Linear Regression Model: 
$$ f(X) = \beta_0 + \sum_{j=1}^{p} X_j \beta_j $$
Regression Trees Model: 
$$ f(x) = \sum_{m=1}^{M} c_m · 1(x \in R_m) $$
Linear regression is a good choice if the relationship between predictive variables and response variables can be well fitted by linear models. On the other hand, the tree method is more appropriate if the relationship is highly nonlinear and complex. 


Trees VS. Linear Models

1) Decision trees are easily explainable to others

2) Compared to other statistical methods like regression, some believe that trees are much more similar to human decision making

3) Decision trees can be displayed graphically, which make them easy to interpret

4) There is no need to create dummy variables when handling qualitative predictors


Advantages to Using Decision Trees

1) Although trees may be easier to understand and interpret, they do not provide the same level of predictive accuracy compared to traditional regression and classification methods

2) Small changes in data can cause large changes in the final estimated tree

3) Not fit for continuous variables - While working with continuous numerical variables, decision tree loses information, when it categorizes variables in different categories.

4) Overfitting - Can create over-complex trees that do not generalize the data well and suffer from high variance. This problem gets solved by setting constraints on model parameters and pruning.


Disadvantages to Using Decision Trees

1. Regression trees
2. Classification trees

Types of decision trees

The algorithm required to build a decision tree involves 3 steps:

1. Start with an empty tree

2. Split on the feature with the highest information gain

3. Recurse with step 2

Creating a minimally sized tree which minimizes training error is actually an extremely hard, (NP-Hard) problem). To approximate this optimal solution, we build a tree by greedily splitting on the feature which results in the most information gain, i.e., pick the feature to split on that more certainly leads to a classification. 


Learning Decision Trees

Approaches for improving decision trees' predictions 

Decision trees can be applied to both regression and classificatioin problems, and can thus be divided into

- Regression trees
- Classification trees

Decision trees applied to regression and classificatioin problems

https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html

Post pruning decision trees with cost complexity pruning

Max depth controls the maximum depth or the number of split points the decision can have and it's the most common parameter used to reduce tree complexity and thus reduce overfitting. The min samples leaf parameter is the threshold that controls what the minimum number of data instances has to be in a leaf to avoid splitting it further. This setting also reduces tree complexity. Finally, max leaf nodes limits the total number of nodes that are leaves of the tree.

Scikit learn key decision tree parameters

- N_estimators: sets # of small decision trees to use in the ensemble 
   - More trees can lead to overfitting
   - Chosen to best exploit the speed and memory capabilities
- Learning_rate: controls emphasis on fixing errors from previous iteration 
   - Adjusted given n_estimators
- The above two are typically tuned together 
   - Small learning rate requires more n estimators to maintain model complexity 
- Max_depth: typically set to a small value (3-5) for most applications
Can effect model complexity


Decision tree key parameters

- Training builds a series of small decision trees 
- Each tree attempts to correct errors from previous stage 
- The learning rate controls how hard each tree tries to correct remaining mistakes from previous round
   - High learning rate: more complex trees 
   - Low learning rate: simpler trees 


Gradient Boosted Decision Trees 

A decision tree of depth 2 is visualized below. Using the 'value' attribute of each leaf, find the accuracy score for the tree of depth 2 and the accuracy score for a tree of depth 1. What is the improvement in accuracy between the model of depth 1 and the model of depth 2?

Find the Accuracy Score of a Decision Tree

A decision tree is composed of several key components that define its structure and partition logic:

* **Nodes**: Split points where the dataset is partitioned based on feature values.
* **Split point**: The threshold or decision rule that separates one class or range from another.
* **Pure node**: A node where all observations belong to a single class, representing perfect classification.
* **Mixed node**: A node containing a mixture of classes that requires further splitting.
* **Leaf nodes**: The terminal nodes at the end of a branch that represent the final predicted class or value.
* **Informativeness of split**: A measure of the information gained from a split, used to select the optimal split point.

Decision Tree Terminology

A decision tree divides the prediction space into a series of simple regions according to the stratifying criteria. Decision trees can be applied to both regression and classification problems and can thus be of two types.

University of Michigan - Ann Arbor

- Generalized Additive Models
- Trees
- Random Forests
- Boosting
- Support Vector Machines
- Naive Bayes Classifier

Computer Intensive Classifiers

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

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

Support Vector Machines are intended for binary classification (cases where there are two classes). The main goal is to find a hyperplane that best divides a dataset into two classes. This is quantified by achieving the maximum distance between data points of both classes separated by the hyperplane. The further away the data points are from the hyperplane, the more confidently we are able to classify future data points into their respective class.

Support Vector Machines

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

Decision Tree

Tree based methods aims to stratify or segment the predictor space (two predictors expressed in x y plane) into several simple regions. We can make a prediction for an observation by taking the mean or mode of the training observations in the region to which the observation belongs.

Tree based methods can be used both for 
-	Regression 
-	Classification


        


Tree-based Methods 

An Artificial Neural Network (ANN) is a computational model inspired by the structure of biological neural networks in the brain, made up of interconnected neuron-like units organized in layers. Rather than following task-specific rules, an ANN learns to recognize patterns in data from examples, adjusting the connections between its units during training.

Artificial Neural Networks

Reference to Artificial Neural Networks

A method of supervised classification that takes many features, assumed to be independent, and uses Bayes' Theorem to determine given those independent variables how likely the likelihood of another event occurring.

It's called "Naive" because it assumes that features are conditionally independent, given the class, i.e., for all instances of a given class, the features have little or no correlation with each other.

(Naive) Bayes Classifier

Generalized Additive Models (GAMs), similar to multiple linear regression models, attempt to predict a response value ($$Y$$) using many predictor variables ($$X_1, \dots, X_p$$). GAMs maintain their additivity by applying separate non-linear functions to each predictor variable ($$f_j(X_j)$$) and adding them together. They can be applied to both quantitative and qualitative responses.

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