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Introduction To Machine Learning

Machine learning regression models are essential for predicting continuous outcomes based on input features. Below are the equations for some common regression techniques:

### 1. Simple Linear Regression
Simple linear regression is the basic form of regression where the target variable \( Y \) is modeled as a linear relationship with the input feature \( X \). $$ Y = \beta_0 + \beta_1 X + \epsilon $$ Where: - \( Y \): The target variable - \( X \): The independent variable - \( \beta_0 \): Intercept (constant) - \( \beta_1 \): Coefficient (slope of the line) - \( \epsilon \): Error term

### 2. Multiple Linear Regression
In multiple linear regression, the model takes multiple input features \( X_1, X_2, \dots, X_n \), and the target variable \( Y \) is modeled as: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_n X_n + \epsilon $$ ### 3. Polynomial Regression
Polynomial regression is an extension of linear regression where the relationship between the independent variable \( X \) and the target variable \( Y \) is modeled as an \( n \)-degree polynomial: $$ Y = \beta_0 + \beta_1 X + \beta_2 X^2 + \dots + \beta_n X^n + \epsilon $$ ### 4. Ridge Regression (L2 Regularization)
Ridge regression introduces a regularization term to the linear regression equation to reduce overfitting by penalizing large coefficients: $$ \min_{\beta} \sum_{i=1}^{n} \left( y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right)^2 + \lambda \sum_{j=1}^{p} \beta_j^2 $$ ### 5. Lasso Regression (L1 Regularization)
Lasso regression adds an L1 penalty, encouraging sparsity in the coefficients (some coefficients become zero): $$ \min_{\beta} \sum_{i=1}^{n} \left( y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right)^2 + \lambda \sum_{j=1}^{p} |\beta_j| $$ ### 6. Elastic Net Regression
Elastic Net combines both L1 (Lasso) and L2 (Ridge) penalties, offering a compromise between the two: $$ \min_{\beta} \sum_{i=1}^{n} \left( y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right)^2 + \lambda_1 \sum_{j=1}^{p} |\beta_j| + \lambda_2 \sum_{j=1}^{p} \beta_j^2 $$ ### 7. Support Vector Regression (SVR)
Support vector regression attempts to fit the data within a margin \( \epsilon \): $$ \min_{\beta} \frac{1}{2} \|\beta\|^2 \quad \text{subject to} \quad |y_i - (\beta_0 + \beta_1 X)| \leq \epsilon $$ ### 8. Decision Tree Regression
The decision tree regression model splits the input space into regions and fits a constant value \( c_j \) in each region \( R_j \): $$ Y = \sum_{j=1}^{J} c_j I(X \in R_j) $$ ### 9. Random Forest Regression
Random Forest is an ensemble method that fits multiple decision trees and averages their predictions: $$ Y = \frac{1}{M} \sum_{m=1}^{M} T_m(X) $$

These equations provide a foundation for understanding how machine learning regression models predict continuous outcomes based on input data.

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