# hxlr

##### Heteroscedastic Extended Logistic Regression

This is a wrapper function for `clm`

(from package
ordinal) to fit (heteroscedastic) extended logistic regression (HXLR)
models (Messner et al. 2013).

- Keywords
- regression

##### Usage

`hxlr(formula, data, subset, na.action, weights, thresholds, link, control, …)`

##### Arguments

- formula
a formula expression of the form

`y ~ x | z`

where`y`

is the response and`x`

and`z`

are regressor variables for the location and the scale of the latend distribution respectively. Response can either be a continuous variable or a factor.- data
an optional data frame containing the variables occurring in the formulas.

- subset
an optional vector specifying a subset of observations to be used for fitting.

- na.action
a function which indicates what should happen when the data contain

`NA`

s. Default is na.omit- weights
optional case weights in fitting.

- thresholds
vector of (transformed) thresholds that are used to cut the continuous response into categories. Data frames or matrices with multiple columns are allowed as well. Then each column is used as separate predictor variable for the intercept model.

- link
link function, i.e., the type of location-scale distribution assumed for the latent distribution. Default is

`logit`

.- control
a list of control parameters passed to

`optim`

. Default is`hxlr.control`

- …
arguments to be used to form the default

`control`

argument if it is not supplied directly.

##### Details

Extended logistic regression (Wilks 2009) extends binary logistic regression to multi-category responses by including the thresholds, that are used to cut a continuous variable into categories, in the regression equation. Heteroscedastic extended logistic regression (Messner et al. 2013) extends this model further and allows to add additional predictor variables that are used to predict the scale of the latent logistic distribution.

##### Value

An object of class `"hxlr"`

, i.e., a list with the following elements.

list of CLM coefficients for intercept, location, and scale model.

list of fitted location and scale parameters.

output from optimization from `optim`

.

Optimization method used for `optim`

.

list of control parameters passed to `optim`

starting values of coefficients used in the optimization.

case weights used for fitting.

number of observations.

number of observations with non-zero weights.

log-likelihood.

covariance matrix.

logical variable whether optimization has converged or not.

number of iterations in optimization.

function call.

the formula supplied.

the `terms`

objects used.

list of levels of the factors used in fitting for location and scale respectively.

the thresholds supplied.

##### References

Messner JW, Mayr GJ, Zeileis A, Wilks DS (2014). Extending Extended Logistic
Regression to Effectively Utilize the Ensemble Spread. *Monthly Weather
Review*, **142**, 448--456. 10.1175/MWR-D-13-00271.1.

Wilks DS (2009). Extending Logistic Regression to Provide
Full-Probability-Distribution MOS Forecasts.
*Meteorological Applications*, **368**, 361--368.

##### See Also

##### Examples

```
# NOT RUN {
data("RainIbk")
## mean and standard deviation of square root transformed ensemble forecasts
RainIbk$sqrtensmean <-
apply(sqrt(RainIbk[,grep('^rainfc',names(RainIbk))]), 1, mean)
RainIbk$sqrtenssd <-
apply(sqrt(RainIbk[,grep('^rainfc',names(RainIbk))]), 1, sd)
## climatological deciles
q <- unique(quantile(RainIbk$rain, seq(0.1, 0.9, 0.1)))
## fit ordinary extended logistic regression with ensemble mean as
## predictor variable
XLR <- hxlr(sqrt(rain) ~ sqrtensmean, data = RainIbk, thresholds = sqrt(q))
## print
XLR
## summary
summary(XLR)
## fit ordinary extended logistic regression with ensemble mean
## and standard deviation as predictor variables
XLRS <- hxlr(sqrt(rain) ~ sqrtensmean + sqrtenssd, data = RainIbk,
thresholds = sqrt(q))
## fit heteroscedastic extended logistic regression with ensemble
## standard deviation as predictor for the scale
HXLR <- hxlr(sqrt(rain) ~ sqrtensmean | sqrtenssd, data = RainIbk,
thresholds = sqrt(q))
## compare AIC of different models
AIC(XLR, XLRS, HXLR)
## XLRS and HXLR are nested in XLR -> likelihood-ratio-tests
if(require("lmtest")) {
lrtest(XLR, XLRS)
lrtest(XLR, HXLR)
}
# }
# NOT RUN {
###################################################################
## Cross-validation and bootstrapping RPS for different models
## (like in Messner 2013).
N <- NROW(RainIbk)
## function that returns model fits
fits <- function(data, weights = rep(1, N)) {
list(
"XLR" = hxlr(sqrt(rain) ~ sqrtensmean, data = data,
weights = weights, thresholds = sqrt(q)),
"XLR:S" = hxlr(sqrt(rain) ~ sqrtensmean + sqrtenssd, data = data,
weights = weights, thresholds = sqrt(q)),
"XLR:SM" = hxlr(sqrt(rain) ~ sqrtensmean + I(sqrtensmean*sqrtenssd),
data = data, weights = weights, thresholds = sqrt(q)),
"HXLR" = hxlr(sqrt(rain) ~ sqrtensmean | sqrtenssd, data = data,
weights = weights, thresholds = sqrt(q)),
"HXLR:S" = hxlr(sqrt(rain) ~ sqrtensmean + sqrtenssd | sqrtenssd,
data = data, weights = weights, thresholds = sqrt(q))
)
}
## cross validation
id <- sample(1:10, N, replace = TRUE)
obs <- NULL
pred <- list(NULL)
for(i in 1:10) {
## splitting into test and training data set
trainIndex <- which(id != i)
testIndex <- which(id == i)
## weights that are used for fitting the models
weights <- as.numeric(table(factor(trainIndex, levels = c(1:N))))
## testdata
testdata <- RainIbk[testIndex,]
## observations
obs <- c(obs, RainIbk$rain[testIndex])
## estimation
modelfits <- fits(RainIbk, weights)
## Prediction
pred2 <- lapply(modelfits, predict, newdata = testdata, type = "cumprob")
pred <- mapply(rbind, pred, pred2, SIMPLIFY = FALSE)
}
names(pred) <- c(names(modelfits))
## function to compute RPS
rps <- function(pred, obs) {
OBS <- NULL
for(i in 1:N)
OBS <- rbind(OBS, rep(0:1, c(obs[i] - 1, length(q) - obs[i] + 1)))
apply((OBS-pred)^2, 1, sum)
}
## compute rps
RPS <- lapply(pred, rps, obs = as.numeric(cut(obs, c(-Inf, q, Inf))))
## bootstrapping mean rps
rpsall <- NULL
for(i in 1:250) {
index <- sample(length(obs), replace = TRUE)
rpsall <- rbind(rpsall, sapply(RPS, function(x) mean(x[index])))
}
rpssall <- 1 - rpsall/rpsall[,1]
boxplot(rpssall[,-1], ylab = "RPSS", main = "RPSS relative to XLR")
abline(h = 0, lty = 2)
# }
```

*Documentation reproduced from package crch, version 1.0-4, License: GPL-2 | GPL-3*