Grows a univariate or multivariate quantile regression forest and returns its conditional quantile and density values. Can be used for both training and testing purposes.

quantreg(formula, data, object, newdata,
  method = "local", splitrule = NULL, prob = NULL, prob.epsilon = NULL,
  oob = TRUE, fast = FALSE, maxn = 1e3, ...)



A symbolic description of the model to be fit. Must be specified unless object is given.


Data frame containing the y-outcome and x-variables in the model. Must be specified unless object is given.


(Optional) A previously grown quantile regression forest.


(Optional) Test data frame used for prediction. Note that prediction on test data must always be done with the quantreg function and not the predict function. See example below.


Method used to calculate quantiles. Three methods are provided: (1) A variation of the method used in Meinshausen (2006) based on forest weight (method = "forest"); (2) The Greenwald-Khanna algorithm, suited for big data, and specified by any one of the following: "gk", "GK", "G-K", "g-k"; (3) The default method, method = "local", which uses the local adjusted cdf approach of Zhang et al. (2019). This does not rely on forest weights and is reasonably fast. See below for further discussion.


The default action is local adaptive quantile regression splitting, but this can be over-ridden by the user. Not applicable to multivariate forests. See details below.


Target quantile probabilities when training. If left unspecified, uses percentiles (1 through 99) for method = "forest", and for Greenwald-Khanna selects equally spaced percentiles optimized for accuracy (see below).


Greenwald-Khanna allowable error for quantile probabilities when training.


Return OOB (out-of-bag) quantiles? If false, in-bag values are returned.


Use fast random forests,, in place of rfsrc? Improves speed but may be less accurate.


Maximum number of unique y training values used when calculating the conditional density.


Further arguments to be passed to the rfsrc function used for fitting the quantile regression forest.


The most common method for calculating RF quantiles uses the method described in Meinshausen (2006) using forest weights. The forest weights method employed here (specified using method="forest"), however differs in that quantiles are estimated using a weighted local cumulative distribution function estimator. For this reason, results may differ from Meinshausen (2006). Moreover, results may also differ as the default splitting rule uses local adaptive quantile regression splitting instead of CART regression mean squared splitting which was used by Meinshausen (2006). Note that local adaptive quantile regression splitting is not available for multivariate forests which reverts to the default multivariate composite splitting rule. In multivariate regression, users however do have the option to over-ride this using Mahalanobis splitting by setting splitrule="mahalanobis"

A second method for estimating quantiles uses the Greenwald-Khanna (2001) algorithm (invoked by method="gk", "GK", "G-K" or "g-k"). While this will not be as accurate as forest weights, the high memory efficiency of Greenwald-Khanna makes it feasible to implement in big data settings unlike forest weights.

The Greenwald-Khanna algorithm is implemented roughly as follows. To form a distribution of values for each case, from which we sample to determine quantiles, we create a chain of values for the case as we grow the forest. Every time a case lands in a terminal node, we insert all of its co-inhabitants to its chain of values.

The best case scenario is when tree node size is 1 because each case gets only one insert into its chain for that tree. The worst case scenario is when node size is so large that trees stump. This is because each case receives insertions for the entire in-bag population.

What the user needs to know is that Greenwald-Khanna can become slow in counter-intutive settings such as when node size is large. The easy fix is to change the epsilon quantile approximation that is requested. You will see a significant speed-up just by doubling prob.epsilon. This is because the chains stay a lot smaller as epsilon increases, which is exactly what you want when node sizes are large. Both time and space requirements for the algorithm are affected by epsilon.

The best results for Greenwald-Khanna come from setting the number of quantiles equal to 2 times the sample size and epsilon to 1 over 2 times the sample size which is the default values used if left unspecified. This will be slow, especially for big data, and less stringent choices should be used if computational speed is of concern.

Finally, the default method, method="local", implements the locally adjusted cdf estimator of Zhang et al. (2019). This does not use forest weights and is reasonably fast and can be used for large data. However, this relies on the assumption of homogeneity of the error distribution, i.e. that errors are iid and therefore have equal variance. While this is reasonably robust to departures of homogeneity, there are instances where this may perform poorly; see Zhang et al. (2019) for details. If hetereogeneity is suspected we recommend method="forest".


Returns the object quantreg containing quantiles for each of the requested probabilities (which can be conveniently extracted using get.quantile). Also contains the conditional density (and conditional cdf) for each case in the training data (or test data if provided) evaluated at each of the unique grow y-values. The conditional density can be used to calculate conditional moments, such as the mean and standard deviation. Use get.quantile.stat as a way to conveniently obtain these quantities. For multivariate forests, returned values will be a list of length equal to the number of target outcomes.


