Package 'distr6'

Description

distr6 is an object oriented (OO) interface, primarily used for interacting with probability distributions in R. Additionally distr6 includes functionality for composite distributions, a symbolic representation for mathematical sets and intervals, basic methods for common kernels and numeric methods for distribution analysis. distr6 is the official R6 upgrade to the distr family of packages.

Details

The main features of distr6 are:

• Currently implements 45 probability distributions (and 11 Kernels) including all distributions in the R stats package. Each distribution has (where possible) closed form analytic expressions for basic statistical methods.

• Decorators that add further functionality to probability distributions including numeric results for useful modelling functions such as p-norms and k-moments.

• Wrappers for composite distributions including convolutions, truncation, mixture distributions and product distributions.

To learn more about distr6, start with the distr6 vignette:

vignette("distr6", "distr6")

And for more advanced usage see the complete tutorials at

https://xoopr.github.io/distr6/index.html #nolint

Author(s)

Maintainer: Raphael Sonabend [email protected] (ORCID)

Authors:

• Franz Kiraly [email protected]

Other contributors:

• Peter Ruckdeschel [email protected] (Author of distr) [contributor]

• Matthias Kohl [email protected] (Author of distr) [contributor]

• Nurul Ain Toha [email protected] [contributor]

• Shen Chen [email protected] [contributor]

• Jordan Deenichin [email protected] [contributor]

• Chengyang Gao [email protected] [contributor]

• Chloe Zhaoyuan Gu [email protected] [contributor]

• Yunjie He [email protected] [contributor]

• Xiaowen Huang [email protected] [contributor]

• Shuhan Liu [email protected] [contributor]

• Runlong Yu [email protected] [contributor]

• Chijing Zeng [email protected] [contributor]

• Qian Zhou [email protected] [contributor]

• Michal Lauer [email protected] [contributor]

• John Zobolas [email protected] (ORCID) [contributor]

See Also

Useful links:

• https://xoopr.github.io/distr6/

• https://github.com/xoopr/distr6/

• Report bugs at https://github.com/xoopr/distr6/issues

Description

Extract a WeightedDiscrete or Matdist or Arrdist from a Arrdist.

Usage

## S3 method for class 'Arrdist'
ad[i = NULL, j = NULL]

Arguments

Value

If length(i) == 1 and length(j) == 1 then returns a WeightedDiscrete otherwise if j is NULL returns an Arrdist. If length(i) is greater than 1 or NULL returns a Matdist if length(j) == 1.

Examples

pdf <- runif(400)
arr <- array(pdf, c(20, 10, 2), list(NULL, sort(sample(1:20, 10)), NULL))
arr <- aperm(apply(arr, c(1, 3), function(x) x / sum(x)), c(2, 1, 3))
darr <- as.Distribution(arr, fun = "pdf")
# WeightDisc
darr[1, 1]
# Matdist
darr[1:2, 1]
# Arrdist
darr[1:3, 1:2]
darr[1, 1:2]

Description

Extract a WeightedDiscrete or Matdist from a Matdist.

Usage

## S3 method for class 'Matdist'
md[i]

Arguments

Value

If length(i) == 1 then returns a WeightedDiscrete otherwise returns a Matdist.

Examples

m <- as.Distribution(
  t(apply(matrix(runif(200), 20, 10, FALSE,
                  list(NULL, sort(sample(1:20, 10)))), 1,
          function(x) x / sum(x))),
  fun = "pdf"
)
m[1]
m[1:2]

Description

Once a VectorDistribution has been constructed, use [ to extract one or more Distributions from inside it.

Usage

## S3 method for class 'VectorDistribution'
vecdist[i]

Arguments

Examples

v <- VectorDistribution$new(distribution = "Binom", params = data.frame(size = 1:2, prob = 0.5))
v[1]
v["Binom1"]

Description

Mathematical and statistical functions for the Arcsine distribution, which is commonly used in the study of random walks and as a special case of the Beta distribution.

Details

The Arcsine distribution parameterised with lower, $a$, and upper, $b$, limits is defined by the pdf,

$f(x) = 1/(\pi\sqrt{(x-a)(b-x))}$

for $-\infty < a \le b < \infty$.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on $[a, b]$.

Default Parameterisation

Arc(lower = 0, upper = 1)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Arcsine

Public fields

Active bindings

Methods

Public methods

• Arcsine$new()

• Arcsine$mean()

• Arcsine$mode()

• Arcsine$variance()

• Arcsine$skewness()

• Arcsine$kurtosis()

• Arcsine$entropy()

• Arcsine$pgf()

• Arcsine$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Arcsine$new(lower = NULL, upper = NULL, decorators = NULL)

Arguments

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions.

