
surv(x, ...)

Arguments

Value

A function function(t, log.p = FALSE, ...) returning the survival probability (or log-survival probability) at t.

Examples

x <- lognormal(0, 1)
S <- surv(x)
S(1)
S(2)

Survival function for a Weibull distribution.

Description

Returns a function that computes S(t) = P(X > t) for the Weibull distribution.

Usage

## S3 method for class 'weibull_dist'
surv(x, ...)

Arguments

Value

A function function(t, log.p = FALSE, ...) returning the survival probability (or log-survival probability) at t.

Examples

x <- weibull_dist(shape = 2, scale = 3)
S <- surv(x)
S(1)
S(3)

Construct a uniform distribution object.

Description

Creates an S3 object representing a continuous uniform distribution on the interval [min, max]. The PDF is f(x) = 1/(max - min) for min \le x \le max.

Usage

uniform_dist(min = 0, max = 1)

Arguments

Value

A uniform_dist object with classes c("uniform_dist", "univariate_dist", "continuous_dist", "dist").

Examples

x <- uniform_dist(min = 0, max = 10)
mean(x)
vcov(x)
format(x)

Variance of a beta distribution.

Description

Computes \alpha\beta / ((\alpha+\beta)^2 (\alpha+\beta+1)).

Usage

## S3 method for class 'beta_dist'
vcov(object, ...)

Arguments

Value

The variance (scalar).

Examples

vcov(beta_dist(2, 5))

Retrieve the variance of a chi_squared object.

Description

Retrieve the variance of a chi_squared object.

Usage

## S3 method for class 'chi_squared'
vcov(object, ...)

Arguments

Value

The variance, 2 * df

Examples

vcov(chi_squared(10))

Variance-covariance for non-dist objects (degenerate distributions).

Description

Variance-covariance for non-dist objects (degenerate distributions).

Usage

## Default S3 method:
vcov(object, ...)

Arguments

Value

0 (degenerate distributions have no variance)

Examples

vcov(42)  # returns 0

Method for obtaining the variance-covariance matrix (or scalar)

Description

Method for obtaining the variance-covariance matrix (or scalar)

Usage

## S3 method for class 'edist'
vcov(object, n = 10000, ...)

Arguments

Value

The variance-covariance matrix of the edist object

Examples

set.seed(1)
z <- normal(0, 1) * exponential(2)
vcov(z)

Method for obtaining the variance of empirical_dist object x.

Description

Method for obtaining the variance of empirical_dist object x.

Usage

## S3 method for class 'empirical_dist'
vcov(object, ...)

Arguments

Value

The sample variance-covariance matrix.

Examples

ed <- empirical_dist(c(1, 2, 3, 4, 5))
vcov(ed) # sample variance

ed_mv <- empirical_dist(matrix(rnorm(20), ncol = 2))
vcov(ed_mv) # 2x2 covariance matrix

Retrieve the variance of a exponential object.

Description

Retrieve the variance of a exponential object.

Usage

## S3 method for class 'exponential'
vcov(object, ...)

Arguments

Value

The variance-covariance matrix of the normal object

Examples

x <- exponential(rate = 2)
vcov(x)

Retrieve the variance of a gamma_dist object.

Description

Retrieve the variance of a gamma_dist object.

Usage

## S3 method for class 'gamma_dist'
vcov(object, ...)

Arguments

Value

The variance, shape / rate^2

Examples

vcov(gamma_dist(shape = 3, rate = 2))

Variance of a log-normal distribution.

Description

Computes (\exp(sdlog^2) - 1) \exp(2 \cdot meanlog + sdlog^2).

Usage

## S3 method for class 'lognormal'
vcov(object, ...)

Arguments

Value

The variance (scalar).

Examples

vcov(lognormal(0, 1))

Variance of a mixture distribution.

Description

Uses the law of total variance: Var(X) = E[Var(X|K)] + Var(E[X|K]).

Usage

## S3 method for class 'mixture'
vcov(object, ...)

Arguments

Value

The variance (scalar for univariate mixtures).

Examples

m <- mixture(list(normal(0, 1), normal(10, 1)), c(0.5, 0.5))
vcov(m)

Retrieve the variance-covariance matrix of an mvn object.

Description

Retrieve the variance-covariance matrix of an mvn object.

Usage

## S3 method for class 'mvn'
vcov(object, ...)

Arguments

Value

The variance-covariance matrix of the mvn object

Examples

X <- mvn(c(0, 0), diag(2))
vcov(X)

Retrieve the variance-covariance matrix (or scalar) of a normal object.

Description

Retrieve the variance-covariance matrix (or scalar) of a normal object.

Usage

## S3 method for class 'normal'
vcov(object, ...)

Arguments

Value

The variance-covariance matrix of the normal object

Examples

x <- normal(0, 4)
vcov(x)

Variance of a Poisson distribution.

Description

For the Poisson distribution, the variance equals lambda.

Usage

## S3 method for class 'poisson_dist'
vcov(object, ...)

Arguments

Value

The variance (scalar), equal to lambda.

Examples

vcov(poisson_dist(5))

Variance of a uniform distribution.

Description

Computes (max - min)^2 / 12.

Usage

## S3 method for class 'uniform_dist'
vcov(object, ...)

Arguments

Value

The variance (scalar).

Examples

vcov(uniform_dist(0, 10))

Method for obtaining the variance of univariate_dist object.

Description

Method for obtaining the variance of univariate_dist object.

Usage

## S3 method for class 'univariate_dist'
vcov(object, ...)

Arguments

Value

Numeric scalar; the variance of the distribution.

Examples

vcov(normal(0, 4))    # 4
vcov(exponential(2))  # 0.25