[R] randomForest() for regression produces offset predictions

[Joshua Knowles]

Hi all,
 
I have observed that when using the randomForest package to do regression, the 
predicted values of the dependent variable given by a trained forest are not 
centred and have the wrong slope when plotted against the true values.
 
This means that the R^2 value obtained by squaring the Pearson correlation are 
better than those obtained by computing the coefficient of determination 
directly. The R^2 value obtained by squaring the Pearson can, however, be 
exactly reproduced by the coeff. of det. if the predicted values are first 
linearly transformed (using lm() to find the required intercept and slope).
 
Does anyone know why the randomForest behaves in this way - producing offset 
predictions? Does anyone know a fix for the problem?
 
(By the way, the feature is there even if the original dependent variable 
values are initially transformed to have zero mean and unit variance.)
 
As an example, here is some simple R code that uses the available swiss 
dataset to show the effect I am observing.

Thanks for any help.
 
--
#### EXAMPLE OF RANDOM FOREST REGRESSION
 
library(randomForest)
data(swiss)
swiss
 
#Build the random forest to predict Infant Mortality
rf.rf<-randomForest(Infant.Mortality ~ ., data=swiss)
 
#And predict the training set again
pred<-c(predict(rf.rf,swiss))
actual<-swiss$Infant.Mortality
 
#Plotting predicted against actual values shows the effect (uncomment to see
this)
#plot(pred,actual)
#abline(0,1)
 
# calculate R^2 as pearson coefficient squared
R2one<-cor(pred,actual)^2
 
# calculate R^2 value as fraction of variance explained
residOpt<-(actual-pred)
residnone<-(actual-mean(actual))
R2two<-1-var(residOpt,na.rm=TRUE)/var(residnone, na.rm=TRUE)
 
# now fit a line through the predicted and true values and
# use this to normalize the data before calculating R^2
 
fit<-lm(actual ~ pred)
coef(fit)
pred2<-pred*coef(fit)[2]+coef(fit)[1]
residOpt<-(actual-pred2)
R2three<-1-var(residOpt,na.rm=TRUE)/var(residnone, na.rm=TRUE)
 
cat("Pearson squared = ",R2one,"\n")
cat("Coeff of determination = ", R2two, "\n")
cat("Coeff of determination after linear fitting = ", R2three, "\n")
 
## END
 

-- 
Joshua Knowles .. j.knowles at manchester.ac.uk
BBSRC David Phillips Fellow
School of Computer Science
The University of Manchester
http://dbkgroup.org/knowles/