Nonparametric Regression with Bayesian Additive Regression Kernels
In bark: Bayesian Additive Regression Kernels

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Bayesian Additive Regression Kernel (BARK) models are a flexible Bayesian nonparametric model for regression and classification problems where the unknown mean function is represented as a weighted sum of multivariate Gaussian kernel functions, \begin{equation} f(\mathbf{x}) = \sum_j \beta_j \prod_d \exp( \lambda_d (x_d - \chi_{jd})^2) \end{equation} that allows nonlinearities, interactions and feature selection using Levy random fields to construct a prior on the unknown function. Each kernel is centered at location parameters $\chi_{jd}$ with precision parameters $\lambda_d$ - these precision parameters capture the importance of each of the $d$ dimensional predictor variables and by setting a $\lambda_d$ to zero may remove important variables that are not important.

Installation

To get the latest version of {r bark}, install from github (needs compilation))

devtools::install_github("merliseclyde/bark")

library(bark)

Example

We will illustrate feature selection in a simple simulated example from Friedman

set.seed(42)
traindata <- data.frame(sim_Friedman2(200, sd=125))
testdata <- data.frame(sim_Friedman2(1000, sd=0))

set.seed(42)
fit.bark.d <- bark(y ~ ., data = traindata,
                   testdata= testdata,
                   classification=FALSE, 
                   selection = FALSE,
                   common_lambdas = FALSE,
#                   fixed = list(eps = .25, gam = 2.5),
                   nburn = 100,
                   nkeep = 250,
                   printevery = 10^10)

mean((fit.bark.d$yhat.test.mean-testdata$y)^2)

set.seed(42)
fit.bark.sd <- bark(y ~ ., data=traindata,
                    testdata = testdata,
                    classification=FALSE, 
                    selection = TRUE,
                    common_lambdas = FALSE,
                    fixed = list(eps = .5, gam = 5),
                    nburn = 100,
                    nkeep = 250,
                    printevery = 10^10)

mean((fit.bark.sd$yhat.test.mean-testdata$y)^2)

bark is similar to SVM, however it allows different kernel smoothing parameters for every dimension of the inputs $x$ using the option common_lambdas = FALSE as well as selection of inputs by allowing the kernel smoothing parameters to be zero using the option selection = TRUE.

The plot below shows posterior draws of the $\lambda$ for the simulated data under the two scenarios allowing different $\lambda_d$ by dimension with and without selection.

boxplot(as.data.frame(fit.bark.d$theta.lambda))

boxplot(as.data.frame(fit.bark.sd$theta.lambda))

While the plots of the $\lambda_j$ without selection (top) and with selection (bottom) are similar, the additional shrinkage of values towards zero has lead to an improvement in RMSE.

Comparison

We will compare {r bark} to two other popular methods, {r BART} and {r SVM} that provide flexible models that are also non-linear in the input variables.

bart.available =  suppressMessages(require(BART))
svm.available  =  suppressMessages(require(e1071))
io.available  =  suppressMessages(require(fdm2id))

SVM

if (svm.available) {
  friedman2.svm = svm(y ~ ., data=traindata, type="eps-regression")
  pred.svm = predict(friedman2.svm, testdata)
  mean((pred.svm - testdata$y)^2)
}

BART

if (bart.available) {
 y.loc = match("y", colnames(traindata))
 friedman2.bart = wbart(x.train = as.matrix(traindata[ , -y.loc]), 
                        y.train =  traindata$y)
  pred.bart =   predict(friedman2.bart, 
                        as.matrix(testdata[ , -y.loc]))
  yhat.bart = apply(pred.bart, 2, mean)
  mean((yhat.bart - testdata$y)^2)
}

Classification Example

The data are generated so that only the first 2 dimensions matter

set.seed(42)
n = 500
circle2 = data.frame(sim_circle(n, dim = 5))
train = sample(1:n, size = floor(n/2), rep=FALSE)

plot(x.1 ~ x.2, data=circle2, col=y+1)

set.seed(42)
circle2.bark = bark(y ~ ., data=circle2, subset=train,
                    testdata = circle2[-train,],
                    classification = TRUE,
                    selection = TRUE,
                    common_lambdas = FALSE,
                    fixed = list(eps = .5, gam = 5),
                    nburn = 100,
                    nkeep = 250,
                    printevery = 10^10)

#Classify
#
mean((circle2.bark$yhat.test.mean > 0) != circle2[-train, "y"])

The plot below shows posterior draws of the $\lambda$ for the simulated data.

boxplot(as.data.frame(circle2.bark$theta.lambda))

SVM

if (svm.available) {
  circle2.svm = svm(y ~ ., data=circle2[train,], type="C")
  pred.svm = predict(circle2.svm, circle2[-train,])
  mean(pred.svm != circle2[-train, "y"])
}

if (bart.available) {
  y.loc = match("y", colnames(circle2))
  circle.bart = pbart(x.train = as.matrix(circle2[train, -y.loc]), 
                            y.train =  circle2[train, y.loc])
  pred.bart =   predict(circle.bart, as.matrix(circle2[-train, -y.loc]))
  mean((pred.bart$prob.test.mean > .5) != circle2[-train, y.loc])
}