# Chapter 2 Random Forests (RF)

Drawing by Phil Cutler.

## 2.1 Introduction

Any tutorial on Random Forests (RF) should also include a review of decicion trees, as these are models that are ensembled together to create the Random Forest model – or put another way, the “trees that comprise the forest.” Much of the complexity and detail of the Random Forest algorithm occurs within the individual decision trees and therefore it’s important to understand decision trees to understand the RF algorithm as a whole. Therefore, before proceeding, it is recommended that you read through the accompanying Classification and Regression Trees Tutorial.

## 2.2 History

The Random Forest algorithm is preceeded by the Random SubspaceMethod (aka “attribute bagging”), which accounts for half of the source of randomness in a Random Forest. The Random Subspace Method is an ensemble method that consists of several classifiers each operating in a subspace of the original feature space. The outputs of the models are then combined, usually by a simple majority vote. Tin Kam Ho applied the random subspace method to decision trees in 1995.

Leo Breiman and Adele Culter combined Breiman’s bagging idea with the random subspace method to create a “Random Forest”, a name which is trademarked by the duo. Due to the trademark, the algorithm is sometimes called Random Decision Forests.

The introduction of random forests proper was first made in a paper by Leo Breiman [1]. This paper describes a method of building a forest of uncorrelated trees using a CART like procedure, combined with randomized node optimization and bagging. In addition, this paper combines several ingredients, some previously known and some novel, which form the basis of the modern practice of random forests, in particular:

The report also offers the first theoretical result for random forests in the form of a bound on the generalization error which depends on the strength of the trees in the forest and their correlation.

Although Brieman’s implemenation of Random Forests used his CART algorithm to construct the decision trees, many modern implementations of Random Forest use entropy-based algorithms for constructing the trees.

## 2.3 Bagging

Bagging (Bootstrap aggregating) was proposed by Leo Breiman in 1994 to improve the classification by combining classifications of randomly generated training sets. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the model averaging approach.

• Bagging or bootstrap aggregation averages a noisy fitted function, refit to many bootstrap samples to reduce it’s variance.
• Bagging can dramatically reduce the variance of unstable procedures (like trees), leading to improved prediction, however any simple, interpretable, model structure (like that of a tree) is lost.
• Bagging produces smoother decision boundaries than trees.

The training algorithm for random forests applies the general technique of bootstrap aggregating, or bagging, to tree learners. Given a training set $$X = x_1, ..., x_n$$ with responses $$Y = y_1, ..., y_n$$, bagging repeatedly ($$B$$ times) selects a random sample with replacement of the training set and fits trees to these samples:

For $$b = 1, ..., B$$: 1. Sample, with replacement, $$n$$ training examples from $$X$$, $$Y$$; call these $$X_b$$, $$Y_b$$. 2. Train a decision or regression tree, $$f_b$$, on $$X_b$$, $$Y_b$$.

After training, predictions for unseen samples $$x'$$ can be made by averaging the predictions from all the individual regression trees on $$x'$$:

${\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}{\hat {f}}_{b}(x')$

or by taking the majority vote in the case of decision trees.

## 2.4 Random Forest Algorithm

The above procedure describes the original bagging algorithm for trees. Random Forests differ in only one way from this general scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a random subset of the features. This process is sometimes called “feature bagging”.

• Random Forests correct for decision trees’ habit of overfitting to their training set.
• Random Forest is an improvement over bagged trees that “de-correlates” the trees even further, reducing the variance.
• At each tree split, a random sample of $$m$$ features is drawn and only those $$m$$ features are considered for splitting.
• Typically $$m = \sqrt{p}$$ or $$\log_2p$$ where $$p$$ is the original number of features.
• For each tree gown on a bootstrap sample, the error rates for observations left out of the bootstrap sample is monitored. This is called the “out-of- bag” or OOB error rate.
• Each tree has the same (statistical) expectation, so increasing the number of trees does not alter the bias of bagging or the Random Forest algorithm.

## 2.5 Decision Boundary

This is an example of a decision boundary in two dimensions of a (binary) classification Random Forest. The black circle is the Bayes Optimal decision boundary and the blue square-ish boundary is learned by the classification tree.

