BART is a Bayesian “sum-of-trees” model in which each tree is constrained by a prior to be a weak learner.
For numeric response \(y = f(x) + \epsilon\), where \(\epsilon \sim N(0, \sigma^2)\).
For binary response \(y\), \(P(Y = 1 \mid x) = \Phi(f(x))\), where \(\Phi\) denotes the standard normal cdf (probit link).
bart(
x.train, y.train, x.test = matrix(0.0, 0, 0),
sigest = NA, sigdf = 3, sigquant = 0.90,
k = 2.0,
power = 2.0, base = 0.95, splitprobs = 1 / numvars,
binaryOffset = 0.0, weights = NULL,
ntree = 200,
ndpost = 1000, nskip = 100,
printevery = 100, keepevery = 1, keeptrainfits = TRUE,
usequants = FALSE, numcut = 100, printcutoffs = 0,
verbose = TRUE, nchain = 1, nthread = 1, combinechains = TRUE,
keeptrees = FALSE, keepcall = TRUE, sampleronly = FALSE,
seed = NA_integer_,
proposalprobs = NULL,
keepsampler = keeptrees)bart2(
formula, data, test, subset, weights, offset, offset.test = offset,
sigest = NA_real_, sigdf = 3.0, sigquant = 0.90,
k = NULL,
power = 2.0, base = 0.95, split.probs = 1 / num.vars,
n.trees = 75L,
n.samples = 500L, n.burn = 500L,
n.chains = 4L, n.threads = min(dbarts::guessNumCores(), n.chains),
combineChains = FALSE,
n.cuts = 100L, useQuantiles = FALSE,
n.thin = 1L, keepTrainingFits = TRUE,
printEvery = 100L, printCutoffs = 0L,
verbose = TRUE, keepTrees = FALSE,
keepCall = TRUE, samplerOnly = FALSE,
seed = NA_integer_,
proposal.probs = NULL,
keepSampler = keepTrees,
...)
# S3 method for bart
plot(
x,
plquants = c(0.05, 0.95), cols = c('blue', 'black'),
...)
# S3 method for bart
predict(
object, newdata, offset, weights,
type = c("ev", "ppd", "bart"),
combineChains = TRUE, ...)
extract(object, ...)
# S3 method for bart
extract(
object,
type = c("ev", "ppd", "bart", "trees"),
sample = c("train", "test"),
combineChains = TRUE, ...)
# S3 method for bart
fitted(
object,
type = c("ev", "ppd", "bart"),
sample = c("train", "test"),
...)
# S3 method for bart
residuals(object, ...)
bart and bart2 return lists assigned the class bart. For applicable quantities, ndpost / keepevery samples are returned. In the numeric \(y\) case, the list has components:
yhat.trainA array/matrix of posterior samples. The \((i, j, k)\) value is the \(j\)th draw of the posterior of \(f\) evaluated at the \(k\)th row of x.train (i.e. \(f^*(x_k)\)) corresponding to chain \(i\). When nchain is one or combinechains is TRUE, the result is a collapsed down to a matrix.
yhat.testSame as yhat.train but now the \(x\)s are the rows of the test data.
yhat.train.meanVector of means of yhat.train across columns and chains, with length equal to the number of training observations.
yhat.test.meanVector of means of yhat.test across columns and chains.
sigmaMatrix of posterior samples of sigma, the residual/error standard deviation. Dimensions are equal to the number of chains times the numbers of samples unless nchain is one or combinechains is TRUE.
first.sigmaBurn-in draws of sigma.
varcountA matrix with number of rows equal to the number of kept draws and each column corresponding to a training variable. Contains the total count of the number of times that variable is used in a tree decision rule (over all trees).
sigestThe rough error standard deviation (\(\sigma\)) used in the prior.
yThe input dependent vector of values for the dependent variable. This is used in plot.bart.
fitOptional sampler object which stores the values of the tree splits. Required for using predict and only stored if keeptrees or keepsampler is TRUE.
n.chainsInformation that can be lost if combinechains is TRUE is tracked here.
kOptional matrix of posterior samples of k. Only present when k is modeled, i.e. there is a hyperprior.
first.kBurn-in draws of k, if modeled.
In the binary \(y\) case, the returned list has the components yhat.train, yhat.test, and varcount as above. In addition the list has a binaryOffset component giving the value used.
