NonParametric Regression using Bayesian Additive Regression Kernels

BARK is a Bayesian sum-of-kernels model.
For numeric response \(y\), we have \(y = f(x) + \epsilon\), where \(\epsilon \sim N(0,\sigma^2)\).
For a binary response \(y\), \(P(Y=1 | x) = F(f(x))\), where \(F\) denotes the standard normal cdf (probit link).
In both cases, \(f\) is the sum of many Gaussian kernel functions. The goal is to have very flexible inference for the unknown function \(f\). BARK uses an approximation to a Cauchy process as the prior distribution for the unknown function \(f\).

Feature selection can be achieved through the inference on the scale parameters in the Gaussian kernels. BARK accepts four different types of prior distributions, e, d, enabling either soft shrinkage or se, sd, enabling hard shrinkage for the scale parameters.

Arguments

x.train: Explanatory variables for training (in sample) data.
Must be a matrix of doubles, with (as usual) rows corresponding to observations and columns to variables.
y.train: Dependent variable for training (in sample) data.
If y is numeric a continuous response model is fit (normal errors).
If y is a logical (or just has values 0 and 1), then a binary response model with a probit link is fit.
x.test: Explanatory variables for test (out of sample) data.
Should have same structure as x.train.
type: BARK type, e, d, se, or sd, default choice is se.
e: BARK with equal weights.
d: BARK with different weights.
se: BARK with selection and equal weights.
sd: BARK with selection and different weights.
classification: TRUE/FALSE logical variable, indicating a classification or regression problem.
keepevery: Every keepevery draw is kept to be returned to the user
nburn: Number of MCMC iterations (nburn*keepevery) to be treated as burn in.
nkeep: Number of MCMC iterations kept for the posterior inference.
nkeep*keepevery iterations after the burn in.
printevery: As the MCMC runs, a message is printed every printevery draws.
keeptrain: Logical, whether to keep results for training samples.
fixed: A list of fixed hyperparameters, using the default values if not specified.
alpha = 1: stable index, must be 1 currently.
eps = 0.5: approximation parameter.
gam = 5: intensity parameter.
la = 1: first argument of the gamma prior on kernel scales.
lb = 2: second argument of the gamma prior on kernel scales.
pbetaa = 1: first argument of the beta prior on plambda.
pbetab = 1: second argument of the beta prior on plambda.
n: number of training samples, automatically generates.
p: number of explanatory variables, automatically generates.
meanJ: the expected number of kernels, automatically generates.
tune: A list of tuning parameters, not expected to change.
lstep: the stepsize of the lognormal random walk on lambda.
frequL: the frequency to update L.
dpow: the power on the death step.
upow: the power on the update step.
varphistep: the stepsize of the lognormal random walk on varphi.
phistep: the stepsize of the lognormal random walk on phi.
theta: A list of the starting values for the parameter theta, use defaults if nothing is given.

Value

bark returns a list, including:

fixed: Fixed hyperparameters
tune: Tuning parameters used
theta.last: The last set of parameters from the posterior draw
theta.nvec: A matrix with nrow(x.train)\(+1\) rows and (nkeep) columns, recording the number of kernels at each training sample
theta.varphi: A matrix with nrow(x.train) \(+1\) rows and (nkeep) columns, recording the precision in the normal gamma prior distribution for the regression coefficients
theta.beta: A matrix with nrow(x.train)\(+1\) rows and (nkeep) columns, recording the regression coefficients
theta.lambda: A matrix with ncol(x.train) rows and (nkeep) columns, recording the kernel scale parameters
thea.phi: The vector of length nkeep, recording the precision in regression Gaussian noise (1 for the classification case)
yhat.train: A matrix with nrow(x.train) rows and (nkeep) columns. Each column corresponds to a draw \(f^*\) from the posterior of \(f\) and each row corresponds to a row of x.train. The \((i,j)\) value is \(f^*(x)\) for the \(j^{th}\) kept draw of \(f\) and the \(i^{th}\) row of x.train.
For classification problems, this is the value of the expectation for the underlying normal random variable.
Burn-in is dropped
yhat.test: Same as yhat.train but now the x's are the rows of the test data
yhat.train.mean: train data fits = row mean of yhat.train
yhat.test.mean: test data fits = row mean of yhat.test

Details

BARK is implemented using a Bayesian MCMC method. At each MCMC interaction, we produce a draw from the joint posterior distribution, i.e. a full configuration of regression coefficients, kernel locations and kernel parameters etc.