Source: Elements of Statistical Learning

## 2.6 Random Forest by Randomization (aka “Extra-Trees”)

In Extremely Randomized Trees (aka Extra- Trees) [2], randomness goes one step further in the way splits are computed. As in Random Forests, a random subset of candidate features is used, but instead of looking for the best split, thresholds (for the split) are drawn at random for each candidate feature and the best of these randomly-generated thresholds is picked as the splitting rule. This usually allows to reduce the variance of the model a bit more, at the expense of a slightly greater increase in bias.

Extremely Randomized Trees is implemented in the extraTrees R package and also available in the h2o R package as part of the h2o.randomForest() function via the histogram_type = "Random" argument.

## 2.7 Out-of-Bag (OOB) Estimates

In random forests, there is no need for cross-validation or a separate test set to get an unbiased estimate of the test set error. It is estimated internally, during the run, as follows:

• Each tree is constructed using a different bootstrap sample from the original data. About one-third of the cases are left out of the bootstrap sample and not used in the construction of the kth tree.
• Put each case left out in the construction of the kth tree down the kth tree to get a classification. In this way, a test set classification is obtained for each case in about one-third of the trees.
• At the end of the run, take j to be the class that got most of the votes every time case n was oob. The proportion of times that j is not equal to the true class of n averaged over all cases is the oob error estimate. This has proven to be unbiased in many tests.

## 2.8 Variable Importance

In every tree grown in the forest, put down the OOB cases and count the number of votes cast for the correct class. Now randomly permute the values of variable m in the oob cases and put these cases down the tree. Subtract the number of votes for the correct class in the variable-$$m$$-permuted OOB data from the number of votes for the correct class in the untouched OOB data. The average of this number over all trees in the forest is the raw importance score for variable $$m$$.

If the values of this score from tree to tree are independent, then the standard error can be computed by a standard computation. The correlations of these scores between trees have been computed for a number of data sets and proved to be quite low, therefore we compute standard errors in the classical way, divide the raw score by its standard error to get a $$z$$-score, ands assign a significance level to the $$z$$-score assuming normality.

If the number of variables is very large, forests can be run once with all the variables, then run again using only the most important variables from the first run.

For each case, consider all the trees for which it is oob. Subtract the percentage of votes for the correct class in the variable-$$m$$-permuted OOB data from the percentage of votes for the correct class in the untouched OOB data. This is the local importance score for variable m for this case.

Variable importance in Extremely Randomized Trees is explained here.

## 2.9 Overfitting

Leo Brieman famously claimed that “Random Forests do not overfit.” This is perhaps not exactly the case, however they are certainly more robust to overfitting than a Gradient Boosting Machine (GBM). Random Forests can be overfit by growing trees that are “too deep”, for example. However, it is hard to overfit a Random Forest by adding more trees to the forest – typically that will increase accuracy (at the expense of computation time).

## 2.10 Missing Data

Missing values do not neccessarily have to be imputed in a Random Forest implemenation, although some software packages will require it.

## 2.11 Practical Uses

Here is a short article called, The Unreasonable Effectiveness of Random Forests, by Ahmed El Deeb, about the utility of Random Forests. It summarizes some of the algorithm’s pros and cons nicely.

## 2.13 Random Forest Software in R

The oldest and most well known implementation of the Random Forest algorithm in R is the randomForest package. There are also a number of packages that implement variants of the algorithm, and in the past few years, there have been several “big data” focused implementations contributed to the R ecosystem as well.

Here is a non-comprehensive list:

• randomForest::randomForest
• h2o::h2o.randomForest
• DistributedR::hpdRF_parallelForest
• party::cForest: A random forest variant for response variables measured at arbitrary scales based on conditional inference trees.
• randomForestSRC implements a unified treatment of Breiman’s random forests for survival, regression and classification problems.
• quantregForest can regress quantiles of a numeric response on exploratory variables via a random forest approach.
• ranger
• Rborist
• The caret package wraps a number of different Random Forest packages in R (full list here):
• Conditional Inference Random Forest (party::cForest)
• Oblique Random Forest (obliqueRF)
• Parallel Random Forest (randomForest + foreach)
• Random Ferns (rFerns)
• Random Forest (randomForest)
• Random Forest (ranger)
• Quantile Random Forest (quantregForest)
• Random Forest by Randomization (extraTrees)
• Random Forest Rule-Based Model (inTrees)
• Random Forest with Additional Feature Selection (Boruta)
• Regularized Random Forest (RRF)
• Rotation Forest (rotationForest)
• Weighted Subspace Random Forest (wsrf)
• The mlr package wraps a number of different Random Forest packages in R:
• Conditional Inference Random Forest (party::cForest)
• Rotation Forest (rotationForest)
• Parallel Forest (ParallelForest)
• Survival Forest (randomForestSRC)
• Random Ferns (rFerns)
• Random Forest (randomForest)
• Random Forest (ranger)
• Synthetic Random Forest (randomForestSRC)
• Random Uniform Forest (randomUniformForest)