Usage

Arcsine$mean(...)

Arguments

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Arcsine$mode(which = "all")

Arguments

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Arcsine$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution.

Usage

Arcsine$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Arcsine$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Arcsine$entropy(base = 2, ...)

Arguments

Method pgf()

The probability generating function is defined by

$pgf_X(z) = E_X[exp(z^x)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Arcsine$pgf(z, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

Arcsine$clone(deep = FALSE)

Arguments

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other continuous distributions: BetaNoncentral, Beta, Cauchy, ChiSquaredNoncentral, ChiSquared, Dirichlet, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Gompertz, Gumbel, InverseGamma, Laplace, Logistic, Loglogistic, Lognormal, MultivariateNormal, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull

Other univariate distributions: Arrdist, Bernoulli, BetaNoncentral, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Description

Mathematical and statistical functions for the Arrdist distribution, which is commonly used in matrixed Bayesian estimators such as Kaplan-Meier with confidence bounds over arbitrary dimensions.

Details

The Arrdist distribution is defined by the pmf,

$f(x_{ijk}) = p_{ijk}$

for $p_{ijk}, i = 1,\ldots,a, j = 1,\ldots,b; \sum_i p_{ijk} = 1$.

This is a generalised case of Matdist with a third dimension over an arbitrary length. By default all results are returned for the median curve as determined by (dim(a)[3L] + 1)/2 where a is the array and assuming third dimension is odd, this can be changed by setting the which.curve parameter.

Given the complexity in construction, this distribution is not mutable (cannot be updated after construction).

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on $x_{111},...,x_{abc}$.

Default Parameterisation

Arrdist(array(0.5, c(2, 2, 2), list(NULL, 1:2, NULL)))

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Arrdist

Public fields

Active bindings

Methods

Public methods

• Arrdist$new()

• Arrdist$strprint()

• Arrdist$mean()

• Arrdist$median()

• Arrdist$mode()

• Arrdist$variance()

• Arrdist$skewness()

• Arrdist$kurtosis()

• Arrdist$entropy()

• Arrdist$mgf()

• Arrdist$cf()

• Arrdist$pgf()

• Arrdist$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Arrdist$new(pdf = NULL, cdf = NULL, which.curve = 0.5, decorators = NULL)

Arguments

Method strprint()

Printable string representation of the Distribution. Primarily used internally.

Usage

Arrdist$strprint(n = 2)

Arguments

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions. If distribution is improper (F(Inf) != 1, then E_X(x) = Inf).

Usage

Arrdist$mean(...)

Arguments

Method median()

Returns the median of the distribution. If an analytical expression is available returns distribution median, otherwise if symmetric returns self$mean, otherwise returns self$quantile(0.5).

Usage

Arrdist$median()

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Arrdist$mode(which = 1)

Arguments

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned. If distribution is improper (F(Inf) != 1, then var_X(x) = Inf).

Usage

Arrdist$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. If distribution is improper (F(Inf) != 1, then sk_X(x) = Inf).

Usage

Arrdist$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3. If distribution is improper (F(Inf) != 1, then k_X(x) = Inf).

Usage

Arrdist$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions. If distribution is improper then entropy is Inf.

Usage

Arrdist$entropy(base = 2, ...)

Arguments

Method mgf()

The moment generating function is defined by

$mgf_X(t) = E_X[exp(xt)]$

where X is the distribution and $E_X$ is the expectation of the distribution X. If distribution is improper (F(Inf) != 1, then mgf_X(x) = Inf).

Usage

Arrdist$mgf(t, ...)

Arguments

Method cf()

The characteristic function is defined by

$cf_X(t) = E_X[exp(xti)]$

where X is the distribution and $E_X$ is the expectation of the distribution X. If distribution is improper (F(Inf) != 1, then cf_X(x) = Inf).

Usage

Arrdist$cf(t, ...)

Arguments

Method pgf()

The probability generating function is defined by

$pgf_X(z) = E_X[exp(z^x)]$

where X is the distribution and $E_X$ is the expectation of the distribution X. If distribution is improper (F(Inf) != 1, then pgf_X(x) = Inf).

