Categorical: Categorical Distribution Class

Description Details Value Distribution support Default Parameterisation Omitted Methods Also known as Super classes Public fields Active bindings Methods References See Also Examples

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,…,k; ∑ 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

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

Methods

Public methods

Inherited methods

Method new()

Creates a new instance of this R6 class.

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

list()
Categories in the distribution, see examples.

probs

numeric()
Probabilities of respective categories occurring.

decorators

(character())
Decorators to add to the distribution during construction.

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) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

Usage
Categorical$mean(...)
Arguments
...

Unused.


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
which

(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.


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
...

Unused.


Method skewness()

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

sk_X = E_X[((x - μ)/σ)^3]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution.

Usage
Categorical$skewness(...)
Arguments
...

Unused.


Method kurtosis()

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

k_X = E_X[((x - μ)/σ)^4]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

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

(logical(1))
If TRUE (default) excess kurtosis returned.

...

Unused.


Method entropy()

The entropy of a (discrete) distribution is defined by

- ∑ (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
base

(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)

...

Unused.


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
t

(integer(1))
t integer to evaluate function at.

...

Unused.


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
t

(integer(1))
t integer to evaluate function at.

...

Unused.


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
z

(integer(1))
z integer to evaluate probability generating function at.

...

Unused.


Method clone()

The objects of this class are cloneable with this method.

Usage
Categorical$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

References

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

See Also

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

Other univariate distributions: Arcsine, 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, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

Examples

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## ------------------------------------------------
## 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)

distr6 documentation built on Sept. 6, 2021, 9:10 a.m.