pdpgbart: Partial Dependence Plots for PGBART

In pgbart: Bayesian Additive Regression Trees Using Particle Gibbs Sampler and Gibbs/Metropolis-Hastings Sampler

Description

Display the effect of a single variable (pdpgbart) or pair of variables (pd2pgbart). Note that if response y is a binary with P(Y=1 | x) = F(f(x)), F the standard normal cdf, then the plots are all on the f scale.

Usage

pdpgbart(
   x.train, y.train,
   xind=1:ncol(x.train), levs=NULL, levquants=c(.05,(1:9)/10,0.95),
   pl=TRUE,  plquants=c(.05,.95),
    ...)

## S3 method for class 'pdpgbart'
plot(x,
     xind = seq_len(length(x$fd)),
     plquants = c(0.05, 0.95), cols = c('black', 'blue'),
     ...)
     
pd2pgbart(
   x.train, y.train,
   xind=1:2, levs=NULL, levquants=c(.05,(1:9)/10,.95),
   pl=TRUE, plquants=c(.05,.95), 
   ...)

## S3 method for class 'pd2pgbart'
plot(x,
     plquants = c(0.05, 0.95), contour.color = 'white',
     justmedian = TRUE,
     ...)

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 is created if q=2, where q is the number of levels of the factor.
y.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.
xind	Integer vector indicating which variables are to be plotted. In pdpgbart, corresponds to the variables (columns of x.train) for which a plot is to be constructed. In plotpdpgbart, corresponds to the indices in list returned by pdpgbart for which plot is to be constructed. In pd2pgbart, the indicies of a pair of variables (columns of x.train) to plot.
levs	Gives the values of a variable at which the plot is to be constructed. Must be a list, where the ith component gives the values for the ith variable. In pdpgbart, it should have same length as xind. In pd2pgbart, it should have length 2. See also argument levquants.
levquants	If levs in NULL, the values of each variable used in the plot are set to the quantiles (in x.train) indicated by levquants. Must be a vector of numeric type.
pl	For pdpgbart and pd2pgbart, if TRUE, plot is subsequently made (by calling plot.*).
plquants	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.
...	Additional arguments. In pdbart and pd2bart, arguments are passed on to pgbart_train. In plot.pdbart, they are passed on to plot. In plot.pd2bart, they are passed on to image.
x	For plot.*, object is returned from pdpgbart or pd2pgbart.
cols	Vector of two colors. The first color is for the median of f, while the second color is for the upper and lower quantiles.
contour.color	Color for contours plotted on top of the image.
justmedian	A logical where if TRUE just one plot is created for the median of f(x) draws. If FALSE, three plots are created one for the median and two additional ones for the lower and upper quantiles. In this case, mfrow is set to c(1,3).

Details

We divide the predictor vector x into a subgroup of interest, x_s and the complement x_c = x - x_s. A prediction f(x) can then be written as f(x_s, x_c). To estimate the effect of x_s on the prediction, Friedman suggests the partial dependence function

f_s(x_s) = (1/n) ∑_{i=1}\^n f(x_s,x_{ic})

where x_{ic} is the ith observation of x_c in the data. Note that (x_s, x_{ic}) will generally not be one of the observed data points. Using pgbart it is straightforward to then estimate and even obtain uncertainty bounds for f_s(x_s). A draw of f*_s(x_s) from the induced pgbart posterior on f_s(x_s) is obtained by simply computing f*_s(x_s) as a byproduct of each MCMC draw f*. The median (or average) of these MCMC draws f*_s(x_s) then yields an estimate of f_s(x_s), and lower and upper quantiles can be used to obtain intervals for f_s(x_s).

In pdpgbart x_s consists of a single variable in x and in pd2pgbart it is a pair of variables.

This is a computationally intensive procedure. For example, in pdbart, to compute the partial dependence plot for 5 x_s values, we need to compute f(x_s, x_c) for all possible (x_s, x_{ic}) and there would be 5n of these where n is the sample size. All of that computation would be done for each kept pgbart draw. For this reason running pgbart with keepevery larger than 1 (eg. 10) makes the procedure much faster.

Value

The plot methods produce the plots and don't return anything.

pdpgbart and pd2pgbart return lists with components given below. The list returned by pdpgbart is assigned class pdpgbart and the list returned by pd2pgbart is assigned class pd2pgbart.

fd	A matrix whose (i, j) value is the ith draw of f_s(x_s) for the jth value of x_s. “fd” is for “function draws”. For pdpgbart, fd is actually a list whose kth component is the matrix described above corresponding to the kth variable chosen by argument xind. The number of columns in each matrix will equal the number of values given in the corresponding component of argument levs (or number of values in levquants). For pd2pgbart, fd is a single matrix. The columns correspond to all possible pairs of values for the pair of variables indicated by xind. That is, all possible (x_i, x_j) where x_i is a value in the levs component corresponding to the first x and x_j is a value in the levs components corresponding to the second one. The first x changes first.
levs	The list of levels used, each component corresponding to a variable. If argument levs was supplied it is unchanged. Otherwise, the levels in levs are as constructed using argument levquants.
xlbs	A vector of character strings which are the plotting labels used for the variables.

The remaining components returned in the list are the same as in the value of pgbart_train. They are simply passed on from the pgbart run used to create the partial dependence plot.

References

Lakshminarayanan B, Roy D, Teh Y W. (2015) Particle Gibbs for Bayesian Additive Regression Trees Artificial Intelligence and Statistics, 553-561.

Chipman, H., George, E., and McCulloch R. (2010) Bayesian Additive Regression Trees. The Annals of Applied Statistics, 4,1, 266-298.

Friedman, J. H. (2001) Greedy Function Approximation: A Gradient Boosting Machine. The Annals of Statistics, 29, 1189–1232.

Examples

## simulate data
f <- function(x) { return(0.5 * x[,1] + 2 * x[,2] * x[,3]) }
sigma <- 0.2
n <- 100
set.seed(27)
x <- matrix(2 * runif(n * 3) -1, ncol = 3)
colnames(x) <- c('rob', 'hugh', 'ed')
Ey <- f(x)
y <- rnorm(n, Ey, sigma)
## first two plot regions are for pdbart, third for pd2bart
par(mfrow = c(1, 3))
## pdbart: one dimensional partial dependence plot
set.seed(99)
pdb1 <-
  pdpgbart(
    x, y, xind=c(1,2),
    levs=list(seq(-1,1,.2), seq(-1,1,.2)), pl=FALSE,
    keepevery=10, ntree=5, nskip=100, ndpost=200
  )
plot(pdb1,ylim=c(-.6,.6))
## pd2bart: two dimensional partial dependence plot
set.seed(99)
pdb2 <-
  pd2pgbart(x, y, xind = c(2, 3),
          levquants = c(0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95),
          pl = FALSE, ntree = 5, keepevery = 10, verbose = FALSE
  )
plot(pdb2)

pgbart documentation built on May 2, 2019, 8:42 a.m.