Note that in the binary \(y\), case yhat.train and yhat.test are \(f(x) + \mathrm{binaryOffset}\). For draws of the probability \(P(Y = 1 | x)\), apply the normal cdf (pnorm) to these values.
The plot method sets mfrow to c(1, 2) and makes two plots. The first plot is the sequence of kept draws of \(\sigma\) including the burn-in draws. Initially these draws will decline as BART finds a good fit and then level off when the MCMC has burnt in. The second plot has \(y\) on the horizontal axis and posterior intervals for the corresponding \(f(x)\) on the vertical axis.
Explanatory variables for training (in sample) data. May be a matrix or a data frame, with rows corresponding to observations and columns to variables. If a variable is a factor in a data frame, it is replaced with dummies. Note that \(q\) dummies are created if \(q > 2\) and one dummy is created if \(q = 2\), where \(q\) is the number of levels of the factor.
Dependent variable for training (in sample) data. If y.train is numeric a continous response model is fit (normal errors). If y.train is a binary factor or has only values 0 and 1, then a binary response model with a probit link is fit.
Explanatory variables for test (out of sample) data. Should have same column structure as x.train. bart will generate draws of \(f(x)\) for each \(x\) which is a row of x.test.
For continuous response models, an estimate of the error variance, \(\sigma^2\), used to calibrate an inverse-chi-squared prior used on that parameter. If not supplied, the least-squares estimate is derived instead. See sigquant for more information. Not applicable when \(y\) is binary.
Degrees of freedom for error variance prior. Not applicable when \(y\) is binary.
The quantile of the error variance prior that the rough estimate (sigest) is placed at. The closer the quantile is to 1, the more aggresive the fit will be as you are putting more prior weight on error standard deviations (\(\sigma\)) less than the rough estimate. Not applicable when \(y\) is binary.
For numeric \(y\), k is the number of prior standard deviations \(E(Y|x) = f(x)\) is away from \(\pm 0.5\). The response (y.train) is internally scaled to range from \(-0.5\) to \(0.5\). For binary \(y\), k is the number of prior standard deviations \(f(x)\) is away from \(\pm 3\). In both cases, the bigger \(k\) is, the more conservative the fitting will be. The value can be either a fixed number, or the a hyperprior of the form chi(degreesOfFreedom = 1.25, scale = Inf). For bart2, the default of NULL uses the value 2 for continuous reponses and a chi hyperprior for binary ones. The default chi hyperprior is improper, and slightly penalizes small values of k.
Power parameter for tree prior.
Base parameter for tree prior.
Prior and transition probabilities of variables used to generate splits. Can be missing/empty/NULL for equiprobability, a numeric vector of length equal to the number variables, or a named numeric vector with only a subset of the variables specified and a .default named value. Values given for factor variables are replicated for each resulting column in the generated model matrix. numvars and num.vars symbols will be rebound before execution to the number of columns in the model matrix.
Used for binary \(y\). When present, the model is \(P(Y = 1 \mid x) = \Phi(f(x) + \mathrm{binaryOffset})\), allowing fits with probabilities shrunk towards values other than \(0.5\).
An optional vector of weights to be used in the fitting process. When present, BART fits a model with observations \(y \mid x \sim N(f(x), \sigma^2 / w)\), where \(f(x)\) is the unknown function.
The number of trees in the sum-of-trees formulation.
The number of posterior draws after burn in, ndpost / keepevery will actually be returned.
Number of MCMC iterations to be treated as burn in.
As the MCMC runs, a message is printed every printevery draws.
Every keepevery draw is kept to be returned to the user. Useful for “thinning” samples.
If TRUE the draws of \(f(x)\) for \(x\) corresponding to the rows of x.train are returned.
When TRUE, determine tree decision rules using estimated quantiles derived from the x.train variables. When FALSE, splits are determined using values equally spaced across the range of a variable. See details for more information.
The maximum number of possible values used in decision rules (see usequants, details). If a single number, it is recycled for all variables; otherwise must be a vector of length equal to ncol(x.train). Fewer rules may be used if a covariate lacks enough unique values.
The number of cutoff rules to printed to screen before the MCMC is run. Given a single integer, the same value will be used for all variables. If 0, nothing is printed.
Logical; if FALSE supress printing.
Integer specifying how many independent tree sets and fits should be calculated.
Integer specifying how many threads to use. Depending on the CPU architecture, using more than the number of chains can degrade performance for small/medium data sets. As such some calculations may be executed single threaded regardless.
