Linear Regression Linear regression is a statistical method used to find a linear relationship between a dependent variable and one or more independent varia...
Linear regression is a statistical method used to find a linear relationship between a dependent variable and one or more independent variables. It involves minimizing the sum of squared errors between predicted and actual values.
Key components:
Coefficients: The slope and intercept of the linear line that best fits the data.
Residuals: The difference between the predicted and actual values.
MSE (Mean Squared Error): A measure of how well the linear model fits the data. Lower MSE indicates a better fit.
Assumptions:
Linear relationship between the variables.
Normally distributed residuals.
Homoscedasticity of errors (constant variance).
Ridge regression is an extension of linear regression that penalizes large coefficients to reduce overfitting. It balances between fitting the data and penalizing complex models.
Key components:
L2 penalty: A measure of the coefficient's magnitude.
Coefficient selection: Shrinks coefficients towards zero, focusing on the most important features.
Minimize L2 norm: Encourages models with smaller, more generalizable coefficients.
Benefits:
Reduces overfitting and improvesgeneralizability.
Increases interpretability of models.
Lasso regression is a regularization technique that adds a penalty term to the loss function. It encourages sparsity in the coefficients, resulting in models with fewer non-zero coefficients.
Key components:
L1 penalty: A measure of the absolute value of each coefficient.
Elastic net penalty: Encourages sparse coefficients by penalizing the absolute value of the coefficients.
Minimize L1 norm: Encourages solutions with many zeros.
Benefits:
Sparse and simple models.
Improves interpretability and model clarity.
Robust to outliers.
Comparison:
| Feature | Linear Regression | Ridge Regression | Lasso Regression |
|---|---|---|---|
| Penalty | None | L2 penalty | L1 penalty |
| Coefficient selection | Yes | No | Yes |
| Interpretability | High | Low | High |
| Robustness to outliers | Low | High | High |