Thus, unlike a lot of other modelling methods in R, we do not produce a single model object from which fits and summaries may be extracted. The output consists of values \(f^*(x)\) (and \(\sigma^*\) in the numeric case) where * denotes a particular draw. The \(x\) is either a row from the training data (x.train)

References

Ouyang, Zhi (2008) Bayesian Additive Regression Kernels. Duke University. PhD dissertation, page 58.

Examples

# Simulate regression example
#  Friedman 2 data set, 200 noisy training, 1000 noise free testing
#  Out of sample MSE in SVM (default RBF): 6500 (sd. 1600)
#  Out of sample MSE in BART (default):    5300 (sd. 1000)
traindata <- sim_Friedman2(200, sd=125)
testdata <- sim_Friedman2(1000, sd=0)
# example with a very small number of iterations to illustrate the method
fit.bark.d <- bark_mat(traindata$x, traindata$y, testdata$x,
                  nburn=10, nkeep=10, keepevery=10,
                  classification=FALSE, type="d")
#> [1] "Starting BARK-d for this regression problem"
boxplot(data.frame(fit.bark.d$theta.lambda))

mean((fit.bark.d$yhat.test.mean-testdata$y)^2)
#> [1] 8295.832
# \donttest{
 # Simulate classification example
 #  Circle 5 with 2 signals and three noisy dimensions
 #  Out of sample erorr rate in SVM (default RBF): 0.110 (sd. 0.02)
 #  Out of sample error rate in BART (default):    0.065 (sd. 0.02)
 traindata <- sim_circle(200, dim=5)
 testdata <- sim_circle(1000, dim=5)
 fit.bark.se <- bark_mat(traindata$x, traindata$y, testdata$x, classification=TRUE, type="se")
#> [1] "Starting BARK-se for this classification problem"
#> [1] "burning iteration 1000, J=16, max(nj)=1"
#> [1] "burning iteration 2000, J=16, max(nj)=2"
#> [1] "burning iteration 3000, J=21, max(nj)=2"
#> [1] "burning iteration 4000, J=18, max(nj)=2"
#> [1] "burning iteration 5000, J=18, max(nj)=1"
#> [1] "burning iteration 6000, J=15, max(nj)=1"
#> [1] "burning iteration 7000, J=10, max(nj)=2"
#> [1] "burning iteration 8000, J=16, max(nj)=2"
#> [1] "burning iteration 9000, J=15, max(nj)=1"
#> [1] "burning iteration 10000, J=12, max(nj)=2"
#> [1] "posterior mcmc iteration 1000, J=9, max(nj)=2"
#> [1] "posterior mcmc iteration 2000, J=14, max(nj)=1"
#> [1] "posterior mcmc iteration 3000, J=25, max(nj)=2"
#> [1] "posterior mcmc iteration 4000, J=17, max(nj)=2"
#> [1] "posterior mcmc iteration 5000, J=14, max(nj)=2"
#> [1] "posterior mcmc iteration 6000, J=14, max(nj)=2"
#> [1] "posterior mcmc iteration 7000, J=15, max(nj)=1"
#> [1] "posterior mcmc iteration 8000, J=15, max(nj)=2"
#> [1] "posterior mcmc iteration 9000, J=15, max(nj)=1"
#> [1] "posterior mcmc iteration 10000, J=13, max(nj)=2"
 boxplot(data.frame(fit.bark.se$theta.lambda))

 mean((fit.bark.se$yhat.test.mean>0)!=testdata$y)
#> [1] 0.01
# }