I will take an attempt to explain:
Suppose our training data set is represented by T and suppose data set has M features (or attributes or variables).
T = {(X1,y1), (X2,y2), ... (Xn, yn)}
and
Xi is input vector {xi1, xi2, ... xiM}
yi is the label (or output or class).
summary of RF:
Random Forests algorithm is a classifier based on primarily two methods -
- Bagging
- Random subspace method.
Suppose we decide to have S number of trees in our forest then we first create S datasets of "same size as original" created from random resampling of data in T with-replacement (n times for each dataset). This will result in {T1, T2, ... TS} datasets. Each of these is called a bootstrap dataset. Due to "with-replacement" every dataset Ti can have duplicate data records and Ti can be missing several data records from original datasets. This is called Bootstrapping. (en.wikipedia.org/wiki/Bootstrapping_(statistics))
Bagging is the process of taking bootstraps & then aggregating the models learned on each bootstrap.
Now, RF creates S trees and uses m (=sqrt(M) or =floor(lnM+1)) random subfeatures out of M possible features to create any tree. This is called random subspace method.
So for each Ti bootstrap dataset you create a tree Ki. If you want to classify some input data D = {x1, x2, ..., xM} you let it pass through each tree and produce S outputs (one for each tree) which can be denoted by Y = {y1, y2, ..., ys}. Final prediction is a majority vote on this set.
Out-of-bag error:
After creating the classifiers (S trees), for each (Xi,yi) in the original training set i.e. T, select all Tk which does not include (Xi,yi). This subset, pay attention, is a set of boostrap datasets which does not contain a particular record from the original dataset. This set is called out-of-bag examples. There are n such subsets (one for each data record in original dataset T). OOB classifier is the aggregation of votes ONLY over Tk such that it does not contain (xi,yi).
Out-of-bag estimate for the generalization error is the error rate of the out-of-bag classifier on the training set (compare it with known yi's).
Why is it important?
The study of error estimates for bagged classifiers in Breiman
[1996b], gives empirical evidence to show that the out-of-bag estimate
is as accurate as using a test set of the same size as the training
set. Therefore, using the out-of-bag error estimate removes the need
for a set aside test set.1
(Thanks @Rudolf for corrections. His comments below.)
