Quantile Regression Forests
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.
## S3 method for class 'rfsrc' quantreg(formula, data, object, newdata, method = "local", splitrule = NULL, prob = NULL, prob.epsilon = NULL, oob = TRUE, fast = FALSE, maxn = 1e3, ...)
formula |
A symbolic description of the model to be fit.
Must be specified unless |
data |
Data frame containing the y-outcome and x-variables in
the model. Must be specified unless |
object |
(Optional) A previously grown quantile regression forest. |
method |
Method used to calculate quantiles. Three methods are
provided. Forest weighted averaging ( |
splitrule |
The default action is local adaptive quantile regression splitting, but this can be over-ridden by the user. |
prob |
Target quantile probabilities when training. If left unspecified,
uses percentiles (1 through 99) for |
prob.epsilon |
Greenwald-Khanna allowable error for quantile probabilities when training. |
newdata |
Test data (optional) over which conditional quantiles are evaluated over. |
oob |
Return OOB (out-of-bag) quantiles? If false, in-bag values are returned. |
fast |
Use fast random forests, |
maxn |
Maximum number of unique y training values used when calculating the conditional density. |
... |
Further arguments to be passed to the |
The most common method for calculating RF quantiles uses forest
weights (Meinshausen, 2006). However we note that the forest weighted
method used here (specified using method="forest") differs from
Meinshuasen (2006) in two important ways: (1) local adaptive quantile
regression splitting is used instead of CART regression mean squared
splitting, and (2) quantiles are estimated using a weighted local
cumulative distribution function estimator. For this reason, results
may differ from Meinshausen (2006).
A second method 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 is the
default procedure used here as it does not rely on forest weights and
is therefore fast and can be used for large data. However be aware
this relies on the assumption of homogeneity of the error
distribution, i.e. that errors are iid and therefore have equal
variance. Now while 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" instead.
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.
## ------------------------------------------------------------
## regression example
## ------------------------------------------------------------
## standard call
o <- quantreg(mpg ~ ., mtcars)
## extract conditional quantiles
print(get.quantile(o))
print(get.quantile(o, c(.25, .50, .75)))
## extract conditional mean and standard deviation
print(get.quantile.stat(o))
## 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
print(get.quantile(o.tst))
print(get.quantile.stat(o.tst))
## ------------------------------------------------------------
## quantile regression for Boston Housing using forest method
## ------------------------------------------------------------
if (library("mlbench", logical.return = TRUE)) {
## quantile regression with mse splitting
data(BostonHousing)
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
## ------------------------------------------------------------
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, m.target = "mpg")
plot.quantreg(o, m.target = "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)
print(qo.confusion)
## 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)
plot.quantreg(o)
## ------------------------------------------------------------
## 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)
plot.quantreg(o)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.