Since there are so many different Random Forest implementations available, there have been several benchmarks to compare the performance of popular implementations, including implementations outside of R. A few examples:

1. Benchmarking Random Forest Classification by Erin LeDell, 2013
2. Benchmarking Random Forest Implementations by Szilard Pafka, 2015
3. Ranger publication by Marvin N. Wright and Andreas Ziegler, 2015

### 2.13.1 randomForest

Authors: Fortran original by LeoBreiman and Adele Cutler, R port by AndyLiaw and Matthew Wiener.

Backend: Fortran

Features:

• This package wraps the original Fortran code by Leo Breiman and Adele Culter and is probably the most widely known/used implemenation in R.
• Although it’s single-threaded, smaller forests can be trained in parallel by writing custom foreach or parall el code, then combined into a bigger forest using the randomForest::combine() function.
• Row weights unimplemented (been on the wishlist for as long as I can remember).
• Uses CART trees split by Gini Impurity.
• Categorical predictors are allowed to have up to 53 categories.
• Multinomial response can have no more than 32 categories.
• Supports R formula interface (but I’ve read some reports that claim it’s slower when the formula interface is used).
# randomForest example
# install.packages("randomForest")
# install.packages("cvAUC")
library(randomForest)
library(cvAUC)
# Load binary-response dataset

# convert to regular data frames
train <- as.data.frame(train)
test <- as.data.frame(test)

# Dimensions
dim(train)
## [1] 10000    29
dim(test)
## [1] 5000   29

# Columns
names(train)
##  [1] "response" "x1"       "x2"       "x3"       "x4"       "x5"
##  [7] "x6"       "x7"       "x8"       "x9"       "x10"      "x11"
## [13] "x12"      "x13"      "x14"      "x15"      "x16"      "x17"
## [19] "x18"      "x19"      "x20"      "x21"      "x22"      "x23"
## [25] "x24"      "x25"      "x26"      "x27"      "x28"
# Identity the response column
ycol <- "response"

# Identify the predictor columns
xcols <- setdiff(names(train), ycol)

# Convert response to factor (required by randomForest)
train[,ycol] <- as.factor(train[,ycol])
test[,ycol] <- as.factor(test[,ycol])
# Train a default RF model with 500 trees
set.seed(1)  # For reproducibility
system.time(
model <- randomForest(
x = train[,xcols],
y = train[,ycol],
xtest = test[,xcols],
ntree = 500
)
)
##    user  system elapsed
##  18.460   0.052  18.566
# Generate predictions on test dataset
preds <- model$test$votes[, 2]
labels <- test[,ycol]

# Compute AUC on the test set
cvAUC::AUC(predictions = preds, labels = labels)
## [1] 0.7834743

### 2.13.2 caret method “parRF”

Authors: Max Kuhn

Backend: Fortran (wraps the randomForest package)

This is a wrapper for the randomForest package that parallelizes the tree building.

library(caret)
library(doParallel)
library(e1071)
# set up parallel environment
cl <- makeCluster(4)
registerDoParallel(cl)

# Train a "parRF" model using caret
model <- caret::train(
x = train[,xcols],
y = train[,ycol],
method = "parRF",
preProcess = NULL,
weights = NULL,
metric = "Accuracy",
maximize = TRUE,
trControl = trainControl(),
tuneGrid = NULL,
tuneLength = 3
)

# good practice to shut down parallel environment
stopCluster(cl) 

### 2.13.3 h2o

Authors: Jan Vitek, Arno Candel, H2O.ai contributors

Backend: Java

Features:

• Distributed and parallelized computation on either a single node or a multi- node cluster.
• Automatic early stopping based on convergence of user-specied metrics to user- specied relative tolerance.
• Data-distributed, which means the entire dataset does not need to fit into memory on a single node.
• Uses histogram approximations of continuous variables for speedup.
• Uses squared error to determine optimal splits.
• Automatic early stopping based on convergence of user-specied metrics to user- specied relative tolerance.
• Support for exponential families (Poisson, Gamma, Tweedie) and loss functions in addition to binomial (Bernoulli), Gaussian and multinomial distributions, such as Quantile regression (including Laplace).
• Grid search for hyperparameter optimization and model selection.
• Model export in plain Java code for deployment in production environments.
• GUI for training & model eval/viz (H2O Flow).

