GAMLSS: Create a GAMLSS Distribution

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

Create a GAMLSS Distribution

Description

A single class and corresponding methods encompassing all distributions from the gamlss.dist package using the workflow from the distributions3 package.

Usage

GAMLSS(family, mu, sigma, tau, nu)

Arguments

family	character. Name of a GAMLSS family provided by gamlss.dist, e.g., NO or BI for the normal or binomial distribution, respectively.
mu	numeric. GAMLSS mu parameter. Can be a scalar or a vector or missing if not part of the family.
sigma	numeric. GAMLSS sigma parameter. Can be a scalar or a vector or missing if not part of the family.
tau	numeric. GAMLSS tau parameter. Can be a scalar or a vector or missing if not part of the family.
nu	numeric. GAMLSS nu parameter. Can be a scalar or a vector or missing if not part of the family.

Details

The S3 class GAMLSS provides a unified workflow based on the distributions3 package (see Zeileis et al. 2022) for all distributions from the gamlss.dist package. The idea is that from fitted gamlss model objects predicted probability distributions can be obtained for which moments (mean, variance, etc.), probabilities, quantiles, etc. can be obtained with corresponding generic functions.

The constructor function GAMLSS sets up a distribution object, representing a distribution from the GAMLSS (generalized additive model of location, scale, and shape) framework by the corresponding parameters plus a family attribute, e.g., NO for the normal distribution or BI for the binomial distribution. There can be up to four parameters, called mu (often some sort of location parameter), sigma (often some sort of scale parameter), tau and nu (other parameters, e.g., capturing shape, etc.).

All parameters can also be vectors, so that it is possible to define a vector of GAMLSS distributions from the same family with potentially different parameters. All parameters need to have the same length or must be scalars (i.e., of length 1) which are then recycled to the length of the other parameters.

Note that not all distributions use all four parameters, i.e., some use just a subset. In that case, the corresponding arguments in GAMLSS should be unspecified, NULL, or NA.

For the GAMLSS distribution objects there is a wide range of standard methods available to the generics provided in the distributions3 package: pdf and log_pdf for the (log-)density (PDF), cdf for the probability from the cumulative distribution function (CDF), quantile for quantiles, random for simulating random variables, and support for the support interval (minimum and maximum). Internally, these methods rely on the usual d/p/q/r functions provided in gamlss.dist, see the manual pages of the individual families. The methods is_discrete and is_continuous can be used to query whether the distributions are discrete on the entire support or continuous on the entire support, respectively.

See the examples below for an illustration of the workflow for the class and methods.

Value

A GAMLSS object, inheriting from distribution.

References

Zeileis A, Lang MN, Hayes A (2022). “distributions3: From Basic Probability to Probabilistic Regression.” Presented at useR! 2022 - The R User Conference. Slides, video, vignette, code at https://www.zeileis.org/news/user2022/.

See Also

gamlss.family

Examples

## package and random seed
library("distributions3")
set.seed(6020)

## three Weibull distributions
X <- GAMLSS("WEI", mu = c(1, 1, 2), sigma = c(1, 2, 2))
X

## moments
mean(X)
variance(X)

## support interval (minimum and maximum)
support(X)
is_discrete(X)
is_continuous(X)

## simulate random variables
random(X, 5)

## histograms of 1,000 simulated observations
x <- random(X, 1000)
hist(x[1, ], main = "WEI(1,1)")
hist(x[2, ], main = "WEI(1,2)")
hist(x[3, ], main = "WEI(2,2)")

## probability density function (PDF) and log-density (or log-likelihood)
x <- c(2, 2, 1)
pdf(X, x)
pdf(X, x, log = TRUE)
log_pdf(X, x)

## cumulative distribution function (CDF)
cdf(X, x)

## quantiles
quantile(X, 0.5)

## cdf() and quantile() are inverses
cdf(X, quantile(X, 0.5))
quantile(X, cdf(X, 1))

## all methods above can either be applied elementwise or for
## all combinations of X and x, if length(X) = length(x),
## also the result can be assured to be a matrix via drop = FALSE
p <- c(0.05, 0.5, 0.95)
quantile(X, p, elementwise = FALSE)
quantile(X, p, elementwise = TRUE)
quantile(X, p, elementwise = TRUE, drop = FALSE)

## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
  "theoretical" = mean(X),
  "empirical" = rowMeans(random(X, 1000))
)

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.