Linear Regression Overview

Introduction to Linear Regression

• First machine learning model discussed.
• A model in data science:
  • Mathematical representation of a real-world process.
  • Input-output relationship example: Pizza consumption based on time since last meal.

Importance of Models

• Help understand the nature of processes being modeled.
• Enable prediction of outputs based on input features.
• Predicting unknowns provides economic value.

Example: Housing Prices

• Input: Size of a house.
• Output: Price of the house.
• Training examples observed for individual houses to establish a model.

Simple Linear Regression Model

• Linear relationship can be expressed as:
  [ y = a_0 + a_1 x ]
  • Where:
    • ( y ) = output (price of house)
    • ( x ) = input feature (size of house)
    • ( a_0, a_1 ) = model parameters

Multiple Features

• General linear equation with multiple features:
  [ y = a_0 + a_1 x_1 + a_2 x_2 + ... + a_n x_n ]
  • ( x_i ) = features
  • ( a_i ) = model parameters
  • ( y ) = target variable.

Cost Function

• Objective: Fit a straight line through training examples.
• Error term for each training example:
  • ( e_i = y_{predicted} - y_{actual} )
• Cost function defined as: [ J = \frac{1}{2m} \sum_{i=1}^{m} e_i^2 ]
  • ( m ) = number of training examples.

Minimizing the Cost Function

• Best fitting model minimizes the cost function.
• Cost function measures the distance of model from data points.
• Gradient descent algorithm used to minimize the cost function:
  • Iteratively adjust model parameters to find the minimum.

Gradient Descent Algorithm Steps

1. Calculate the slope at the current parameter values.
2. Update parameters using a small step size ( \alpha ).
3. Update the cost function with new parameters.
4. Repeat steps 1-3 multiple times until convergence.

Learning Rate ( \alpha )

• Determines the step size during gradient descent.
• If ( \alpha ) is too high, the algorithm may oscillate and not converge.
• Ideal value: approximately 0.01 to balance speed and convergence.
• Learning rate is a hyperparameter that impacts model performance.

Conclusion

• Understanding the theory behind the cost function and gradient descent is essential.
• Code implementations available to train linear regression models without deep mathematical understanding.
• Congratulations on learning about linear regression!