  The effect of training data size on machine learning methods has been well
investigated over the past two decades. The predictive performance of tree
based machine learning methods, in general, improves with a decreasing rate as
the size of training data increases. We investigate this in optimal trees
ensemble (OTE) where the method fails to learn from some of the training
observations due to internal validation. Modified tree selection methods are
thus proposed for OTE to cater for the loss of training observations in
internal validation. In the first method, corresponding out-of-bag (OOB)
observations are used in both individual and collective performance assessment
for each tree. Trees are ranked based on their individual performance on the
OOB observations. A certain number of top ranked trees is selected and starting
from the most accurate tree, subsequent trees are added one by one and their
impact is recorded by using the OOB observations left out from the bootstrap
sample taken for the tree being added. A tree is selected if it improves
predictive accuracy of the ensemble. In the second approach, trees are grown on
random subsets, taken without replacement-known as sub-bagging, of the training
data instead of bootstrap samples (taken with replacement). The remaining
observations from each sample are used in both individual and collective
assessments for each corresponding tree similar to the first method. Analysis
on 21 benchmark datasets and simulations studies show improved performance of
the modified methods in comparison to OTE and other state-of-the-art methods.
