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Linear Models

1. What are Linear Models?

Linear models are a class of models that make predictions using a linear function of the input features. The prediction is computed as a weighted sum of the input features plus a bias term. They have been extensively studied over more than a century and remain widely used due to their simplicity, interpretability, and effectiveness in many scenarios.


2. Mathematical Formulation

For regression, the general form of a linear model's prediction is:

y^=w0x0+w1x1++wpxp+b

where;

  • y^ is the predicted output,
  • xi is the i-th input feature,
  • wi is the learned weight coefficient for feature xi,
  • b is the intercept (bias term),
  • p is the number of features.

In vector form:

y^=wTx+b

where w=(w0,w1,...,wp) and x=(x0,x1,...,xp).


3. Interpretation and Intuition

  • The prediction is a linear combination of features — each feature contributes proportionally to its weight.
  • The model captures linear relationships between features and targets.
  • Despite simplicity, when data has a large number of features, linear models can approximate complex functions (even perfectly fit training data if number of features ≥ number of samples).

4. Linear Models for Regression

Ordinary Least Squares (OLS) / Linear Regression

·         The classic linear regression model estimates w and b by minimizing the sum of squared differences between observed and predicted values.

·         Objective: Minimize the residual sum of squares minw,bi=1N(yiy^i)2 where yi are true outputs and y^i are predicted outputs.

·         This results in a convex optimization problem with a closed-form solution using linear algebra.


5. Linear Models for Classification

  • Linear models are also extensively used for classification tasks.
  • For example, Logistic Regression models the probability of a class as a logistic function applied to the linear combination of features.
  • Similarly, Linear Support Vector Machines (SVMs) seek a separating hyperplane defined by a linear function.

6. When Do Linear Models Perform Well?

  • Particularly effective when the number of features is large relative to the number of samples, as they can fit complex combinations of features.
  • Efficient to train on very large datasets where training more complex models is computationally prohibitive.
  • Often serve as baseline models or components in more complex pipelines.

7. Limitations and Failure Cases

  • In low-dimensional spaces or when the true decision boundary is non-linear, linear models may underperform.
  • They can't naturally handle complex, non-linear relationships unless combined with feature transformations or kernel methods (e.g., kernelized SVMs).
  • Feature scaling and careful regularization are necessary to avoid overfitting or underfitting.

8. Key Variants

  • Ordinary Least Squares (OLS): Minimizes squared error, no regularization.
  • Ridge Regression: Adds L2 regularization to penalize large weights.
  • Lasso Regression: Adds L1 regularization for feature selection/sparsity.
  • Elastic Net: Combines L1 and L2 penalties.
  • Variants apply different techniques for parameter estimation and complexity control.

9. Summary

  • Linear models predict through a weighted sum of features.
  • They are computationally efficient and interpretable.
  • Perform well with many features or large datasets.
  • May be outperformed in non-linear or low-dimensional contexts.
  • Integral to classical and modern machine learning workflows.

 

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