Usage

Arrdist$pgf(z, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

Arrdist$clone(deep = FALSE)

Arguments

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other discrete distributions: Bernoulli, Binomial, Categorical, Degenerate, DiscreteUniform, EmpiricalMV, Empirical, Geometric, Hypergeometric, Logarithmic, Matdist, Multinomial, NegativeBinomial, WeightedDiscrete

Other univariate distributions: Arcsine, Bernoulli, BetaNoncentral, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Examples

x <- Arrdist$new(pdf = array(0.5, c(3, 2, 4),
                 dimnames = list(NULL, 1:2, NULL)))
Arrdist$new(cdf = array(c(0.5, 0.5, 0.5, 1, 1, 1), c(3, 2, 4),
                        dimnames = list(NULL, 1:2, NULL))) # equivalently

# d/p/q/r
x$pdf(1)
x$cdf(1:2) # Assumes ordered in construction
x$quantile(0.42) # Assumes ordered in construction
x$rand(10)

# Statistics
x$mean()
x$variance()

summary(x)

# Changing which.curve
arr <- array(runif(90), c(3, 2, 5), list(NULL, 1:2, NULL))
arr <- aperm(apply(arr, c(1, 3), function(x) x / sum(x)), c(2, 1, 3))
arr[, , 1:3]
x <- Arrdist$new(arr)
x$mean() # default 0.5 quantile (in this case index 3)
x$setParameterValue(which.curve = 3) # equivalently
x$mean()
# 1% quantile
x$setParameterValue(which.curve = 0.01)
x$mean()
# 5th index
x$setParameterValue(which.curve = 5)
x$mean()
# mean
x$setParameterValue(which.curve = "mean")
x$mean()

Description

Coerces matrices to a VectorDistribution containing WeightedDiscrete distributions or a Matdist. Number of distributions are the number of rows in the matrix, number of x points are number of columns in the matrix.

Usage

as.Distribution(obj, fun, decorators = NULL, vector = FALSE)

## S3 method for class 'matrix'
as.Distribution(obj, fun, decorators = NULL, vector = FALSE)

## S3 method for class 'array'
as.Distribution(obj, fun, decorators = NULL, vector = FALSE)

Arguments

Value

A VectorDistribution or Matdist

Examples

pdf <- runif(200)
mat <- matrix(pdf, 20, 10, FALSE, list(NULL, 1:10))
mat <- t(apply(mat, 1, function(x) x / sum(x)))

# coercion to matrix distribution
as.Distribution(mat, fun = "pdf")

# coercion to vector of weighted discrete distributions
as.Distribution(mat, fun = "pdf", vector = TRUE)

Description

Helper functions to quickly convert compatible objects to a MixtureDistribution.

Usage

as.MixtureDistribution(object, weights = "uniform")

Arguments

Description

Helper functions to quickly convert compatible objects to a ProductDistribution.

Usage

as.ProductDistribution(object)

Arguments

Description

Helper functions to quickly convert compatible objects to a VectorDistribution.

Usage

as.VectorDistribution(object)

Arguments

Description

Mathematical and statistical functions for the Bernoulli distribution, which is commonly used to model a two-outcome scenario.

Details

The Bernoulli distribution parameterised with probability of success, $p$, is defined by the pmf,

$f(x) = p, \ if \ x = 1$

$f(x) = 1 - p, \ if \ x = 0$

for probability $p$.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on $\{0,1\}$.

Default Parameterisation

Bern(prob = 0.5)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Bernoulli

Public fields

Active bindings

Methods

Public methods

• Bernoulli$new()

• Bernoulli$mean()

• Bernoulli$mode()

• Bernoulli$median()

• Bernoulli$variance()

• Bernoulli$skewness()

• Bernoulli$kurtosis()

• Bernoulli$entropy()

• Bernoulli$mgf()

• Bernoulli$cf()

• Bernoulli$pgf()

• Bernoulli$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Bernoulli$new(prob = NULL, qprob = NULL, decorators = NULL)

Arguments

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions.

Usage

Bernoulli$mean(...)

Arguments

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Bernoulli$mode(which = "all")

Arguments

Method median()

Returns the median of the distribution. If an analytical expression is available returns distribution median, otherwise if symmetric returns self$mean, otherwise returns self$quantile(0.5).

Usage

Bernoulli$median()

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Bernoulli$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution.

Usage

Bernoulli$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Bernoulli$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Bernoulli$entropy(base = 2, ...)

Arguments

Method mgf()

The moment generating function is defined by

$mgf_X(t) = E_X[exp(xt)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Bernoulli$mgf(t, ...)

