## ----message=FALSE, echo=FALSE------------------------------------------------
library("gbm3")
set.seed(557165559)

## ----shrinkViters, fig.cap="**Figure 1**. Out-of-sample predictive performance by number of iterations and shrinkage. Smaller values of the shrinkage parameter offer improved predictive performance, but with decreasing marginal improvement", echo=FALSE, cache=TRUE, out.width="100%", dpi=300----
# N <- 10000
# X1 <- runif(N)
# X2 <- 2*runif(N)
# X3 <- ordered(sample(letters[1:4],N,replace=TRUE),levels=letters[4:1])
# X4 <- factor(sample(letters[1:6],N,replace=TRUE))
# X5 <- factor(sample(letters[1:3],N,replace=TRUE))
# X6 <- 3*runif(N)
# mu <- c(-1,0,1,2)[as.numeric(X3)]
# 
# SNR <- 10 # signal-to-noise ratio
# Y <- X1**1.5 + 2 * (X2**.5) + mu
# sigma <- sqrt(var(Y)/SNR)
# Y <- Y + rnorm(N,0,sigma)
# 
# ## introduce some missing values
# X1[sample(1:N,size=500)] <- NA
# X4[sample(1:N,size=300)] <- NA
# 
# data <- data.frame(Y=Y,X1=X1,X2=X2,X3=X3,X4=X4,X5=X5,X6=X6)
# 
# ## fit initial model
# shrink <- c(0.1,0.05,0.01,0.005,0.001)
# err <- vector("list",length(shrink))
# 
# for(i in 1:length(shrink))
# {
#     gbm1 <- gbm(Y~X1+X2+X3+X4+X5+X6,
#                 data=data,
#                 distribution="gaussian",
#                 n.trees=10000,
#                 shrinkage=shrink[i],
#                 interaction.depth=3,
#                 bag.fraction = 0.5,
#                 train.fraction = 0.2,
#                 n.minobsinnode = 10,
#                 keep.data=FALSE,
#                 verbose=FALSE)
#     err[[i]] <- gbm1$valid.error
# }
# 
# ylim <- range(unlist(lapply(err,range)))
# mins <- min(sapply(err,min))
# ylim <- c(0.185,0.21)
# 
# plot(0,0,ylim=ylim,xlim=c(0,10000),type="n",xlab="Iterations",ylab="Squared error")
# for(i in 1:length(shrink))
#     {
#         x <- which.min(err[[i]])
#         y <- err[[i]][x]
#         j <- round(seq(1,10000,length=500))
#         j <- sort(c(j,x))
# 
#         k <- 1:length(j)
#         lines(j[k],err[[i]][j][k],col=i)
#         rug(x, col=i)
#         text(x,y-0.0005,as.character(shrink[i]),adj=1)
#     }
# 
# abline(h=min(mins))

knitr::include_graphics("shrinkViters.png")

## ----oobperf2, echo = FALSE, message=FALSE, fig.align='center', fig.cap='**Figure 2**. Out-of-sample predictive performance of four methods of selecting the optimal number of iterations. The vertical axis plots performance relative the best. The boxplots indicate relative performance across thirteen real datasets from the UCI repository. See `demo(OOB-reps)`.', out.width="80%"----

knitr::include_graphics("oobperf2.png")