gamlss2

Extracting Fitted or Predicted Probability Distributions from gamlss2 Models

Description

Methods for gamlss2 model objects for extracting fitted (in-sample) or predicted (out-of-sample) probability distributions as distributions3 objects.

Usage

## S3 method for class 'gamlss2'
prodist(object, ...)

Arguments

object A model object of class gamlss2.

… Arguments passed on to predict.gamlss2, e.g., newdata.

Details

To facilitate making probabilistic forecasts based on gamlss2 model objects, the prodist method extracts fitted or predicted probability distribution objects. Internally, the predict.gamlss2 method is used first to obtain the distribution parameters (mu, sigma, tau, nu, or a subset thereof). Subsequently, the corresponding distribution object is set up using the GAMLSS class from the gamlss.dist package, enabling the workflow provided by the distributions3 package (see Zeileis et al. 2022).

Note that these probability distributions only reflect the random variation in the dependent variable based on the model employed (and its associated distributional assumption for the dependent variable). This does not capture the uncertainty in the parameter estimates.

Value

An object of class GAMLSS 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/.

Examples

library("gamlss2")


## packages, code, and data
library("distributions3")
data("cars", package = "datasets")

## fit heteroscedastic normal GAMLSS model
## stopping distance (ft) explained by speed (mph)
m <- gamlss2(dist ~ s(speed) | s(speed), data = cars, family = NO)

GAMLSS-RS iteration  1: Global Deviance = 407.3647 eps = 0.125474     
GAMLSS-RS iteration  2: Global Deviance = 406.1879 eps = 0.002888     
GAMLSS-RS iteration  3: Global Deviance = 405.7966 eps = 0.000963     
GAMLSS-RS iteration  4: Global Deviance = 405.7908 eps = 0.000014     
GAMLSS-RS iteration  5: Global Deviance = 405.7908 eps = 0.000000

## obtain predicted distributions for three levels of speed
d <- prodist(m, newdata = data.frame(speed = c(10, 20, 30)))
print(d)

                                  1                                   2 
"GDF NO(mu = 23.12, sigma = 10.05)" "GDF NO(mu = 58.82, sigma = 18.56)" 
                                  3 
"GDF NO(mu = 95.57, sigma = 34.28)"

## obtain quantiles (works the same for any distribution object 'd' !)
quantile(d, 0.5)

       1        2        3 
23.11757 58.82097 95.57069

quantile(d, c(0.05, 0.5, 0.95), elementwise = FALSE)

     q_0.05    q_0.5    q_0.95
1  6.586727 23.11757  39.64840
2 28.284971 58.82097  89.35696
3 39.180036 95.57069 151.96134

quantile(d, c(0.05, 0.5, 0.95), elementwise = TRUE)

         1          2          3 
  6.586727  58.820966 151.961337

## visualization
plot(dist ~ speed, data = cars)
nd <- data.frame(speed = 0:240/4)
nd$dist <- prodist(m, newdata = nd)
nd$fit <- quantile(nd$dist, c(0.05, 0.5, 0.95))
matplot(nd$speed, nd$fit, type = "l", lty = 1, col = "slategray", add = TRUE)

## moments
mean(d)

       1        2        3 
23.11757 58.82097 95.57069

variance(d)

        1         2         3 
 101.0032  344.6431 1175.3296

## simulate random numbers
random(d, 5)

         r_1      r_2       r_3       r_4       r_5
1   8.657994 27.50428  18.94314  31.51712  19.16080
2  76.247476 45.87117  72.70055  61.95744  34.79499
3 144.423690 73.13722 133.38308 103.56546 148.34871

## density and distribution
pdf(d, 50 * -2:2)

        d_-100        d_-50          d_0        d_50        d_100
1 1.024798e-34 1.273422e-13 0.0028169995 0.001109383 7.777801e-15
2 2.750602e-18 7.430283e-10 0.0001419917 0.019195539 1.835767e-03
3 9.985419e-10 1.414926e-06 0.0002389622 0.004810090 1.153999e-02

cdf(d, 50 * -2:2)

        p_-100        p_-50          p_0       p_50     p_100
1 8.352306e-35 1.727573e-13 0.0107171145 0.99626197 1.0000000
2 5.890414e-18 2.289984e-09 0.0007662613 0.31733979 0.9867278
3 5.831408e-09 1.087434e-05 0.0026542456 0.09188323 0.5513996

## Poisson example
data("FIFA2018", package = "distributions3")
m2 <- gamlss2(goals ~ s(difference), data = FIFA2018, family = PO)

GAMLSS-RS iteration  1: Global Deviance = 355.3922 eps = 0.045332     
GAMLSS-RS iteration  2: Global Deviance = 355.3922 eps = 0.000000

d2 <- prodist(m2, newdata = data.frame(difference = 0))
print(d2)

                   1 
"GDF PO(mu = 1.237)"

quantile(d2, c(0.05, 0.5, 0.95))

[1] 0 1 3

## note that log_pdf() can replicate logLik() value
sum(log_pdf(prodist(m2), FIFA2018$goals))

[1] -177.6961

logLik(m2)

'log Lik.' -177.6961 (df=2.005144)