BART for dichotomous outcomes with parallel computation and stratified random sampling
BART is a Bayesian “sum-of-trees” model.
For numeric response y, we have
y = f(x) + e,
where e ~ 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.
rs.pbart( x.train, y.train, x.test=matrix(0.0,0,0), C=floor(length(y.train)/2000), k=2.0, power=2.0, base=.95, binaryOffset=0, ntree=50L, numcut=100L, ndpost=1000L, nskip=100L, keepevery=1L, printevery=100, keeptrainfits=FALSE, transposed=FALSE, mc.cores = 2L, nice = 19L, seed = 99L )
x.train |
Explanatory variables for training (in sample) data. |
y.train |
Dependent variable for training (in sample) data. |
x.test |
Explanatory variables for test (out of sample) data. |
C |
The number of shards to break the data into and analyze separately. |
k |
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 |
Power parameter for tree prior. |
base |
Base parameter for tree prior. |
binaryOffset |
Used for binary y. |
ntree |
The number of trees in the sum. |
numcut |
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. |
ndpost |
The number of posterior draws returned. |
nskip |
Number of MCMC iterations to be treated as burn in. |
keepevery |
Every keepevery draw is kept to be returned to the user. |
printevery |
As the MCMC runs, a message is printed every printevery draws. |
keeptrainfits |
Whether to keep |
transposed |
When running |
seed |
Setting the seed required for reproducible MCMC. |
mc.cores |
Number of cores to employ in parallel. |
nice |
Set the job niceness. The default niceness is 19: niceness goes from 0 (highest) to 19 (lowest). |
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).
rs.pbart returns an object of type pbart which is
essentially a list.
yhat.shard |
Estimates generated from the individual shards rather than from the whole. This object is only useful for assessing convergence. A matrix with ndpost 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. |
yhat.train |
Estimates generated from the whole if A matrix with ndpost 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. |
yhat.test |
Estimates generated from the whole if Same as yhat.train but now the x's are the rows of the test data. |
varcount |
a matrix with ndpost 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. |
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.
##simulate from Friedman's five-dimensional test function
##Friedman JH. Multivariate adaptive regression splines
##(with discussion and a rejoinder by the author).
##Annals of Statistics 1991; 19:1-67.
f = function(x) #only the first 5 matter
sin(pi*x[ , 1]*x[ , 2]) + 2*(x[ , 3]-.5)^2+x[ , 4]+0.5*x[ , 5]-1.5
sigma = 1.0 #y = f(x) + sigma*z where z~N(0, 1)
k = 50 #number of covariates
thin = 25
ndpost = 2500
nskip = 100
C = 10
m = 10
n = 10000
set.seed(12)
x.train=matrix(runif(n*k), n, k)
Ey.train = f(x.train)
y.train=(Ey.train+sigma*rnorm(n)>0)*1
table(y.train)/n
x <- x.train
x4 <- seq(0, 1, length.out=m)
for(i in 1:m) {
x[ , 4] <- x4[i]
if(i==1) x.test <- x
else x.test <- rbind(x.test, x)
}
## parallel::mcparallel/mccollect do not exist on windows
if(.Platform$OS.type=='unix') {
##test BART with token run to ensure installation works
post = rs.pbart(x.train, y.train,
C=C, mc.cores=4, keepevery=1,
seed=99, ndpost=1, nskip=1)
}
## Not run:
post = rs.pbart(x.train, y.train, x.test=x.test,
C=C, mc.cores=8, keepevery=thin,
seed=99, ndpost=ndpost, nskip=nskip)
str(post)
par(mfrow=c(2, 2))
M <- nrow(post$yhat.test)
pred <- matrix(nrow=M, ncol=10)
for(i in 1:m) {
h <- (i-1)*n+1:n
pred[ , i] <- apply(pnorm(post$yhat.test[ , h]), 1, mean)
}
pred <- apply(pred, 2, mean)
plot(x4, qnorm(pred), xlab=expression(x[4]),
ylab='partial dependence function', type='l')
i <- floor(seq(1, n, length.out=10))
j <- seq(-0.5, 0.4, length.out=10)
for(h in 1:10) {
auto.corr <- acf(post$yhat.shard[ , i[h]], plot=FALSE)
if(h==1) {
max.lag <- max(auto.corr$lag[ , 1, 1])
plot(1:max.lag+j[h], auto.corr$acf[1+(1:max.lag), 1, 1],
type='h', xlim=c(0, max.lag+1), ylim=c(-1, 1),
ylab='auto-correlation', xlab='lag')
}
else
lines(1:max.lag+j[h], auto.corr$acf[1+(1:max.lag), 1, 1],
type='h', col=h)
}
for(j in 1:10) {
if(j==1)
plot(pnorm(post$yhat.shard[ , i[j]]),
type='l', ylim=c(0, 1),
sub=paste0('N:', n, ', k:', k),
ylab=expression(Phi(f(x))), xlab='m')
else
lines(pnorm(post$yhat.shard[ , i[j]]),
type='l', col=j)
}
geweke <- gewekediag(post$yhat.shard)
j <- -10^(log10(n)-1)
plot(geweke$z, pch='.', cex=2, ylab='z', xlab='i',
sub=paste0('N:', n, ', k:', k),
xlim=c(j, n), ylim=c(-5, 5))
lines(1:n, rep(-1.96, n), type='l', col=6)
lines(1:n, rep(+1.96, n), type='l', col=6)
lines(1:n, rep(-2.576, n), type='l', col=5)
lines(1:n, rep(+2.576, n), type='l', col=5)
lines(1:n, rep(-3.291, n), type='l', col=4)
lines(1:n, rep(+3.291, n), type='l', col=4)
lines(1:n, rep(-3.891, n), type='l', col=3)
lines(1:n, rep(+3.891, n), type='l', col=3)
lines(1:n, rep(-4.417, n), type='l', col=2)
lines(1:n, rep(+4.417, n), type='l', col=2)
text(c(1, 1), c(-1.96, 1.96), pos=2, cex=0.6, labels='0.95')
text(c(1, 1), c(-2.576, 2.576), pos=2, cex=0.6, labels='0.99')
text(c(1, 1), c(-3.291, 3.291), pos=2, cex=0.6, labels='0.999')
text(c(1, 1), c(-3.891, 3.891), pos=2, cex=0.6, labels='0.9999')
text(c(1, 1), c(-4.417, 4.417), pos=2, cex=0.6, labels='0.99999')
par(mfrow=c(1, 1))
##dev.copy2pdf(file='geweke.rs.pbart.pdf')
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.