Unlike classification models, classification trees do not suffer from overfitting in the event that a large B value is used. However, this requires choosing "a value of B sufficiently large that the error has settled down" (ISLR p.317).

Overfitting in classification trees

Similar to bagging for regression trees, bagging can be extended to classification problems as well. There exist various approaches, but the most simple involves picking the "majority vote" using $B$ trees for each test observation, which represents the most common class among those $B$ predictions.

University of Michigan - Ann Arbor

Bootstrap aggregation, or bagging, is a way to help construct more powerful prediction models by reducing high variance of a statistical learning method. It is often used to improve the performance of decision trees. In order to reduce the variance, $B$ different bootstrapped training data sets are produced (typically taken from a single training data set). It then uses the $b$th bootstrapped training data set to train the method to produce  $\hat{f}^{*b}(x)$. Afterwards, all the predictions are then averaged in order to get  $\hat{f}_{bag}(x) =  \frac{1}{B} \sum_{b=1}^B \hat{f}^{*b}(x)$. 

Bagging

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Bagging can be applied to decision trees, particularly with regression. It follows the same overall process: B regression trees are produced using B bootstrapped training sets. Afterwards, the predictions of all B regression trees are averaged in order to obtain a model that has reduced the variance of the B individual decision trees.

Bagging for Regression Trees

Bagging for Classification Trees

Out-of-Bag (OOB) observations give way to estimating the test error of a bagged model without cross-validation. OOB means that the observations were not used to fit a given bagged tree. Using the trees that included OOB observations, it is possible to predict the response for the $i$th observation, which returns close to $B/3$ predictions. 
In the case of regression trees, these predictions can then be averaged in order to obtain a single prediction for the $i$th observation. 
With classification trees, the single prediction for the $i$th observation can be found by taking a majority vote. As a result, the OOB prediction can be used to estimate error; with regression trees, the overall OOB MSE can be found, and similarly, classification error for classification trees. 

Out-of-Bag Error Estimation

One downside of bagging is the tradeoff between prediction accuracy and model interpretability. Although bagging increases prediction accuracy, it does so at the expense of interpretability. We can no longer represent a bagged model as a single tree. It is also no longer clear which variables are most important to the model once bagging has been integrated. 

However, there are ways to determine the most important predictors. For regression trees, we can record the total amount the RSS is decreased as a result of the splits for a specific predictor. This value is then averaged over all $B$ trees and the predictor is considered important if that value is large. Likewise, with classification trees, we can record the total amount that the Gini index is decreased as a result of the splits for a specific predictor, and average it over $B$ trees.

Variable Importance Measures

*Random forests* is quite similar to *bagging* with *decision trees* except that there is an improvement on spliting trees. They both use bootstrapped subset of training oberservations for a tree and repeated that process several times for the average/most common result. 

The difference lies in that, random forest only use randomly selected subsets of all predictors for spliting trees. This improvement *decorrelates* the trees and solves the problem of bagging that,  if there is a very strong predictor, most bagged trees might look similar and averaging could not reduce much variance for the highly correlated quantities. However, this can be addressed by decorrelating the trees thus reducing the variance and making the trees more reliable.

Random Forests vs. Bagging with Decision Trees

Boosting is heavily dependent on the trees formed before it, whereas in bagging, each tree is formed independently and varies instead based on the sample of predictors chosen. 

Boosting vs Bagging

Below is a video that I found helpful about understanding bagging and decision trees. The coursera course the video is a part of may be helpful as well.

[SAS]. Bagging [Video File]. Retrieved from [www.coursera.org/lecture/machine-learning-sas/bagging-Rj2Fq](https://www.coursera.org/lecture/machine-learning-sas/bagging-Rj2Fq) 

Coursera: Bagging using Decision Trees

Step 1: Here you replace the original data with new data. The new data usually have a fraction of the original data's columns and rows, which then can be used as hyper-parameters in the bagging model.

Step 2: You build classifiers on each dataset. Generally, you can use the same classifier for making models and predictions.

Step 3: Lastly, you use an average value to combine the predictions of all the classifiers, depending on the problem. Generally, these combined values are more robust than a single model.

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