Decision Tree Algorithms

Decision trees are widely used machine learning algorithm and can be used for both classification and regression tasks. These models work by splitting data into subsets based on feature and this splitting is called as decision making and each leaf node tells us prediction. This splitting creates a tree-like structure. They are easy to interpret and visualize for understanding the decision-making process.

In machine learning we have various types of decision trees and in this article we will explore them that so that we can use them in machine learning for various task.

Types of Decision Tree Algorithms

The different decision tree algorithms are listed below:

• ID3(Iterative Dichotomiser 3)
• C4.5
• CART(Classification and Regression Trees)
• CHAID (Chi-Square Automatic Interaction Detection)
• MARS(Multivariate Adaptive Regression Splines)
• Conditional Inference Trees

Each of them have their own working and advantages that we will learn now.

1. ID3 (Iterative Dichotomiser 3)

ID3 is a classic decision tree algorithm commonly used for classification tasks. It works by greedily choosing the feature that maximizes the information gain at each node. It calculates entropy and information gain for each feature and selects the feature with the highest information gain for splitting.

Entropy denoted by H(D) for dataset D, is calculated using the formula:

H(D) = \Sigma^n _{i=1}\;p_{i}\; log_{2}(p_{i})

Information gain quantifies the reduction in entropy after splitting the dataset on a feature:

Information\; Gain = H(D) - \Sigma^V_{v=1} \frac{|D_{v}|}{|D|}H (D_{v})

ID3 recursively splits the dataset using the feature with the highest information gain until all examples in a node belong to the same class or no features remain to split. After the tree is constructed it prune branches that don't significantly improve accuracy to reduce overfitting.

ID3 has limitations: it tends to overfit the training data and cannot directly handle continuous attributes. These issues are addressed by other algorithms like C4.5 and CART.

For its implementation you can refer to the article: Iterative Dichotomiser 3 (ID3) Algorithm From Scratch

2. C4.5

C4.5 uses a modified version of information gain called the gain ratio to reduce the bias towards features with many values. The gain ratio is computed by dividing the information gain by the intrinsic information which measures the amount of data required to describe an attribute’s values:

Gain Ratio = \frac{Split\; gain}{Gain\;\;information}

It addresses several limitations of ID3 including its inability to handle continuous attributes and its tendency to overfit the training set. It handles continuous attributes by first sorting the attribute values and then selecting the midpoint between adjacent values as a potential split point. The split that maximizes information gain or gain ratio is chosen.

It can also generate rules from the decision tree by converting each path from the root to a leaf into a rule, which can be used to make predictions on new data.

This algorithm improves accuracy and reduces overfitting by using gain ratio and post-pruning. While effective for both discrete and continuous attributes, C4.5 may still struggle with noisy data and large feature sets.

C4.5 has limitations:

• It can be prone to overfitting especially in noisy datasets even if uses pruning techniques.
• Performance may degrade when dealing with datasets that have many features.

3. CART (Classification and Regression Trees)

CART is a widely used decision tree algorithm that is used for classification and regression tasks.

1. For classification CART splits data based on the Gini impurity which measures the likelihood of incorrectly classified randomly selected data. The feature that minimizes the Gini impurity is selected for splitting at each node.

Gini Impurity (for Classification) :

Gini(D) = 1 - \Sigma^n _{i=1}\; p^2_{i}
where p_i​ is the probability of class i in dataset D.

2. For regression CART builds regression trees by minimizing the variance of the target variable within each subset. The split that reduces the variance the most is chosen.

To reduce overfitting, CART uses cost-complexity pruning after tree construction. This method involves minimizing a cost function that combines the impurity and tree complexity by adding a complexity parameter to the impurity measure. It builds binary trees where each internal node has exactly two child nodes simplifying the splitting process and making the resulting tree easier to interpret.

For its implementation you can refer to the article: Implementing CART (Classification And Regression Tree) in Python

4. CHAID (Chi-Square Automatic Interaction Detection)

CHAID uses chi-square tests to determine the best splits especially for categorical variables. It recursively divides the data into smaller subsets until each subset contains only data points of the same class or within a specified range of values. It chooses feature for splitting with highest chi-squared statistic indicating the strong relationship with the target variable. This approach is particularly useful for analyzing large datasets with many categorical features.

Chi-Square Statistic Formula:

X^2 = \Sigma \frac{(O_{i} - E_{i})^2}{E_{i}}

Where

• O_i represents the observed frequency
  E_i represents the expected frequency in each category.

It compares the observed distribution to the expected distribution to determine if there is a significant difference.

CHAID can be applied to both classification and regression tasks. In classification, algorithm assigns a class label to new data points by following the tree from the root to a leaf node with leaf node’s class label being assigned to data. In regression it predicts the target variable by averaging the values at the leaf node.

5. MARS (Multivariate Adaptive Regression Splines)

MARS is an extension of the CART algorithm. It uses splines to model non-linear relationships between variables. It constructs a piecewise linear model where the relationship between the input and output variables is linear but with variable slopes at different points, known as knots. It automatically selects and positions these knots based on the data distribution and the need to capture non-linearities.

Basis Functions: Each basis function in MARS is a simple linear function defined over a range of the predictor variable. The function is described as:

h(x) = \Bigg \{ x - t \;\; if \; x>t \\ t-x \;\; if x \leq t \Bigg\}

Where

• x is a predictor variable
• t is the knot function.

Knot Function: The knots are the points where the piecewise linear functions connect. MARS places these knots to best represent the data's non-linear structure.

MARS begins by constructing a model with a single piece and then applies forward stepwise selection to iteratively add pieces that reduce the error. The process continues until the model reaches a desired complexity. It is particularly effective for modeling complex relationships in data and is widely used in regression tasks.

6. Conditional Inference Trees

Conditional Inference Trees uses statistical tests to choose splits based on the relationship between features and the target variable. It use permutation tests to select the feature that best splits the data while minimizing bias.

The algorithm follows a recursive approach. At each node it evaluates the statistical significance of potential splits using tests like the Chi-squared test for categorical features and the F-test for continuous features. The feature with the strongest relationship to the target is selected for the split. The process continues until the data cannot be further split or meets predefined stopping criteria.

Summarizing all Algorithms

Here’s a short summary of all decision tree algorithms we have learned so far:

1. ID3: Uses information gain to split data and works well for classification but it is prone to overfitting and struggles with continuous data.
2. C4.5: Advance version of ID3 with gain ratio for both discrete and continuous data but struggle with noisy data.
3. CART: Used for both classification and regression task. It minimizes Gini impurity for classification and MSE for regression with pruning technique to prevent overfitting.
4. CHAID: Uses chi-square tests for splitting and is effective for large categorical datasets but not for continuous data.
5. MARS: Extended version of CART using piecewise linear functions to model non-linear relationships but it is computationally expensive.
6. Conditional Inference Trees: Uses statistical hypothesis testing for unbiased splits and handles various data types but it is slower than others.

Decision tree algorithms offer interpretable approach for both classification and regression tasks. While each algorithm brings its own strengths understanding their underlying mechanisms is crucial for selecting the best algorithm for a given problem for better accuracy of model.