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Bagging, aka bootstrap aggregation, is a relatively simple way to increase the power of a predictive statistical model by taking multiple random samples(with replacement) from your training data set, and using each of these samples to construct a separate model and separate predictions for your test set. These predictions are then averaged to create a, hopefully more accurate, final prediction value.

One can quickly intuit that this technique will be more useful when the predictors are more unstable. In other words, if the random samples that you draw from your training set are very different, they will generally lead to very different sets of predictions. This greater variability will lead to a stronger final result. When the samples are extremely similar, all of the predictions derived from the samples will likewise be extremely similar, making bagging a bit superfluous.

Taking smaller samples from your training set will induce greater instability, but taking samples that are too small will result in useless models. Generally, some fraction of your training set between 1/50th and 1/2 will be useful for bagging purposes(of course, this greatly depends on how many observations are in your training set). The smaller your bagging samples, the more samples you will need to collect and the more models you will need to generate to create more stability in the final predictors.

While there are some libraries that will take care of bagging for you in R, writing your own function allows you a greater degree of control and understanding over the process, and it is a relatively quick exercise.

Okay, enough theoretical framework. Lets jump into the code. Anyone who wants more theory can consult this paper about bagging.

I’m going to construct a relatively trivial example where a dependent variable, y, can be predicted by some combination of the independent variables x1, x2, and x3.

```set.seed(10)
y<-c(1:1000)
x1<-c(1:1000)*runif(1000,min=0,max=2)
x2<-c(1:1000)*runif(1000,min=0,max=2)
x3<-c(1:1000)*runif(1000,min=0,max=2)
```
As you can see, y is a sequence of the values from 1 to 1000. x1, x2, and x3 are permutations of y, but with random errors added. runif generates a specified number of random numbers from 0 to 1, unless a min and max are specified, in which case the numbers fall between those values. Each of the x sequences will roughly approximate y, but with random errors thrown in. The set.seed function is simply to ensure that the subsequent random number generation proceeds in a predictable fashion, so that your results match mine.

Fitting a linear model to the variables results in an R squared of .7042:
```lm_fit<-lm(y~x1+x2+x3)
summary(lm_fit)
```
Now we will see how well the x values predict y. First, we designate a random sample of y to be our "test" set. The rest will be the training set.
```set.seed(10)
all_data<-data.frame(y,x1,x2,x3)
positions <- sample(nrow(all_data),size=floor((nrow(all_data)/4)*3))
training<- all_data[positions,]
testing<- all_data[-positions,]
```
The above code places all of our variables into a data frame, then randomly selects 3/4 of the data to be the training set, and places the rest into the testing set.

We are now able to generate predictions for the testing set by creating a linear model on the training set and applying it to the testing set. We are also able to calculate the prediction error by subtracting the actual values from the predicted values (the error calculation here is root mean squared error):
```lm_fit<-lm(y~x1+x2+x3,data=training)
predictions<-predict(lm_fit,newdata=testing)
error<-sqrt((sum((testing\$y-predictions)^2))/nrow(testing))
```
The calculated error should be 161.15.

The next step is to run a function that implements bagging. In order to do this, I will be using the foreach package. Although I will not use it in parallel mode, this code is designed for parallel execution, and I highly recommend reading my post about how to do it if you do not know how.
```library(foreach)
length_divisor<-4
iterations<-1000
predictions<-foreach(m=1:iterations,.combine=cbind) %do% {
training_positions <- sample(nrow(training), size=floor((nrow(training)/length_divisor)))
train_pos<-1:nrow(training) %in% training_positions
lm_fit<-lm(y~x1+x2+x3,data=training[train_pos,])
predict(lm_fit,newdata=testing)
}
predictions<-rowMeans(predictions)
error<-sqrt((sum((testing\$y-predictions)^2))/nrow(testing))
```
The above code randomly samples 1/4 of the training set in each iteration, and generates predictions for the testing set based the sample. It will execute the number of time specified by iterations. When iterations was set to 10, I received an error value of 161.10. At 300 iterations, error went to 161.12, at 500 iterations, error went to 161.19, at 1000 iterations, error went to 161.13, and at 5000 iterations, error went to 161.07. Eventually, bagging will converge, and more iterations will not help any further. However, the potential for improvement in results exists. You should be extremely cautious and assess the stability of the results before deploying this approach, however, as too few iterations or too large a length divisor can cause extremely unstable results. This example is trivial, but this can lead to better results in a more "real-world" application.

Finally, we can place this code into a function to wrap it up nicely:
```bagging<-function(training,testing,length_divisor=4,iterations=1000)
{
predictions<-foreach(m=1:iterations,.combine=cbind) %do% {
training_positions <- sample(nrow(training), size=floor((nrow(training)/length_divisor)))
train_pos<-1:nrow(training) %in% training_positions
lm_fit<-lm(y~x1+x2+x3,data=training[train_pos,])
predict(lm_fit,newdata=testing)
}
rowMeans(predictions)
}
```
As you can see, bagging can be a useful tool when used correctly. Although this is a trivial example, you can replace the data and even replace the simple linear model with more powerful models to make this function more useful.