R: Quantile Regression Forests

quantreg.rfsrc {randomForestSRC} | R Documentation

## Quantile Regression Forests

### Description

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.

### Usage

    ## 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, ...)

### Arguments

`formula` | A symbolic description of the model to be fit. Must be specified unless `object` is given.
`data` | Data frame containing the y-outcome and x-variables in the model. Must be specified unless `object` is given.
`object` | (Optional) A previously grown quantile regression forest.
`method` | Method used to calculate quantiles. Three methods are provided. Forest weighted averaging (`method = "forest"`) is the standard method provided in most random forest packages. A second method is the Greenwald-Khanna algorithm which is suited for big data and is specified by any one of the following: "gk", "GK", "G-K", "g-k". The third method (`method = "local"`) is the default method used and uses the local adjusted cdf approach of Zhang et al. (2019). This does not use forest weights and is therefore reasonably fast and can be used for large data - however it relies on the assumption of a homogeneous (equal variance) error distribution which can be a strong assumption and undesireable consequences can result if the assumption is violated. See below for further discussion.
`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 `method = "forest"`, and for Greenwald-Khanna selects equally spaced percentiles optimized for accuracy (see below).
`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, `rfsrcFast`, in place of `rfsrc`? Improves speed but may be less accurate.
`maxn` | 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.

### Details

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.

### Value

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.

### Author(s)

Hemant Ishwaran and Udaya B. Kogalur

### References

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

`rfsrc`

### Examples

    ## ------------------------------------------------------------
    ## 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)

[Package randomForestSRC version 2.11.0 Index]