gbmm: Generalized BART Mixed Model for continuous and binary outcomes

Description

BART is a Bayesian “sum-of-trees” model. For a numeric response $y$ , we have $y = f (x) + ϵ$ , where $ϵ \sim N (0, σ^{2})$ .

$f$ 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.

Usage

gbmm(
      x.train, y.train,
      x.test=matrix(0,0,0), type='wbart',
      u.train=NULL, B=NULL,
      ntype=as.integer(
          factor(type, levels=c('wbart', 'pbart'))),
      sparse=FALSE, theta=0, omega=1,
      a=0.5, b=1, augment=FALSE, rho=NULL,
      xinfo=matrix(0,0,0), usequants=FALSE,
      rm.const=TRUE,
      sigest=NA, sigdf=3, sigquant=0.90,
      k=2, power=2, base=0.95,
      lambda=NA, tau.num=c(NA, 3, 6)[ntype], 
      offset=NULL, 
      ntree=c(200L, 50L, 50L)[ntype], numcut=100L,
      ndpost=1000L, nskip=100L, 
      keepevery=c(1L, 10L, 10L)[ntype],
      printevery=100L, transposed=FALSE,
      hostname=FALSE,
      mc.cores = 1L, ## mc.gbmm only
      nice = 19L,    ## mc.gbmm only
      seed = 99L     ## mc.gbmm only
)
mc.gbmm(
         x.train, y.train,
         x.test=matrix(0,0,0), type='wbart',
         u.train=NULL, B=NULL,
         ntype=as.integer(
             factor(type, levels=c('wbart', 'pbart'))),
         sparse=FALSE, theta=0, omega=1,
         a=0.5, b=1, augment=FALSE, rho=NULL,
         xinfo=matrix(0,0,0), usequants=FALSE,
         rm.const=TRUE,
         sigest=NA, sigdf=3, sigquant=0.90,
         k=2, power=2, base=0.95,
         lambda=NA, tau.num=c(NA, 3, 6)[ntype], 
         offset=NULL, 
         ntree=c(200L, 50L, 50L)[ntype], numcut=100L,
         ndpost=1000L, nskip=100L, 
         keepevery=c(1L, 10L, 10L)[ntype],
         printevery=100L, transposed=FALSE,
         hostname=FALSE,
         mc.cores = 2L, nice = 19L, seed = 99L
)

Arguments

x.train

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 created if $q = 2$ where $q$ is the number of levels of the factor. gbmm will generate draws of $f (x)$ for each $x$ which is a row of x.train.

y.train

Continuous or binary dependent variable for training (in sample) data. If $y$ is numeric, then a continuous BART model is fit (Normal errors). If $y$ is binary (has only 0's and 1's), then a binary BART model with a probit link is fit by default: you can over-ride the default via the argument type to specify a logit BART model.

x.test

Explanatory variables for test (out of sample) data. Should have same structure as x.train. gbmm will generate draws of $f (x)$ for each $x$ which is a row of x.test.

u.train

Integer indices specifying the random effects.

The prior for the standard deviation of the random effects is $U (0, B)$ .

type

You can use this argument to specify the type of fit. 'wbart' for continuous BART or 'pbart' for probit BART.

ntype

The integer equivalent of type where 'wbart' is 1 and 'pbart' is 2.

sparse

Whether to perform variable selection based on a sparse Dirichlet prior rather than simply uniform; see Linero 2016.

theta

Set $t h e t a$ parameter; zero means random.

omega

Set $o m e g a$ parameter; zero means random.

Sparse parameter for $B e t a (a, b)$ prior: $0.5 <= a <= 1$ where lower values inducing more sparsity.