Logical; if TRUE, samples will be returned in arrays of dimensions equal to nchain \(\times\) ndpost \(\times\) number of observations.
Logical; must be TRUE in order to use predict with the result of a bart fit. Note that for models with a large number of observations or a large number of trees, keeping the trees can be very memory intensive.
Logical; if FALSE, returned object will have call set to call("NULL"), otherwise the call used to instantiate BART.
Optional integer specifying the desired pRNG seed. It should not be needed when running single-threaded - set.seed will suffice, and can be used to obtain reproducible results when multi-threaded. See Reproducibility section below.
Named numeric vector or NULL, optionally specifying the proposal rules and their probabilities. Elements should be "birth_death", "change", and "swap" to control tree change proposals, and "birth" to give the relative frequency of birth/death in the "birth_death" step. Defaults are 0.5, 0.1, 0.4, and 0.5 respectively.
Logical that can be used to save the underlying dbartsSampler-class object even if keepTrees is false.
The same as x.train, the name reflecting that a formula object can be used instead.
The same as y.train, the name reflecting that a data frame can be specified when a formula is given instead.
The same as x.train. Can be missing.
A vector of logicals or indicies used to subset of the data. Can be missing.
The same as binaryOffset. Can be missing.
A vector of offsets to be used with test data, in case it is different than the training offset. If offest is missing, defaults to NULL.
An object of class bart, returned from either the function bart or bart2.
Test data for prediction. Obeys all the same rules as x.train but cannot be missing.
Builds the sampler from its arguments and returns it without running it. Useful to use the bart2 interface in more complicated models.
Object of class bart, returned by function bart, which contains the information to be plotted.
In the plots, beliefs about \(f(x)\) are indicated by plotting the posterior median and a lower and upper quantile. plquants is a double vector of length two giving the lower and upper quantiles.
Vector of two colors. First color is used to plot the median of \(f(x)\) and the second color is used to plot the lower and upper quantiles.
The quantity to be returned by generic functions. Options are "ev" - samples from the posterior of the individual level expected value, "bart" - the sum of trees component; same as "ev" for linear models but on the probit scale for binary ones, "ppd" - samples from the posterior predictive distribution, and "trees" - a data frame with tree information for when model was fit with keepTrees equal to TRUE. To synergize with predict.glm, "response" can be used as a synonym for "ev" and "link" can be used as a synonym for "bart". For information on extracting trees, see the subsection below.
Either "train" or "test".
Additional arguments passed on to plot, dbartsControl, or extract when type is "trees". Not used in predict.
Hugh Chipman: hugh.chipman@gmail.com, Robert McCulloch: robert.mcculloch1@gmail.com, Vincent Dorie: vdorie@gmail.com.
BART is an Bayesian MCMC method. At each MCMC interation, we produce a draw from the joint posterior \((f, \sigma) \mid (x, y)\) in the numeric \(y\) case and just \(f\) in the binary \(y\) case.
Thus, unlike a lot of other modeling methods in R, bart does 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) or the test data (x.test).
Decision rules for any tree are of the form \(x \le c\) vs. \(x > c\) for each ‘\(x\)’ corresponding to a column of x.train. usequants determines the means by which the set of possible \(c\) is determined. If usequants is TRUE, then the \(c\) are a subset of the values interpolated half-way between the unique, sorted values obtained from the corresponding column of x.train. If usequants is FALSE, the cutoffs are equally spaced across the range of values taken on by the corresponding column of x.train.
The number of possible values of \(c\) is determined by numcut. If usequants is FALSE, numcut equally spaced cutoffs are used covering the range of values in the corresponding column of x.train. If usequants is TRUE, then for a variable the minimum of numcut and one less than the number of unique elements for that variable are used.
kThe amount of shrinkage of the node parameters is controlled by k. k can be given as either a fixed, positive number, or as any value that can be used to build a supported hyperprior. At present, only \(\chi_\nu s\) priors are supported, where \(\nu\) is a degrees of freedom and \(s\) is a scale. Both values must be positive, however the scale can be infinite which yields an improper prior, which is interpretted as just the polynomial part of the distribution. If \(nu\) is 1 and \(s\) is \(\infty\), the prior is “flat”.
