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

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.

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 We can no longer represent a bagged model as a single tree.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.

Steps of Bagging

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  $\hat{f}_{bag}(x) =  \frac{1}{B} \sum_{b=1}^B \hat{f}^{*b}(x)$. 

University of Michigan - Ann Arbor

Bootstrap resampling is a way to create new samples of the original data. It obtains new datasets by sampling observations from the original set through replacement. These new datasets have the same sample size as the original and are useful in determining estimates for the standard error and confidence intervals of a statistic by comparing the accuracy of the model for each new data set.

Bootstrap method for resampling

Approaches for improving decision trees' predictions 

Weakness of Trees model is predictive accuracy compared to other models. We can tackle this weakness of Trees by:

 - Bagging
 - Random Forrests
 - Boosting

How to improve the weakness of Trees model

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

https://blogs.sas.com/content/iml/2018/12/12/essential-guide-bootstrapping-sas.html

Image Reference: Bootstrap

To create the bootstrap sample, select a random observation from your main data source, and then add the observation to your bootstrap sample. The observation then gets put back into your main data source (this is what allows observations to be selected more than once). This process gets repeated $n$ times until your bootstrap sample is complete.

Bootstrap resampling algorithm

Instead of repeatedly obtaining new data sets from a population, replacement provides an alternative to do just that from an already existing data set. Data sets made through replacement are limited to containing any points in the original data set. Some of the points from the original data set will show up multiple times, while others not at all.

Replacement in Bootstrap method for resampling

Bagging

The Random Forest is a model made up of many decision trees. It trains each one on a slightly different set of the observations, splitting nodes in each tree considering a limited number of the features. The final predictions of the random forest are made by averaging the predictions of each individual tree. In random forest algorithm, we do not need to do pruning for each "weak" decision tree. 

Advantage of Random Forest:
- Less overfitting
- Parallel implementation

Random Forest

Boosting is another approach that can be used for both classification and regression. Boosting improves predictions of a decision tree by learning slowly. Unlike bagging, boosting works sequentially, where each tree grown uses the information from those created before it, and works on modified versions of the original data set. Boosting combines a large number of trees ($\hat{f}^{1}, ...,  \hat{f}^{B}$). Decision trees are fitted based on the residuals of the model rather than the response (Y), then update the residuals by adding it to the fitted function. This means that each model created is slowly reducing the error of the previous ones. This causes an improvement of $\hat{f}$, particularly in places where it does not do as well as others. Decision trees using boosting tend to be very small.


Boosting

Pruning decision trees is the process of removing certain branches which are not beneficial towards the model’s goal. Decision trees frequently over-fit, which leads to an excess of branches. As a result, a decision tree performs extremely well on the training set, but is over-fitting leading to poor test set performance. To prevent this, individuals can ‘prune’ the decision tree leading to fewer branches preventing over-fitting. Types of pruning include minimum error, smallest tree, and early stopping.

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