BART is a Bayesian “sum-of-trees” 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 tree models. The goal is to have very flexible inference for the uknown function \(f\).
In the spirit of “ensemble models”, each tree is constrained by a prior to be a weak learner so that it contributes a small amount to the overall fit.
bart(
x.train, y.train, x.test=matrix(0.0,0,0),
sigest=NA, sigdf=3, sigquant=.90,
k=2.0,
power=2.0, base=.95,
binaryOffset=0,
ntree=200,
ndpost=1000, nskip=100,
printevery=100, keepevery=1, keeptrainfits=TRUE,
usequants=FALSE, numcut=100, printcutoffs=0,
verbose=TRUE)
# S3 method for bart
plot(
x,
plquants=c(.05,.95), cols =c('blue','black'),
...)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 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.
bart returns a list assigned class ‘bart’.
In the numeric \(y\) case, the list has components:
A matrix with (ndpost/keepevery) rows and nrow(x.train) columns.
Each row corresponds to a draw \(f^*\) from the posterior of \(f\)
and each column corresponds to a row of x.train.
The \((i,j)\) value is \(f^*(x)\) for the \(i^{th}\) kept draw of \(f\)
and the \(j^{th}\) row of x.train.
Burn-in is dropped.
Same as yhat.train but now the x's are the rows of the test data.
train data fits = mean of yhat.train columns.
test data fits = mean of yhat.test columns.
post burn in draws of sigma, length = ndpost/keepevery.
burn-in draws of sigma.
a matrix with (ndpost/keepevery) rows and nrow(x.train) columns. Each row is for a draw. For each variable (corresponding to the columns), the total count of the number of times that variable is used in a tree decision rule (over all trees) is given.
The rough error standard deviation (\(\sigma\)) used in the prior.
The input dependent vector of values for the dependent variable.
This is used in plot.bart.
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
\(f(x)\) + binaryOffset. If you want draws of the probability
\(P(Y=1 | x)\) you need to apply the normal cdf (pnorm)
to these values.
Explanatory variables for training (in sample) data.
May be a matrix or a data frame,
with (as usual) 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.
makeind is used to generate the dummies.
bart will generate draws of \(f(x)\) for each \(x\)
which is a row of x.train.
Dependent variable for training (in sample) data.
If y is numeric a continous response model is fit (normal errors).
If y is a factor (or just has 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 structure as x.train.
bart will generate draws of \(f(x)\) for each \(x\) which is a row of x.test.
The prior for the error variance (\(\sigma^2\)) is inverted chi-squared (the standard conditionally conjugate prior). The prior is specified by choosing the degrees of freedom, a rough estimate of the corresponding standard deviation and a quantile to put this rough estimate at. If sigest=NA then the rough estimate will be the usual least squares estimator. Otherwise the supplied value will be used. Not used if y is binary.
Degrees of freedom for error variance prior. Not used if y is binary.
The quantile of the prior that the rough estimate (see 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 used if y is binary.
For numeric y, k is the number of prior standard deviations \(E(Y|x) = f(x)\) is away from +/-.5. The response (y.train) is internally scaled to range from -.5 to .5. For binary y, k is the number of prior standard deviations \(f(x)\) is away from +/-3. In both cases, the bigger k is, the more conservative the fitting will be.
Power parameter for tree prior.
Base parameter for tree prior.
Used for binary \(y\).
The model is \(P(Y=1 | x) = F(f(x) + binaryOffset)\).
The idea is that \(f\) is shrunk towards 0, so the offset allows you to shrink towards
a probability other than .5.
The number of trees in the sum.
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.
A “draw” will consist of values of the error standard deviation (\(\sigma\))
and \(f^*(x)\)
at \(x\) = rows from the train(optionally) and test data, where \(f^*\) denotes
the current draw of \(f\).
If true the draws of \(f(x)\) for \(x\) = rows of x.train are returned.
Decision rules in the tree are of the form \(x \le c\) vs. \(x > c\) for each variable corresponding to a column of x.train. usequants determines how the set of possible c is determined. If usequants is true, then the c are a subset of the values (xs[i]+xs[i+1])/2 where xs is 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 (see usequants). If a single number if given, this is used for all variables. Otherwise a vector with length equal to ncol(x.train) is required, where the \(i^{th}\) element gives the number of c used for the \(i^{th}\) variable in x.train. 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 min(numcut, the number of unique values in the corresponding columns of x.train - 1) c values are used.
The number of cutoff rules c to printed to screen before the MCMC is run. Give a single integer, the same value will be used for all variables. If 0, nothing is printed.
Logical, if FALSE supress printing.
Value returned by 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.
Additional arguments passed on to plot.
Hugh Chipman: hugh.chipman@gmail.com
Robert McCulloch: robert.e.mcculloch@gmail.com.
BART is an Bayesian MCMC method. At each MCMC interation, we produce a draw from the joint posterior \((f,\sigma) | (x,y)\) in the numeric \(y\) case and just \(f\) in the binary \(y\) case.
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) or the test data (x.test).
Chipman, H., George, E., and McCulloch R. (2010) Bayesian Additive Regression Trees. The Annals of Applied Statistics, 4,1, 266-298.
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.
Friedman, J.H. (1991) Multivariate adaptive regression splines. The Annals of Statistics, 19, 1--67.
pdbart
##simulate data (example from Friedman MARS paper)
f = function(x){
10*sin(pi*x[,1]*x[,2]) + 20*(x[,3]-.5)^2+10*x[,4]+5*x[,5]
}
sigma = 1.0 #y = f(x) + sigma*z , z~N(0,1)
n = 100 #number of observations
set.seed(99)
x=matrix(runif(n*10),n,10) #10 variables, only first 5 matter
Ey = f(x)
y=Ey+sigma*rnorm(n)
lmFit = lm(y~.,data.frame(x,y)) #compare lm fit to BART later
##run BART
set.seed(99)
bartFit = bart(x,y,ndpost=200) #default is ndpost=1000, this is to run example fast.
plot(bartFit) # plot bart fit
##compare BART fit to linear matter and truth = Ey
fitmat = cbind(y,Ey,lmFit$fitted,bartFit$yhat.train.mean)
colnames(fitmat) = c('y','Ey','lm','bart')
print(cor(fitmat))
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