README.md

Randomer Forest

• Randomer Forest
  • Repo Contents
  • Description
  • Tested on
  • Hardware Requirements
  • Software Dependencies
  • Installation
• Usage
  • Load the library :
  • Create a forest:
  • Making predictions and determining error rate:
  • Compute similarities:
  • Compute tree strengths and correlations:
  • Train Structured RerF (S-RerF) for image classification:
  • Unsupervised classification (U-RerF)
  • Similarity Randomer Forest (SmerF)
  • Fast-RerF (fpRerF)

Repo Contents

• R: R building blocks for user interface code. Internally called by user interface.
• man: Package documentation
• src: C++ functions called from within R
• tests: testthat tests

Description

R-RerF (aka Randomer Forest (RerF), or Random Projection Forests) is a generalization of the Random Forest (RF) algorithm. RF partitions the input (feature) space via a series of recursive binary hyperplanes. Hyperplanes are constrained to be axis-aligned. In other words, each partition is a test of the form X_i > t, where t is a threshold and X_i is one of p inputs (features) {X₁, …, X_p}. The best axis-aligned split is found by sampling a random subset of the p inputs and choosing the one that best partitions the observed data according to some specified split criterion. RerF relaxes the constraint that the splitting hyperplanes must be axis-aligned. That is, each partition in RerF is a test of the form w₁X₁ + … + w_pX_p > t. The orientations of hyperplanes are sampled randomly via a user-specified distribution on the coefficients w_i, although an empirically validated default distribution is provided. Currently only classification is supported. Regression and unsupervised learning will be supported in the future.

Tested on

• Mac OSX: 10.11 (El Capitan), 10.12 (Sierra), 10.13 (High Sierra)
• Linux: Ubuntu 16.04 and 17.10, CentOS 6
• Windows: 10

Hardware Requirements

Any machine with >= 2 GB RAM

Software Dependencies

• OpenMP (for fpRerF)
• R (>= 3.3.0)
• R packages:
  • dummies
  • compiler
  • RcppArmadillo
  • RcppZiggurat
  • parallel

Installation

• Installation normally takes ~5-10 minutes
• Non-Windows users install the GNU Scientific Library (libgsl0-dev).
• Windows users install Rtools (https://cran.r-project.org/bin/windows/Rtools/)

Stable Release from CRAN:

From within R-

install.packages("rerf")

Development Version from Github:

From terminal:

git clone https://github.com/neurodata/RerF.git
## defaults to the staging branch
cd RerF
Rscript -e "install.packages('R-Project/', type = 'source', repos = NULL)"

Mac OS

• run brew install libomp.
• edit the user Makevars file ~/.R/Makevars:

omploc=$(brew --prefix libomp)

SHLIB_OPENMP_CFLAGS = -Xpreprocessor -fopenmp -I$(omploc)/include
SHLIB_OPENMP_CXXFLAGS = -Xpreprocessor -fopenmp -I$(omploc)/include

CFLAGS   = -Wall -O3 -ffast-math
CXXFLAGS = -Wall -O3 -ffast-math

• then Rscript -e "install.packages('R-Project/', type = 'source', repos = NULL)" from the above instructions.

---

Usage

Runtime for the following examples should be < 1 sec on any machine.

Load the library :

library(rerf)

Create a forest:

To create a forest the minimum data needed is an n by d input matrix (X) and an n length vector of corresponding class labels (Y). Rows correspond to samples and columns correspond to features.

X <- as.matrix(iris[,1:4])
Y <- iris[[5L]]
forest <- RerF(X, Y, seed = 1L, num.cores = 1L)