Arguments

Method cf()

The characteristic function is defined by

$cf_X(t) = E_X[exp(xti)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Bernoulli$cf(t, ...)

Arguments

Method pgf()

The probability generating function is defined by

$pgf_X(z) = E_X[exp(z^x)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Bernoulli$pgf(z, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

Bernoulli$clone(deep = FALSE)

Arguments

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other discrete distributions: Arrdist, Binomial, Categorical, Degenerate, DiscreteUniform, EmpiricalMV, Empirical, Geometric, Hypergeometric, Logarithmic, Matdist, Multinomial, NegativeBinomial, WeightedDiscrete

Other univariate distributions: Arcsine, Arrdist, BetaNoncentral, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Description

Mathematical and statistical functions for the Beta distribution, which is commonly used as the prior in Bayesian modelling.

Details

The Beta distribution parameterised with two shape parameters, $\alpha, \beta$, is defined by the pdf,

$f(x) = (x^{\alpha-1}(1-x)^{\beta-1}) / B(\alpha, \beta)$

for $\alpha, \beta > 0$, where $B$ is the Beta function.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on $[0, 1]$.

Default Parameterisation

Beta(shape1 = 1, shape2 = 1)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Beta

Public fields

Active bindings

Methods

Public methods

• Beta$new()

• Beta$mean()

• Beta$mode()

• Beta$variance()

• Beta$skewness()

• Beta$kurtosis()

• Beta$entropy()

• Beta$pgf()

• Beta$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Beta$new(shape1 = NULL, shape2 = NULL, decorators = NULL)

Arguments

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions.

Usage

Beta$mean(...)

Arguments

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Beta$mode(which = "all")

Arguments

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Beta$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution.

Usage

Beta$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Beta$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Beta$entropy(base = 2, ...)

Arguments

Method pgf()

The probability generating function is defined by

$pgf_X(z) = E_X[exp(z^x)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Beta$pgf(z, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

Beta$clone(deep = FALSE)

Arguments

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other continuous distributions: Arcsine, BetaNoncentral, Cauchy, ChiSquaredNoncentral, ChiSquared, Dirichlet, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Gompertz, Gumbel, InverseGamma, Laplace, Logistic, Loglogistic, Lognormal, MultivariateNormal, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull

Other univariate distributions: Arcsine, Arrdist, Bernoulli, BetaNoncentral, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Description

Mathematical and statistical functions for the Noncentral Beta distribution, which is commonly used as the prior in Bayesian modelling.

Details

The Noncentral Beta distribution parameterised with two shape parameters, $\alpha, \beta$, and location, $\lambda$, is defined by the pdf,

$f(x) = exp(-\lambda/2) \sum_{r=0}^\infty ((\lambda/2)^r/r!) (x^{\alpha+r-1}(1-x)^{\beta-1})/B(\alpha+r, \beta)$

for $\alpha, \beta > 0, \lambda \ge 0$, where $B$ is the Beta function.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on $[0, 1]$.

Default Parameterisation

BetaNC(shape1 = 1, shape2 = 1, location = 0)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> BetaNoncentral

Public fields

Active bindings

Methods

Public methods

• BetaNoncentral$new()

• BetaNoncentral$clone()

Method new()

Creates a new instance of this R6 class.

Usage

BetaNoncentral$new(
  shape1 = NULL,
  shape2 = NULL,
  location = NULL,
  decorators = NULL
)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

BetaNoncentral$clone(deep = FALSE)

Arguments

Author(s)

Jordan Deenichin

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other continuous distributions: Arcsine, Beta, Cauchy, ChiSquaredNoncentral, ChiSquared, Dirichlet, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Gompertz, Gumbel, InverseGamma, Laplace, Logistic, Loglogistic, Lognormal, MultivariateNormal, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull

Other univariate distributions: Arcsine, Arrdist, Bernoulli, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Description

Mathematical and statistical functions for the Binomial distribution, which is commonly used to model the number of successes out of a number of independent trials.

Details

The Binomial distribution parameterised with number of trials, n, and probability of success, p, is defined by the pmf,

$f(x) = C(n, x)p^x(1-p)^{n-x}$

for $n = 0,1,2,\ldots$ and probability $p$, where $C(a,b)$ is the combination (or binomial coefficient) function.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on ${0, 1,...,n}$.

