pdbart {dbarts}

Partial Dependence Plots for BART

Description

Run bart at test observations constructed so that a plot can be created displaying the effect of a single variable (pdbart) or pair of variables (pd2bart). Note that if 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

pdbart(
    x.train, y.train,
    xind = NULL,
    levs = NULL, levquants = c(0.05, seq(0.1, 0.9, 0.1), 0.95),
    pl = TRUE, plquants = c(0.05, 0.95),
    ...)

## S3 method for class 'pdbart'
plot(
    x,
    xind = seq_len(length(x$fd)),
    plquants = c(0.05, 0.95), cols = c('black', 'blue'),
    ...)

pd2bart(
    x.train, y.train,
    xind = NULL,
    levs = NULL, levquants = c(0.05, seq(0.1, 0.9, 0.1), 0.95),
    pl = TRUE, plquants = c(0.05, 0.95),
    ...)

## S3 method for class 'pd2bart'
plot(
    x,
    plquants = c(0.05, 0.95), contour.color = 'white',
    justmedian = TRUE,
    ...)

Arguments

x.train

Explanatory variables for training (in sample) data. Can be any valid input to bart, such as a matrix or a formula. Also accepted are fitted bart models or dbartsSampler with keepTrees equal to TRUE.

y.train

Dependent variable for training (in sample) data. Can be a numeric vector or, when passing x.train as a formula, a data.frame or other object used to find variables. Not required if x.train is a fitted model or sampler.

xind

Integer, character vector, or the right-hand side of a formula indicating which variables are to be plotted. In pdbart, corresponds to the variables (columns of x.train) for which a plot is to be constructed. In plot.pdbart, corresponds to the indices in list returned by pdbart for which plot is to be constructed. In pd2bart, the indicies of a pair of variables (columns of x.train) to plot. If NULL a default of all columns is used for pdbart and the first two columns is used for pd2bart.

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 pdbart, it should have same length as xind. In pd2bart, it should have length 2. See also argument levquants.

levquants

If levs in NULL, the values of each variable used in the plot is set to the quantiles (in x.train) indicated by levquants. Must be a vector of numeric type.

pl

For pdbart and pd2bart, 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 bart. In plot.pdbart, they are passed on to plot. In plot.pd2bart, they are passed on to image.

x

For plot.*, object returned from pdbart or pd2bart.

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 \setminus 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) = \frac{1}{n}\sum_{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 BART 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 BART 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 pdbart x_s consists of a single variable in x and in pd2bart 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 BART draw. For this reason running BART with keepevery larger than 1 (eg. 10) makes the procedure much faster.

Value

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

pdbart and pd2bart return lists with components given below. The list returned by pdbart is assigned class pdbart and the list returned by pd2bart is assigned class pd2bart.

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 pdbart 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 pd2bart, 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 bart. They are simply passed on from the BART run used to create the partial dependence plot. The function plot.bart can be applied to the object returned by pdbart or pd2bart to examine the BART run.

Author(s)

Hugh Chipman: [e-mail].
Robert McCulloch: [e-mail].

References

Chipman, H., George, E., and McCulloch, R. (2006) BART: Bayesian Additive Regression Trees.

Chipman, H., George, E., and McCulloch R. (2006) Bayesian Ensemble Learning.

both of the above at: https://www.rob-mcculloch.org/

Friedman, J.H. (2001) Greedy function approximation: A gradient boosting machine. The Annals of Statistics, 29, 1189–1232.

Examples

## Not run: 
## 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 <- pdbart(
    x, y, xind = c(1, 2),
    levs = list(seq(-1, 1, 0.2), seq(-1, 1, 0.2)),
    pl = FALSE, keepevery = 10, ntree = 100
)
plot(pdb1, ylim = c(-0.6, 0.6))

## pd2bart: two dimensional partial dependence plot
set.seed(99)
pdb2 <- pd2bart(
    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 = 100, keepevery = 10, verbose = FALSE)
plot(pdb2)

## compare BART fit to linear model and truth = Ey
lmFit  <- lm(y ~ ., data.frame(x, y))
fitmat <- cbind(y, Ey, lmFit$fitted, pdb1$yhat.train.mean)
colnames(fitmat) <- c('y', 'Ey', 'lm', 'bart')
print(cor(fitmat))

## example showing the use of a pre-fitted model
df <- data.frame(y, x)
set.seed(99)
bartFit <- bart(
    y ~ rob + hugh + ed, df,
    keepevery = 10, ntree = 100, keeptrees = TRUE)
pdb1 <- pdbart(bartFit, xind = rob + ed, pl = FALSE)

## End(Not run)