Expected output

forest$trees[[1]]
#> $treeMap
#>  [1]   1  -1   2  -2   3   4   5  -3  -4   6  13   7  10  -5   8  -6   9
#> [18]  -7  -8  -9  11 -10  12 -11 -12 -13 -14
#> 
#> $CutPoint
#>  [1]  2.35 -3.55 -9.90  6.95 -4.75 -5.05 -5.35  5.90  1.80 -6.50  6.05
#> [12]  1.65 -1.65
#> 
#> $ClassProb
#>       [,1] [,2] [,3]
#>  [1,]    1    0    0
#>  [2,]    0    0    1
#>  [3,]    0    0    1
#>  [4,]    0    1    0
#>  [5,]    0    0    1
#>  [6,]    0    0    1
#>  [7,]    0    1    0
#>  [8,]    0    0    1
#>  [9,]    0    1    0
#> [10,]    0    0    1
#> [11,]    0    1    0
#> [12,]    0    0    1
#> [13,]    0    0    1
#> [14,]    0    1    0
#> 
#> $matAstore
#>  [1]  3  1  3 -1  4  1  1 -1  2 -1  1  1  3 -1  3 -1  3 -1  1  1  4  1  1
#> [24] -1  1  1  4  1  4 -1
#> 
#> $matAindex
#>  [1]  0  2  6 10 12 14 16 18 20 22 24 26 28 30
#> 
#> $ind
#> NULL
#> 
#> $rotmat
#> NULL
#> 
#> $rotdims
#> NULL
#> 
#> $delta.impurity
#> NULL

forest is a trained forest which is needed for all other rerf functions. Additional parameters and more complex examples of training a forest can be found using the help function (?RerF)

Making predictions and determining error rate:

In the example below, trainIdx is used to subset the iris dataset in order to make a training set and a testing set.

trainIdx <- c(1:40, 51:90, 101:140)
X <- as.matrix(iris[,1:4])
Y <- iris[[5L]]
forest <- RerF(X[trainIdx, ], Y[trainIdx], num.cores = 1L, rank.transform = TRUE, seed = 1)
# Using a set of samples with unknown classification
predictions <- Predict(X[-trainIdx, ], forest, num.cores = 1L, Xtrain = X[trainIdx, ])
error.rate <- mean(predictions != Y[-trainIdx])

Expected output

predictions
#>  [1] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [7] setosa     setosa     setosa     setosa     versicolor versicolor
#> [13] versicolor versicolor versicolor versicolor versicolor versicolor
#> [19] versicolor versicolor virginica  virginica  virginica  virginica 
#> [25] virginica  virginica  virginica  virginica  virginica  virginica 
#> Levels: setosa versicolor virginica
error.rate
#> [1] 0

If a testing set is not available the error rate of a forest can be determined based on the samples held out-of-bag while training (out-of-bag samples are randomly chosen for each tree in the forest).

X <- as.matrix(iris[,1:4])
Y <- iris[[5L]]
forest <- RerF(X, Y, store.oob=TRUE, num.cores = 1L, seed = 1L)
predictions <- OOBPredict(X, forest, num.cores = 1L)
oob.error <- mean(predictions != Y)

Expected output

predictions
#>   [1] setosa     setosa     setosa     setosa     setosa     setosa    
#>   [7] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [13] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [19] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [25] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [31] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [37] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [43] setosa     setosa     setosa     setosa     setosa     setosa    
#>  [49] setosa     setosa     versicolor versicolor versicolor versicolor
#>  [55] versicolor versicolor versicolor versicolor versicolor versicolor
#>  [61] versicolor versicolor versicolor versicolor versicolor versicolor
#>  [67] versicolor versicolor versicolor versicolor virginica  versicolor
#>  [73] versicolor versicolor versicolor versicolor versicolor virginica 
#>  [79] versicolor versicolor versicolor versicolor versicolor virginica 
#>  [85] versicolor versicolor versicolor versicolor versicolor versicolor
#>  [91] versicolor versicolor versicolor versicolor versicolor versicolor
#>  [97] versicolor versicolor versicolor versicolor virginica  virginica 
#> [103] virginica  virginica  virginica  virginica  versicolor virginica 
#> [109] virginica  virginica  virginica  virginica  virginica  virginica 
#> [115] virginica  virginica  virginica  virginica  virginica  versicolor
#> [121] virginica  virginica  virginica  virginica  virginica  virginica 
#> [127] virginica  virginica  virginica  virginica  virginica  virginica 
#> [133] virginica  versicolor virginica  virginica  virginica  virginica 
#> [139] virginica  virginica  virginica  virginica  virginica  virginica 
#> [145] virginica  virginica  virginica  virginica  virginica  virginica 
#> Levels: setosa versicolor virginica
oob.error
#> [1] 0.04

Compute similarities:

Computes pairwise similarities between observations. The similarity between two points is defined as the fraction of trees such that two points fall into the same leaf node (i.e. two samples are similar if they consistently show up in the same leaf node). This function produces an n by n symmetric similarity matrix.