Default Parameterisation

Binom(size = 10, prob = 0.5)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Binomial

Public fields

Active bindings

Methods

Public methods

• Binomial$new()

• Binomial$mean()

• Binomial$mode()

• Binomial$variance()

• Binomial$skewness()

• Binomial$kurtosis()

• Binomial$entropy()

• Binomial$mgf()

• Binomial$cf()

• Binomial$pgf()

• Binomial$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Binomial$new(size = NULL, prob = NULL, qprob = NULL, decorators = NULL)

Arguments

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions.

Usage

Binomial$mean(...)

Arguments

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Binomial$mode(which = "all")

Arguments

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Binomial$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution.

Usage

Binomial$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Binomial$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Binomial$entropy(base = 2, ...)

Arguments

Method mgf()

The moment generating function is defined by

$mgf_X(t) = E_X[exp(xt)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Binomial$mgf(t, ...)

Arguments

Method cf()

The characteristic function is defined by

$cf_X(t) = E_X[exp(xti)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Binomial$cf(t, ...)

Arguments

Method pgf()

The probability generating function is defined by

$pgf_X(z) = E_X[exp(z^x)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Binomial$pgf(z, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

Binomial$clone(deep = FALSE)

Arguments

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other discrete distributions: Arrdist, Bernoulli, Categorical, Degenerate, DiscreteUniform, EmpiricalMV, Empirical, Geometric, Hypergeometric, Logarithmic, Matdist, Multinomial, NegativeBinomial, WeightedDiscrete

Other univariate distributions: Arcsine, Arrdist, Bernoulli, BetaNoncentral, Beta, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Description

Helper function for quickly combining distributions into a Arrdist.

Usage

## S3 method for class 'Arrdist'
c(..., decorators = NULL)

Arguments

Value

Arrdist

Examples

# create three array distributions with different column names
arr <- replicate(3, {
  pdf <- runif(400)
  arr <- array(pdf, c(20, 10, 2), list(NULL, sort(sample(1:20, 10)), NULL))
  arr <- aperm(apply(arr, c(1, 3), function(x) x / sum(x)), c(2, 1, 3))
  as.Distribution(arr, fun = "pdf")
})
do.call(c, arr)

Description

Helper function for quickly combining distributions into a VectorDistribution.

Usage

## S3 method for class 'Distribution'
c(..., name = NULL, short_name = NULL, decorators = NULL)

Arguments

Value

A VectorDistribution

See Also

VectorDistribution

Examples

# Construct and combine
c(Binomial$new(), Normal$new())

# More complicated distributions
b <- truncate(Binomial$new(), 2, 6)
n <- huberize(Normal$new(), -1, 1)
c(b, n)

# Concatenate VectorDistributions
v1 <- VectorDistribution$new(list(Binomial$new(), Normal$new()))
v2 <- VectorDistribution$new(
  distribution = "Gamma",
  params = data.table::data.table(shape = 1:2, rate = 1:2)
)
c(v1, v2)

Description

Helper function for quickly combining distributions into a Matdist.

Usage

## S3 method for class 'Matdist'
c(..., decorators = NULL)

Arguments

Value

Matdist

Examples

# create three matrix distributions with different column names
mats <- replicate(3, {
  pdf <- runif(200)
  mat <- matrix(pdf, 20, 10, FALSE, list(NULL, sort(sample(1:20, 10))))
  mat <- t(apply(mat, 1, function(x) x / sum(x)))
  as.Distribution(mat, fun = "pdf")
})
do.call(c, mats)

Description

Mathematical and statistical functions for the Categorical distribution, which is commonly used in classification supervised learning.

Details

The Categorical distribution parameterised with a given support set, $x_1,...,x_k$, and respective probabilities, $p_1,...,p_k$, is defined by the pmf,

$f(x_i) = p_i$

for $p_i, i = 1,\ldots,k; \sum p_i = 1$.

Sampling from this distribution is performed with the sample function with the elements given as the support set and the probabilities from the probs parameter. The cdf and quantile assumes that the elements are supplied in an indexed order (otherwise the results are meaningless).

The number of points in the distribution cannot be changed after construction.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on $x_1,...,x_k$.