Hemant Ishwaran and Udaya B. Kogalur


Greenwald M. and Khanna S. (2001). Space-efficient online computation of quantile summaries. Proceedings of ACM SIGMOD, 30(2):58-66.

Meinshausen N. (2006) Quantile regression forests, Journal of Machine Learning Research, 7:983-999.

Zhang H., Zimmerman J., Nettleton D. and Nordman D.J. (2019). Random forest prediction intervals. The American Statistician. 4:1-5.

See also


# \donttest{
## ------------------------------------------------------------
## regression example
## ------------------------------------------------------------

## standard call
o <- quantreg(mpg ~ ., mtcars)

## extract conditional quantiles
print(get.quantile(o, c(.25, .50, .75)))

## extract conditional mean and standard deviation

## continuous rank probabiliy score (crps) performance
plot(get.quantile.crps(o), type = "l")

## ------------------------------------------------------------
## train/test regression example
## ------------------------------------------------------------

## train (grow) call followed by test call
o <- quantreg(mpg ~ ., mtcars[1:20,])
o.tst <- quantreg(object = o, newdata = mtcars[-(1:20),])

## extract test set quantiles and conditional statistics

## ------------------------------------------------------------
## quantile regression for Boston Housing using forest method
## ------------------------------------------------------------

if (library("mlbench", logical.return = TRUE)) {

  ## quantile regression with mse splitting
  o <- quantreg(medv ~ ., BostonHousing, method = "forest", nodesize = 1)

  ## continuous rank probabiliy score (crps) 
  plot(get.quantile.crps(o), type = "l")

  ## quantile regression plot
  plot.quantreg(o, .05, .95)
  plot.quantreg(o, .25, .75)

  ## (A) extract 25,50,75 quantiles
  quant.dat <- get.quantile(o, c(.25, .50, .75))

  ## (B) values expected under normality
  quant.stat <- get.quantile.stat(o)
  c.mean <- quant.stat$mean
  c.std <- quant.stat$std
  q.25.est <- c.mean + qnorm(.25) * c.std
  q.75.est <- c.mean + qnorm(.75) * c.std

  ## compare (A) and (B)
  print(head(data.frame(quant.dat[, -2],  q.25.est, q.75.est)))


## ------------------------------------------------------------
## multivariate mixed outcomes example
## quantiles are only returned for the continous outcomes
## ------------------------------------------------------------

dta <- mtcars
dta$cyl <- factor(dta$cyl)
dta$carb <- factor(dta$carb, ordered = TRUE)
o <- quantreg(cbind(carb, mpg, cyl, disp) ~., data = dta)

plot.quantreg(o, = "mpg")
plot.quantreg(o, = "disp")

## ------------------------------------------------------------
## multivariate regression example using Mahalanobis splitting
## ------------------------------------------------------------

dta <- mtcars
o <- quantreg(cbind(mpg, disp) ~., data = dta, splitrule = "mahal")

plot.quantreg(o, = "mpg")
plot.quantreg(o, = "disp")

## ------------------------------------------------------------
## example of quantile regression for ordinal data
## ------------------------------------------------------------

## use the wine data for illustration
data(wine, package = "randomForestSRC")

## run quantile regression
o <- quantreg(quality ~ ., wine, ntree = 100)

## extract "probabilities" = density values
qo.dens <- o$quantreg$density
yunq <- o$quantreg$yunq
colnames(qo.dens) <- yunq

## convert y to a factor 
yvar <- factor(cut(o$yvar, c(-1, yunq), labels = yunq)) 
## confusion matrix
qo.confusion <- get.confusion(yvar, qo.dens)

## normalized Brier score
cat("Brier:", 100 * get.brier.error(yvar, qo.dens), "\n")

## ------------------------------------------------------------
## example of large data using Greenwald-Khanna algorithm 
## ------------------------------------------------------------

## load the data and do quick and dirty imputation
data(housing, package = "randomForestSRC")
housing <- impute(SalePrice ~ ., housing,
         ntree = 50, nimpute = 1, splitrule = "random")

## Greenwald-Khanna algorithm 
## request a small number of quantiles 
o <- quantreg(SalePrice ~ ., housing, method = "gk",
        prob = (1:20) / 20, prob.epsilon = 1 / 20, ntree = 250)
## ------------------------------------------------------------
## using mse splitting with local cdf method for large data
## ------------------------------------------------------------

## load the data and do quick and dirty imputation
data(housing, package = "randomForestSRC")
housing <- impute(SalePrice ~ ., housing,
         ntree = 50, nimpute = 1, splitrule = "random")

## use mse splitting and reduce number of trees
o <- quantreg(SalePrice ~ ., housing, splitrule = "mse", ntree = 250)

# }