X <- as.matrix(iris[,1:4])
Y <- iris[[5L]]
forest <- RerF(X, Y, num.cores = 1L, seed = 1L)
sim.matrix <- ComputeSimilarity(X, forest, num.cores = 1L)

Expected output

sim.matrix[1, ]
#>   [1] 1.000 0.942 0.948 0.946 1.000 0.944 0.982 0.996 0.920 0.962 0.976
#>  [12] 0.986 0.938 0.924 0.828 0.776 0.968 1.000 0.814 1.000 0.994 0.996
#>  [23] 0.982 0.964 0.978 0.948 0.988 0.998 0.998 0.948 0.950 0.992 0.952
#>  [34] 0.866 0.958 0.964 0.916 1.000 0.928 1.000 0.996 0.886 0.948 0.988
#>  [45] 0.962 0.932 0.994 0.948 0.986 0.972 0.000 0.000 0.000 0.002 0.000
#>  [56] 0.000 0.002 0.010 0.000 0.004 0.010 0.000 0.000 0.000 0.000 0.004
#>  [67] 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#>  [78] 0.000 0.000 0.000 0.002 0.002 0.000 0.000 0.008 0.004 0.000 0.000
#>  [89] 0.000 0.002 0.002 0.000 0.000 0.008 0.000 0.000 0.000 0.000 0.014
#> [100] 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [111] 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [122] 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [133] 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [144] 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Compute tree strengths and correlations:

Computes estimates of tree strength and correlation according to the definitions in Breiman’s 2001 Random Forests paper.

set.seed(24)
nsamp <- 30 ## number of training samples per species
trainIdx <- vapply(list(1:50, 51:100, 101:150), sample, outer(1,1:nsamp), size = nsamp)
X <- as.matrix(iris[,1:4])
Y <- iris[[5L]]
forest <- RerF(X[trainIdx, ], Y[trainIdx], num.cores = 1L, seed = 1L)
predictions <- Predict(X[-trainIdx, ], forest, num.cores = 1L, aggregate.output = FALSE)
scor <- StrCorr(predictions, Y[-trainIdx])

Expected output

scor
#> $s
#> [1] 0.8256333
#> 
#> $rho
#> [1] 0.6987141

Compute feature (projection) importance (DEV version only):

Computes the Gini importance for all of the unique projections used to split the data. The returned value is a list with members imp and features. The member imp is a numeric vector of feature importances sorted in decreasing order. The member features is a list the same length as imp of vectors specifying the split projections corresponding to the values in imp. The projections are represented by the vector such that the odd numbered indices indicate the canonical feature indices and the even numbered indices indicate the linear coefficients. For example a vector (1,-1,4,1,5,-1) is the projection -X1 + X4 - X5. Note: it is highly advised to run this only when the splitting features (projections) have unweighted coefficients, such as for the default setting or for RF.

X <- as.matrix(iris[, 1:4]) # feature matrix
Y <- iris$Species # class labels
p <- ncol(X) # number of features in the data
d <- ceiling(sqrt(p)) # number of features to sample at each split

# Here we specify that we want to run the standard random forest algorithm and we want to store the decrease in impurity at each split node. The latter option is required in order to compute Gini feature importance.

forest <- RerF(as.matrix(iris[, 1:4]), iris[[5L]], FUN = RandMatRF,
               paramList = list(p = p, d = d), num.cores = 1L,
               store.impurity = TRUE, seed = 1L)

feature.imp <- FeatureImportance(forest, num.cores = 1L, type = "R")
#> Message: Computing feature importance for RandMatRF.

