bark {bark}

Nonparametric Regression using Bayesian Additive Regression Kernels

Description

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.

Usage

bark(
  formula,
  data,
  subset,
  na.action = na.omit,
  testdata = NULL,
  selection = TRUE,
  common_lambdas = TRUE,
  classification = FALSE,
  keepevery = 100,
  nburn = 100,
  nkeep = 100,
  printevery = 1000,
  keeptrain = FALSE,
  verbose = FALSE,
  fixed = list(),
  tune = list(lstep = 0.5, frequL = 0.2, dpow = 1, upow = 0, varphistep = 0.5, phistep =
    1),
  theta = list()
)

Arguments

formula

model formula for the model with all predictors, Y ~ X. The X variables will be centered and scaled as part of model fitting.

data

a data frame. Factors will be converted to numerical vectors based on the using 'model.matrix'.

subset

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

na.action

a function which indicates what should happen when the data contain NAs. The default is "na.omit".

testdata

Dataframe with test data for out of sample prediction.
Should have same structure as data.

selection

Logical variable indicating whether variable dependent kernel parameters \lambda may be set to zero in the MCMC; default is TRUE.

common_lambdas

Logical variable indicating whether kernel parameters \lambda should be predictor specific or common across predictors; default is TRUE. Note if common_lambdas = TRUE and selection = TRUE this applies just to the non-zero lambda_j.

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.

verbose

Logical, whether to print out messages

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.

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.

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)

Value

bark returns an object of class 'bark' with a list, including:

call

the matched call

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; NULL if testdata are not provided

yhat.train.mean

train data fits = row mean of yhat.train

yhat.test.mean

test data fits = row mean of yhat.test

References

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

See Also

Other bark functions: bark-package-deprecated, bark-package, sim_Friedman1(), sim_Friedman2(), sim_Friedman3(), sim_circle()

Examples

##Simulated 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 <- data.frame(sim_Friedman2(200, sd=125))
testdata <- data.frame(sim_Friedman2(1000, sd=0))
# example with a very small number of iterations to illustrate usage
fit.bark.d <- bark(y ~ ., data=traindata, testdata= testdata,
                   nburn=10, nkeep=10, keepevery=10,
                   classification=FALSE, 
                   common_lambdas = FALSE,
                   selection = FALSE)
boxplot(data.frame(fit.bark.d$theta.lambda))
mean((fit.bark.d$yhat.test.mean-testdata$y)^2)

 ##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(y ~ ., 
                     data=data.frame(traindata), 
                     testdata= data.frame(testdata), 
                     classification=TRUE,
                     nburn=100, nkeep=200, )
 boxplot(as.data.frame(fit.bark.se$theta.lambda))
 mean((fit.bark.se$yhat.test.mean>0)!=testdata$y)