A really good paper, that explains random forests very well : 
[www.stat.berkeley.edu/~breiman/randomforest2001.pdf](https://www.stat.berkeley.edu/~breiman/randomforest2001.pdf) 

University of Michigan - Ann Arbor

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

Decision Trees are **non-parametric**, since we are not making any assumptions about the underlying function $$f$$. Consequently, decision trees have very low structural error (bias). By averaging the output of many trees in a random forest, we are lowering the variance of the model, while keeping the bias the same. 


Why Random Forests?

*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

A study on different data sizes and different forest sizes (# of attributes) found that at a certian point, increasing the forest size becomes much more computationally expensive without any significant performance gain. 

The study found using between 64 and 128 attributes was an optimal number, with fewer being less accurate, and more slower with similar accuracy. 

Random Forests: Selecting Number of Trees

Basic algorithm:

1. Pick N random records from the dataset. 

2. Build a decision tree based on these N records. 

3. Choose the number of trees you want in your algorithm and repeat steps 1 and 2. 

4. In case of a regression problem, for a new record, each tree in the forest predicts a value for Y (output). The final value can be calculated by taking the average of all the values predicted by all the trees in the forest. Or, in the case of a classification problem, each tree in the forest predicts the category to which the new record belongs. Finally, the new record is assigned to the category that wins the majority vote. 

How Random Forest work?

Random Forests

```python
from sklearn.metrics import accuracy_score
import sklearn.ensemble as skens

# build a random forest
rf_model = skens.RandomForestClassifier(n_estimators=10,oob_score=True, criterion='entropy')
rf_model.fit(df_iris_train.ix[:,:4],df_iris_train.species)

# predict the model 
predicted_labels = rf_model.predict(df_iris_test.ix[:,:4])
df_iris_test['predicted_rf_tree'] = predicted_labels

# find accuracy score
accuracy = accuracy_score(df_iris_test.species, predicted_labels)

# utility class to compare the predictions versus ground truth
def comparePlot(input_frame,real_column,predicted_column):
    df_a = input_frame.copy()
    df_b = input_frame.copy()
    df_a['label_source'] = 'Species'
    df_b['label_source'] = 'Classifier'
    df_a['label'] = df_a[real_column]
    df_b['label'] = df_b[predicted_column].apply(lambda x: 'Predict %s'%x)
    df_c = pd.concat((df_a, df_b), axis=0, ignore_index=True)
    sns.lmplot(x='sepal_length', y='sepal_width', col='label_source',
               hue='label', data=df_c, fit_reg=False, size=4);

# compare plot
comparePlot(df_iris_test,"species","predicted_rf_tree")
```

Random Forest Python Code

Below is a representation of under and overfitting of the boundaries of U.S. sates. The data comes from the US Census Bureau. In its original format, the data is a single Keyhole Markup Language (KML) file which contains latitude and longitude coordinates of the borders of US states. The necessary latitude, longitude, and label (state) data were parsed from the KML files using a simple Python script. The main idea here is to understand the bias-variance trade-off and how that relates to under and overfitting. 

Additionally there are two examples of ways they avoided under and overfitting and created a much more accurate map using (gradient) boosting and random forest classifiers.

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