Expected output

feature.imp
#> $imp
#> [1] 23248.250 20118.932  5445.032  1167.640
#> 
#> $features
#> $features[[1]]
#> [1] 4 1
#> 
#> $features[[2]]
#> [1] 3 1
#> 
#> $features[[3]]
#> [1] 1 1
#> 
#> $features[[4]]
#> [1] 2 1
#> 
#> 
#> $type
#> [1] "R"

Train Structured RerF (S-RerF) for image classification:

S-RerF samples and evaluates a set of random features at each split node, where each feature is defined as a random linear combination of intensities of pixels contained in a contiguous patch within an image. Thus, the generated features exploit local structure inherent in images.

To be able to run this example quickly we will consider training and testing on the digits 3 and 5. You can try a differernt subset of digits by changing numsub in the code chunk below.

data(mnist)

## Get a random subsample, 100 each of 3's and 5's
set.seed(320)
threes <- sample(which(mnist$Ytrain %in% 3), 100)
fives  <- sample(which(mnist$Ytrain %in% 5), 100)
numsub <- c(threes, fives)

Ytrain <- mnist$Ytrain[numsub]
Xtrain <- mnist$Xtrain[numsub,]
Ytest <- mnist$Ytest[mnist$Ytest %in% c(3,5)]
Xtest <- mnist$Xtest[mnist$Ytest %in% c(3,5),]

# p is number of dimensions, d is the number of random features to evaluate, iw is image width, ih is image height, patch.min is min width of square patch to sample pixels from, and patch.max is the max width of square patch
p <- ncol(Xtrain)
d <- ceiling(sqrt(p))
iw <- sqrt(p)
ih <- iw
patch.min <- 1L
patch.max <- 5L
forest <- RerF(Xtrain, Ytrain, num.cores = 1L, FUN = RandMatImagePatch,
               paramList = list(p = p, d = d, iw = iw, ih = ih,
                                pwMin = patch.min, pwMax = patch.max),
               seed = 1L)
predictions <- Predict(Xtest, forest, num.cores = 1L)
mnist.error.rate <- mean(predictions != Ytest)

Expected output

mnist.error.rate
#> [1] 0.03575184

Unsupervised classification (U-RerF)

Using the Iris dataset we will show how to use the unsupervised verison.

set.seed(54321)
X <- as.matrix(iris[, 1:4])

u1 <- Urerf(X, trees = 100, Progress = FALSE)

The dissimilarity matrix

m <- as.matrix(u1$sim)
#plot(as.raster(m))

Running h-clust on the resulting dissimiliarity matrix

disSim <- hclust(as.dist(1 - u1$similarityMatrix), method = 'mcquitty')
clusters <- cutree(disSim, k = 3)
table(clusters, truth = as.numeric(iris[[5]]))
#>         truth
#> clusters  1  2  3
#>        1 50  0  0
#>        2  0 50 17
#>        3  0  0 33

Similarity Randomer Forest (SmerF)

Create the similarity matrix for training

train <- sort(c(sample(1:50, 25), sample(51:100, 25), sample(101:150, 25)))
test <- setdiff(1:150, train)

X <- iris[, -5]
Y <- 1 - as.matrix(dist(X[train, ]))
Ytest <- 1 - as.matrix(dist(X[test, ]))

Run SmerF

iris.forest <- RerF(X[train, ], Y, task = 'similarity', num.cores = 4L)
Yhat <- Predict(X[test, ], iris.forest, num.cores = 4L)

Frobenius norm of (Y - \hat{Y})

(f.iris <- norm(Ytest - Yhat, "F"))
#> [1] 15.58063
max(abs(Ytest - Yhat))
#> [1] 0.9428222

Forest Packing with fast-RerF

X <- mnist$Xtrain
Y <- mnist$Ytrain


## runs in under a minute on all of MNIST
system.time({
f <- fpRerF(X, Y, forestType = "binnedBaseRerF", numTreesInForest = 100, numCores = 4)
})
#>    user  system elapsed 
#>   6.767   0.021   1.699

training.pred <- fpPredict(f, X)
testing.pred <- fpPredict(f, mnist$Xtest)

(training.error <- mean(training.pred != Y))
#> [1] 0.0015
(testing.error <- mean(testing.pred != mnist$Ytest))
#> [1] 0.0563

---