For BART on binary outcomes, the degree of overfitting can be highly sensitive to k so it is encouraged to consider a number of values. The default hyperprior for binary BART, chi(1.25, Inf), has been shown to work well in a large number of datasets, however crossvalidation may be helpful. Running for a short time with a flat prior may be helpful to see the range of values of k that are consistent with the data.
bart and rbart_vi support fitted to return the posterior mean of a predicted quantity, as well as predict to return a set of posterior samples for a different sample. In addition, the extract generic can be used to obtain the posterior samples for the training data or test data supplied during the initial fit.
Using predict with a bart object requires that it be fitted with the option keeptrees/keepTrees as TRUE. Keeping the trees for a fit can require a sizeable amount of memory and is off by default.
All generics return values on the scale of expected value of the response by default. This means that predict, extract, and fitted for binary outcomes return probabilities unless specifically the sum-of-trees component is requested (type = "bart"). This is in contrast to yhat.train/yhat.test that are returned with the fitted model.
saveing and loading fitted BART objects for use with predict requires that R's serialization mechanism be able to access the underlying trees, in addition to being fit with keeptrees/keepTrees as TRUE. For memory purposes, the trees are not stored as R objects unless specifically requested. To do this, one must “touch” the sampler's state object before saving, e.g. for a fitted object bartFit, execute invisible(bartFit$fit$state).
Behavior differs when running multi- and single-threaded, as the pseudo random number generators (pRNG) used by R are not thread safe. When single-threaded, R's built-in generator is used; if set at the start, the global .Random.seed will be used and its value updated as samples are drawn. When multi-threaded, the default behavior is to draw new random seeds for each thread using the clock and use thread-specific pRNGs.
This behavior can be modified by setting seed, or by using ... to pass arguments to dbartsControl. For the single-threaded case, a new pRNG is built using that seed that is separate from R's native generator. As such, the global state will not be modified by subsequent calls to the generator. For multi-threaded, the seeds for threads are drawn sequentially using the supplied seed, and will again be separate from R's native generator.
Consequently, the seed argument is not needed when running single-threaded - set.seed will suffice. However, when multi-threaded the seed argument can be used to obtain reproducible results.
When a model is fit with keeptrees (bart) or keepTrees (bart2) equal to TRUE, the generic extract can be used to retrieve a data frame containing the tree fit information. In this case, extract will accept the additional, optional arguments: chainNums, sampleNums, and treeNums. Each should be an integer vector detailing the desired trees to be returned.
The result of extract will be a data frame with columns:
sample, chain, tree - index variables
n - number of observations in node
var - either the index of the variable used for splitting or -1 if the node is a leaf
value - either the value such that observations less than or equal to it are sent down the left path of the tree or the predicted value for a leaf node
The order of nodes in the result corresponds to a depth-first traversal, going down the left-side first. The names of variables used in splitting can be recovered by examining the column names of the fit$data@x element of a fitted bart or bart2 model. See the package vignette “Working with dbarts Saved Trees”.
Chipman, H., George, E., and McCulloch, R. (2009) BART: Bayesian Additive Regression Trees.
Chipman, H., George, E., and McCulloch R. (2006) Bayesian Ensemble Learning. Advances in Neural Information Processing Systems 19, Scholkopf, Platt and Hoffman, Eds., MIT Press, Cambridge, MA, 265-272.
both of the above at: https://www.rob-mcculloch.org
Friedman, J.H. (1991) Multivariate adaptive regression splines. The Annals of Statistics, 19, 1--67.
pdbart
## simulate data (example from Friedman MARS paper)
## y = f(x) + epsilon , epsilon ~ N(0, sigma)
## x consists of 10 variables, only first 5 matter
f <- function(x) {
10 * sin(pi * x[,1] * x[,2]) + 20 * (x[,3] - 0.5)^2 +
10 * x[,4] + 5 * x[,5]
}
set.seed(99)
sigma <- 1.0
n <- 100
x <- matrix(runif(n * 10), n, 10)
Ey <- f(x)
y <- rnorm(n, Ey, sigma)
## run BART
set.seed(99)
bartFit <- bart(x, y)
plot(bartFit)
## compare BART fit to linear matter and truth = Ey
lmFit <- lm(y ~ ., data.frame(x, y))
fitmat <- cbind(y, Ey, lmFit$fitted, bartFit$yhat.train.mean)
colnames(fitmat) <- c('y', 'Ey', 'lm', 'bart')
print(cor(fitmat))
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