I am using quantile regression (for example via gbm
or quantreg
in R) – not focusing on the median but instead an upper quantile (e.g. 75th). Coming from a predictive modeling background, I want to measure how well the model fits on a test set and be able to describe this to a business user. My question is how? In a typical setting with a continuous target I could do the following:
- Calculate the overall RMSE
- Decile the data set by the predicted value and compare the average
actual to the average predicted in each decile. - Etc.
What can be done in this case, where there really is no actual value (i don't think at least) to compare the prediction to?
Here is an example code:
install.packages("quantreg")
library(quantreg)
install.packages("gbm")
library(gbm)
data("barro")
trainIndx<-sample(1:nrow(barro),size=round(nrow(barro)*0.7),replace=FALSE)
train<-barro[trainIndx,]
valid<-barro[-trainIndx,]
modGBM<-gbm(y.net~., # formula
data=train, # dataset
distribution=list(name="quantile",alpha=0.75), # see the help for other choices
n.trees=5000, # number of trees
shrinkage=0.005, # shrinkage or learning rate,
# 0.001 to 0.1 usually work
interaction.depth=5, # 1: additive model, 2: two-way interactions, etc.
bag.fraction = 0.5, # subsampling fraction, 0.5 is probably best
train.fraction = 0.5, # fraction of data for training,
# first train.fraction*N used for training
n.minobsinnode = 10, # minimum total weight needed in each node
cv.folds = 5, # do 3-fold cross-validation
keep.data=TRUE, # keep a copy of the dataset with the object
verbose=TRUE) # don’t print out progress
best.iter<-gbm.perf(modGBM,method="cv")
pred<-predict(modGBM,valid,best.iter)
Now what – since we don't observe the percentile of the conditional distribution?
Add:
I hypothesized several methods and I would like to know if they are correct and if there are better ones – also how to interpret the first:
-
Calculate the average value from the loss functions:
qregLoss<-function(actual, estimate,quantile) { (sum((actual-estimate)*(quantile-((actual-estimate)<0))))/length(actual) }
This is the loss function for quantile regression – but how do we interpret the value?
-
Should we expect that if for example we are calculating the 75th percentile that on a test set, the predicted value should be greater than the actual value around 75% of the time?
Are there other methods formal or heuristic to describe how well the model predicts new cases?
Best Answer
A useful reference may be Haupt, Kagerer, and Schnurbus (2011) discussing the use of quantile-specific measures of predictive accuracy based on cross-validations for various classes of quantile regression models.