Default Parameterisation

Cat(elements = 1, probs = 1)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Categorical

Public fields

Active bindings

Methods

Public methods

• Categorical$new()

• Categorical$mean()

• Categorical$mode()

• Categorical$variance()

• Categorical$skewness()

• Categorical$kurtosis()

• Categorical$entropy()

• Categorical$mgf()

• Categorical$cf()

• Categorical$pgf()

• Categorical$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Categorical$new(elements = NULL, probs = NULL, decorators = NULL)

Arguments

Examples

# Note probabilities are automatically normalised (if not vectorised)
x <- Categorical$new(elements = list("Bapple", "Banana", 2), probs = c(0.2, 0.4, 1))

# Length of elements and probabilities cannot be changed after construction
x$setParameterValue(probs = c(0.1, 0.2, 0.7))

# d/p/q/r
x$pdf(c("Bapple", "Carrot", 1, 2))
x$cdf("Banana") # Assumes ordered in construction
x$quantile(0.42) # Assumes ordered in construction
x$rand(10)

# Statistics
x$mode()

summary(x)

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions.

Usage

Categorical$mean(...)

Arguments

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Categorical$mode(which = "all")

Arguments

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Categorical$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution.

Usage

Categorical$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Categorical$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Categorical$entropy(base = 2, ...)

Arguments

Method mgf()

The moment generating function is defined by

$mgf_X(t) = E_X[exp(xt)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Categorical$mgf(t, ...)

Arguments

Method cf()

The characteristic function is defined by

$cf_X(t) = E_X[exp(xti)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Categorical$cf(t, ...)

Arguments

Method pgf()

The probability generating function is defined by

$pgf_X(z) = E_X[exp(z^x)]$

where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Categorical$pgf(z, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

Categorical$clone(deep = FALSE)

Arguments

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other discrete distributions: Arrdist, Bernoulli, Binomial, Degenerate, DiscreteUniform, EmpiricalMV, Empirical, Geometric, Hypergeometric, Logarithmic, Matdist, Multinomial, NegativeBinomial, WeightedDiscrete

Other univariate distributions: Arcsine, Arrdist, Bernoulli, BetaNoncentral, Beta, Binomial, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Examples

## ------------------------------------------------
## Method `Categorical$new`
## ------------------------------------------------

# Note probabilities are automatically normalised (if not vectorised)
x <- Categorical$new(elements = list("Bapple", "Banana", 2), probs = c(0.2, 0.4, 1))

# Length of elements and probabilities cannot be changed after construction
x$setParameterValue(probs = c(0.1, 0.2, 0.7))

# d/p/q/r
x$pdf(c("Bapple", "Carrot", 1, 2))
x$cdf("Banana") # Assumes ordered in construction
x$quantile(0.42) # Assumes ordered in construction
x$rand(10)

# Statistics
x$mode()

summary(x)

Description

Mathematical and statistical functions for the Cauchy distribution, which is commonly used in physics and finance.

Details

The Cauchy distribution parameterised with location, $\alpha$, and scale, $\beta$, is defined by the pdf,

$f(x) = 1 / (\pi\beta(1 + ((x - \alpha) / \beta)^2))$

for $\alpha \epsilon R$ and $\beta > 0$.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on the Reals.

Default Parameterisation

Cauchy(location = 0, scale = 1)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Cauchy

Public fields

Methods

Public methods

• Cauchy$new()

• Cauchy$mean()

• Cauchy$mode()

• Cauchy$variance()

• Cauchy$skewness()

• Cauchy$kurtosis()

• Cauchy$entropy()

• Cauchy$mgf()

• Cauchy$cf()

• Cauchy$pgf()

• Cauchy$clone()

Method new()

Creates a new instance of this R6 class.

Usage

Cauchy$new(location = NULL, scale = NULL, decorators = NULL)

Arguments

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

$E_X(X) = \sum p_X(x)*x$

with an integration analogue for continuous distributions.

Usage

Cauchy$mean(...)

Arguments

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Cauchy$mode(which = "all")

Arguments

Method variance()

The variance of a distribution is defined by the formula

$var_X = E[X^2] - E[X]^2$

where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Cauchy$variance(...)

Arguments

Method skewness()

The skewness of a distribution is defined by the third standardised moment,

$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution.

Usage

Cauchy$skewness(...)

Arguments

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

$k_X = E_X[\frac{x - \mu}{\sigma}^4]$

where $E_X$ is the expectation of distribution X, $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Cauchy$kurtosis(excess = TRUE, ...)

Arguments

Method entropy()

The entropy of a (discrete) distribution is defined by

$- \sum (f_X)log(f_X)$

where $f_X$ is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Cauchy$entropy(base = 2, ...)

Arguments

Method mgf()

The moment generating function is defined by

$mgf_X(t) = E_X[exp(xt)]$