Sparse parameter for $B e t a (a, b)$ prior; typically, $b = 1$ .

rho

Sparse parameter: typically $r h o = p$ where $p$ is the number of covariates under consideration.

augment

Whether data augmentation is to be performed in sparse variable selection.

xinfo

You can provide the cutpoints to BART or let BART choose them for you. To provide them, use the xinfo argument to specify a list (matrix) where the items (rows) are the covariates and the contents of the items (columns) are the cutpoints.

usequants

If usequants=FALSE, then the cutpoints in xinfo are generated uniformly; otherwise, if TRUE, uniform quantiles are used for the cutpoints.

rm.const

Whether or not to remove constant variables.

sigest

The prior for the error variance ( $s i g m a^{2}$ ) 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.

sigdf

Degrees of freedom for error variance prior. Not used if $y$ is binary.

sigquant

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 ( $s i g m a$ ) 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 +/-0.5. The response, codey.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 +/-3. The bigger k is, the more conservative the fitting will be.

power

Power parameter for tree prior.

base

Base parameter for tree prior.

lambda

The scale of the prior for the variance. If lambda is zero, then the variance is to be considered fixed and known at the given value of sigest. Not used if $y$ is binary.

tau.num

The numerator in the tau definition, i.e., tau=tau.num/(k*sqrt(ntree)).

offset

Continous BART operates on y.train centered by offset which defaults to mean(y.train). With binary BART, the centering is $P (Y = 1 | x) = F (f (x) + o f f s e t)$ where offset defaults to F^{-1}(mean(y.train)). You can use the offset parameter to over-ride these defaults.

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 $i^{t h}$ element gives the number of $c$ used for the $i^{t h}$ 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 $m i n (n u m c u t, t h e n u m b e r o f u n i q u e v a l u e s i n t h e c o r r e s p o n d i n g c o l u m n s o f x . t r a i n - 1)$ values are used.

ndpost

The number of posterior draws returned.

nskip

Number of MCMC iterations to be treated as burn in.

printevery

As the MCMC runs, a message is printed every printevery draws.

keepevery

Every keepevery draw is kept to be returned to the user.

transposed

When running gbmm in parallel, it is more memory-efficient to transpose x.train and x.test, if any, prior to calling mc.gbmm.

hostname

When running on a cluster occasionally it is useful to track on which node each chain is running; to do so set this argument to TRUE.

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).

Value

gbmm returns an object of type gbmm which is essentially a list. In the numeric $y$ case, the list has components:

yhat.train

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 $i^{t h}$ kept draw of $f$ and the $j^{t h}$ row of x.train. 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 = mean of yhat.train columns.

yhat.test.mean

test data fits = mean of yhat.test columns.

sigma

post burn in draws of sigma, length = ndpost.

first.sigma

burn-in draws of sigma.

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.

sigest

The rough error standard deviation ( $σ$ ) used in the prior.

Details

BART is a Bayesian MCMC method. At each MCMC interation, we produce a draw from the joint posterior $(f, σ) | (x, y)$ 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 $σ^{*}$ 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.

For x.train/x.test with missing data elements, gbmm will singly impute them with hot decking. For one or more missing covariates, record-level hot-decking imputation deWaPann11 is employed that is biased towards the null, i.e., nonmissing values from another record are randomly selected regardless of the outcome. Since mc.gbmm runs multiple gbmm threads in parallel, mc.gbmm performs multiple imputation with hot decking, i.e., a separate imputation for each thread. This record-level hot-decking imputation is biased towards the null, i.e., nonmissing values from another record are randomly selected regardless of y.train.

References

Chipman, H., George, E., and McCulloch R. (2010) Bayesian Additive Regression Trees. The Annals of Applied Statistics, 4,1, 266-298 <doi:10.1214/09-AOAS285>.

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.

De Waal, T., Pannekoek, J. and Scholtus, S. (2011) Handbook of statistical data editing and imputation. John Wiley & Sons, Hoboken, NJ.

Friedman, J.H. (1991) Multivariate adaptive regression splines. The Annals of Statistics, 19, 1--67.

Linero, A.R. (2018) Bayesian regression trees for high dimensional prediction and variable selection. JASA, 113(522), 626--636.

Examples

Run this code

# NOT RUN {
##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

##test BART with token run to ensure installation works
set.seed(99)
bartFit = wbart(x,y,nskip=5,ndpost=5)

# }
# NOT RUN {
##run BART
set.seed(99)
bartFit = wbart(x,y)

##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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