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,b∑i=1N(yi−y^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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Mashtishk Vigyan Anusandhan

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