Implementation details are presented in slidedecks by Michal Mahalova and Jan Vitek.

#install.packages("h2o")
library(h2o)
#h2o.shutdown(prompt = FALSE)
h2o.init(nthreads = -1)  #Start a local H2O cluster using nthreads = num available cores
##
## H2O is not running yet, starting it now...
##
## Note:  In case of errors look at the following log files:
##
##
## Starting H2O JVM and connecting: ... Connection successful!
##
## R is connected to the H2O cluster:
##     H2O cluster uptime:         3 seconds 197 milliseconds
##     H2O cluster timezone:       America/New_York
##     H2O data parsing timezone:  UTC
##     H2O cluster version:        3.18.0.4
##     H2O cluster version age:    28 days, 3 hours and 9 minutes
##     H2O cluster total nodes:    1
##     H2O cluster total memory:   1.78 GB
##     H2O cluster total cores:    4
##     H2O cluster allowed cores:  4
##     H2O cluster healthy:        TRUE
##     H2O Connection ip:          localhost
##     H2O Connection port:        54321
##     H2O Connection proxy:       NA
##     H2O Internal Security:      FALSE
##     H2O API Extensions:         XGBoost, Algos, AutoML, Core V3, Core V4
##     R Version:                  R version 3.4.4 (2018-03-15)
# convert data to h2o objects
train <- as.h2o(train)
##
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test <- as.h2o(test)
##
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# Dimensions
dim(train)
## [1] 10000    29
dim(test)
## [1] 5000   29

# Columns
names(train)
##  [1] "response" "x1"       "x2"       "x3"       "x4"       "x5"
##  [7] "x6"       "x7"       "x8"       "x9"       "x10"      "x11"
## [13] "x12"      "x13"      "x14"      "x15"      "x16"      "x17"
## [19] "x18"      "x19"      "x20"      "x21"      "x22"      "x23"
## [25] "x24"      "x25"      "x26"      "x27"      "x28"
# Identity the response column
ycol <- "response"

# Identify the predictor columns
xcols <- setdiff(names(train), ycol)

# Convert response to factor (required by randomForest)
train[,ycol] <- as.factor(train[,ycol])
test[,ycol] <- as.factor(test[,ycol])
# Train a default RF model with 500 trees
system.time(
model <- h2o.randomForest(
x = xcols,
y = ycol,
training_frame = train,
seed = 1, #for reproducibility
ntrees = 500
)
)
##
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##    user  system elapsed
##   0.570   0.027  25.513
# Compute AUC on test dataset
# H2O computes many model performance metrics automatically, including AUC

perf <- h2o.performance(model = model, newdata = test)
h2o.auc(perf)
## [1] 0.785394
# good practice to shut down h2o environment
h2o.shutdown(prompt = FALSE)
## [1] TRUE

### 2.13.4 Rborist

Authors: Mark Seligman

Backend: C++

The Arborist provides a fast, open-source implementation of the Random Forest algorithm. The Arborist achieves its speed through efficient C++ code and parallel, distributed tree construction. This slidedeck provides detail about the implementation and vision of the project.

Features:

• Began as proprietary implementation, but was open-sourced and rewritten following dissolution of venture.
• Project called “Aborist” but R package is called “Rborist”. A Python interface is in development.
• CPU based but a GPU version called Curborist (Cuda Rborist) is in development (unclear if it will be open source).
• Unlimited factor cardinality.
• Emphasizes multi-core but not multi-node.
• Both Python support and GPU support have been “coming soon” since summer 2015, not sure the status of the projects.

### 2.13.5 ranger

Authors: Marvin N. Wright and Andreas Ziegler

Backend: C++

Ranger is a fast implementation of random forest (Breiman 2001) or recursive partitioning, particularly suited for high dimensional data. Classification, regression, probability estimation and survival forests are supported. Classification and regression forests are implemented as in the original Random Forest (Breiman 2001), survival forests as in Random Survival Forests (Ishwaran et al. 2008). For probability estimation forests see Malley et al. (2012).

Features: