Conditional Random Forests

Description

An implementation of the random forest and bagging ensemble algorithms utilizing conditional inference trees as base learners.

Usage

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cforest(formula, data, weights, subset, na.action = na.pass, 
        control = ctree_control(teststat = "quad",
                                testtype = "Univ", mincriterion = 0, ...), 
        ytrafo = NULL, scores = NULL, ntree = 500L, 
        perturb = list(replace = FALSE, fraction = 0.632), 
        mtry = ceiling(sqrt(nvar)), applyfun = NULL, cores = NULL, 
        trace = FALSE, ...)
## S3 method for class 'cforest'
predict(object, newdata = NULL, 
        type = c("response", "prob", "weights", "node"),
        OOB = FALSE, FUN = NULL, simplify = TRUE, ...)

Arguments

formula

a symbolic description of the model to be fit.

data

a data frame containing the variables in the model.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

weights

an optional vector of weights to be used in the fitting process. Non-negative integer valued weights are allowed as well as non-negative real weights. Observations are sampled (with or without replacement) according to probabilities weights / sum(weights). The fraction of observations to be sampled (without replacement) is computed based on the sum of the weights if all weights are integer-valued and based on the number of weights greater zero else. Alternatively, weights can be a double matrix defining case weights for all ncol(weights) trees in the forest directly. This requires more storage but gives the user more control.

na.action

a function which indicates what should happen when the data contain missing value.

control

a list with control parameters, see ctree_control. The default values correspond to those of the default values used by cforest from the party package.

ytrafo

an optional named list of functions to be applied to the response variable(s) before testing their association with the explanatory variables. Note that this transformation is only performed once for the root node and does not take weights into account. Alternatively, ytrafo can be a function of data and weights. In this case, the transformation is computed for every node. This feature is experimental and the user interface likely to change.

scores

an optional named list of scores to be attached to ordered factors.

ntree

Number of trees to grow for the forest.

perturb

a list with arguments replace and fraction determining which type of resampling with replace = TRUE referring to the n-out-of-n bootstrap and replace = FALSE to sample splitting. fraction is the number of observations to draw without replacement.

mtry

number of input variables randomly sampled as candidates at each node for random forest like algorithms. Bagging, as special case of a random forest without random input variable sampling, can be performed by setting mtry either equal to Inf or manually equal to the number of input variables.

applyfun

an optional lapply-style function with arguments function(X, FUN, ...). It is used for computing the variable selection criterion. The default is to use the basic lapply function unless the cores argument is specified (see below).

cores

numeric. If set to an integer the applyfun is set to mclapply with the desired number of cores.

trace

a logical indicating if a progress bar shall be printed while the forest grows.

object

An object as returned by cforest

newdata

An optional data frame containing test data.

type

a character string denoting the type of predicted value returned, ignored when argument FUN is given. For "response", the mean of a numeric response, the predicted class for a categorical response or the median survival time for a censored response is returned. For "prob" the matrix of conditional class probabilities (simplify = TRUE) or a list with the conditional class probabilities for each observation (simplify = FALSE) is returned for a categorical response. For numeric and censored responses, a list with the empirical cumulative distribution functions and empirical survivor functions (Kaplan-Meier estimate) is returned when type = "prob". "weights" returns an integer vector of prediction weights. For type = "where", a list of terminal node ids for each of the trees in the forest ist returned.

OOB

a logical defining out-of-bag predictions (only if newdata = NULL).

FUN

a function to compute summary statistics. Predictions for each node have to be computed based on arguments (y, w) where y is the response and w are case weights.

simplify

a logical indicating whether the resulting list of predictions should be converted to a suitable vector or matrix (if possible).

...

additional arguments.

Details

This implementation of the random forest (and bagging) algorithm differs from the reference implementation in randomForest with respect to the base learners used and the aggregation scheme applied.

Conditional inference trees, see ctree, are fitted to each of the ntree perturbed samples of the learning sample. Most of the hyper parameters in ctree_control regulate the construction of the conditional inference trees.

Hyper parameters you might want to change are:

1. The number of randomly preselected variables mtry, which is fixed to the square root of the number of input variables.

2. The number of trees ntree. Use more trees if you have more variables.

3. The depth of the trees, regulated by mincriterion. Usually unstopped and unpruned trees are used in random forests. To grow large trees, set mincriterion to a small value.

The aggregation scheme works by averaging observation weights extracted from each of the ntree trees and NOT by averaging predictions directly as in randomForest. See Hothorn et al. (2004) and Meinshausen (2006) for a description.

Predictions can be computed using predict. For observations with zero weights, predictions are computed from the fitted tree when newdata = NULL.

Ensembles of conditional inference trees have not yet been extensively tested, so this routine is meant for the expert user only and its current state is rather experimental. However, there are some things available in cforest that can't be done with randomForest, for example fitting forests to censored response variables (see Hothorn et al., 2004, 2006a) or to multivariate and ordered responses. Using the rich partykit infrastructure allows additional functionality in cforest, such as parallel tree growing and probabilistic forecasting (for example via quantile regression forests). Also plotting of single trees from a forest is much easier now.

Unlike cforest, cforest is entirely written in R which makes customisation much easier at the price of longer computing times. However, trees can be grown in parallel with this R only implemention which renders speed less of an issue. Note that the default values are different from those used in package party, most importantly the default for mtry is now data-dependent. predict(, type = "node") replaces the where function and predict(, type = "prob") the treeresponse function.

Moreover, when predictors vary in their scale of measurement of number of categories, variable selection and computation of variable importance is biased in favor of variables with many potential cutpoints in randomForest, while in cforest unbiased trees and an adequate resampling scheme are used by default. See Hothorn et al. (2006b) and Strobl et al. (2007) as well as Strobl et al. (2009).

Value

An object of class cforest.

References

Breiman L (2001). Random Forests. Machine Learning, 45(1), 5–32.

Hothorn T, Lausen B, Benner A, Radespiel-Troeger M (2004). Bagging Survival Trees. Statistics in Medicine, 23(1), 77–91.

Hothorn T, Bühlmann P, Dudoit S, Molinaro A, Van der Laan MJ (2006a). Survival Ensembles. Biostatistics, 7(3), 355–373.

Hothorn T, Hornik K, Zeileis A (2006b). Unbiased Recursive Partitioning: A Conditional Inference Framework. Journal of Computational and Graphical Statistics, 15(3), 651–674.

Hothorn T, Zeileis A (2015). partykit: A Modular Toolkit for Recursive Partytioning in R. Journal of Machine Learning Research, 16, 3905–3909.

Meinshausen N (2006). Quantile Regression Forests. Journal of Machine Learning Research, 7, 983–999.

Strobl C, Boulesteix AL, Zeileis A, Hothorn T (2007). Bias in Random Forest Variable Importance Measures: Illustrations, Sources and a Solution. BMC Bioinformatics, 8, 25. http://www.biomedcentral.com/1471-2105/8/25

Strobl C, Malley J, Tutz G (2009). An Introduction to Recursive Partitioning: Rationale, Application, and Characteristics of Classification and Regression Trees, Bagging, and Random Forests. Psychological Methods, 14(4), 323–348.

Examples

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## basic example: conditional inference forest for cars data
cf <- cforest(dist ~ speed, data = cars)

## prediction of fitted mean and visualization
nd <- data.frame(speed = 4:25)
nd$mean  <- predict(cf, newdata = nd, type = "response")
plot(dist ~ speed, data = cars)
lines(mean ~ speed, data = nd)

## predict quantiles (aka quantile regression forest)
myquantile <- function(y, w) quantile(rep(y, w), probs = c(0.1, 0.5, 0.9))
p <- predict(cf, newdata = nd, type = "response", FUN = myquantile)
colnames(p) <- c("lower", "median", "upper")
nd <- cbind(nd, p)

## visualization with conditional (on speed) prediction intervals
plot(dist ~ speed, data = cars, type = "n")
with(nd, polygon(c(speed, rev(speed)), c(lower, rev(upper)),
  col = "lightgray", border = "transparent"))
points(dist ~ speed, data = cars)
lines(mean ~ speed, data = nd, lwd = 1.5)
lines(median ~ speed, data = nd, lty = 2, lwd = 1.5)
legend("topleft", c("mean", "median", "10% - 90% quantile"),
  lwd = c(1.5, 1.5, 10), lty = c(1, 2, 1),
  col = c("black", "black", "lightgray"), bty = "n")


### we may also use predicted conditional (on speed) densities
mydensity <- function (y, w) approxfun(density(y, weights = w/sum(w))[1:2], rule = 2)
pd <- predict(cf, newdata = nd, type = "response", FUN = mydensity)

## visualization in heatmap (instead of scatterplot)
## with fitted curves as above
dist <- -10:150
dens <- t(sapply(seq_along(pd), function(i) pd[[i]](dist)))
image(nd$speed, dist, dens, xlab = "speed", col = rev(gray.colors(9)))
lines(mean ~ speed, data = nd, lwd = 1.5)
lines(median ~ speed, data = nd, lty = 2, lwd = 1.5)
lines(lower ~ speed, data = nd, lty = 2)
lines(upper ~ speed, data = nd, lty = 2)

### honest (i.e., out-of-bag) cross-classification of
### true vs. predicted classes
data("mammoexp", package = "TH.data")
table(mammoexp$ME, predict(cforest(ME ~ ., data = mammoexp, ntree = 50),
                           OOB = TRUE, type = "response"))

### fit forest to censored response
if (require("TH.data") && require("survival")) {

    data("GBSG2", package = "TH.data")
    bst <- cforest(Surv(time, cens) ~ ., data = GBSG2, ntree = 50)
 
    ### estimate conditional Kaplan-Meier curves
    print(predict(bst, newdata = GBSG2[1:2,], OOB = TRUE, type = "prob"))

    print(bst$nodes[[1]])
}