8. Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between a dependent variable and one or more independent variables. It assumes a linear relationship between these variables.

Core Concept

The goal of linear regression is to find the "best-fitting" straight line through a set of data points. This line is determined by an equation that minimizes the difference between the observed values of the dependent variable and the values predicted by the model.

The general equation for simple linear regression (one independent variable) is:

$y = \beta_0 + \beta_1x + \epsilon$

Where:

• $y$: The dependent variable (what we are trying to predict).
• $x$: The independent variable (the predictor).
• $\beta_0$: The y-intercept (the value of $y$ when $x$ is 0).
• $\beta_1$: The slope of the line (the change in $y$ for a unit change in $x$).
• $\epsilon$: The error term, representing the part of $y$ that cannot be explained by $x$.

For multiple linear regression (two or more independent variables), the equation extends to:

$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n + \epsilon$

Where $x_1, x_2, ..., x_n$ are the different independent variables.

Algorithm Design Steps

Designing an algorithm for linear regression typically involves the following steps:

1. Data Collection and Preparation:

   • Gather relevant data, ensuring it includes both the dependent and independent variables.
   • Clean the data: handle missing values (imputation or removal), outliers, and incorrect entries.
   • Perform feature engineering if necessary (e.g., creating new features from existing ones).
   • Split the data into training and testing sets. The training set is used to train the model, and the testing set is used to evaluate its performance on unseen data.

2. Model Selection:

   • Choose between simple linear regression (one predictor) or multiple linear regression (multiple predictors) based on the problem and data.

3. Parameter Estimation:

   • The most common method for estimating the coefficients ($\beta_0, \beta_1, ...$) is the Ordinary Least Squares (OLS) method. OLS aims to minimize the sum of the squared residuals (the differences between actual and predicted values).
   • Other methods like Gradient Descent can also be used, especially for larger datasets or more complex models, to iteratively find the optimal coefficients.

4. Model Training:

   • Use the training data to calculate the optimal values for $\beta_0, \beta_1, ...$ using the chosen estimation method (e.g., OLS).

5. Model Evaluation:

   • Assess the performance of the trained model using the testing set. Common evaluation metrics include:
     • Mean Squared Error (MSE): The average of the squared errors.
     • Root Mean Squared Error (RMSE): The square root of MSE, providing error in the same units as the dependent variable.
     • R-squared ($R^2$): The proportion of the variance in the dependent variable that is predictable from the independent variable(s). A higher $R^2$ indicates a better fit.
     • Mean Absolute Error (MAE): The average of the absolute differences between actual and predicted values.

6. Prediction:

   • Once the model is trained and evaluated, it can be used to make predictions on new, unseen data by plugging the independent variable values into the regression equation.

7. Interpretation and Refinement:

   • Interpret the coefficients to understand the relationships between variables.
   • Check for assumptions of linear regression (e.g., linearity, independence of errors, homoscedasticity, normality of errors). Violations may require model adjustments or different modeling techniques.

Example (Conceptual)

Imagine you want to predict a student's exam score ($y$) based on the number of hours they studied ($x$).

1. Data: You collect data from several students: hours studied and their corresponding exam scores.
2. Preparation: You ensure no missing data and split the data into training and testing sets.
3. Model: You choose simple linear regression: $score = \beta_0 + \beta_1 \times hours_studied$.
4. Training: Using OLS on the training data, you find the best $\beta_0$ and $\beta_1$. For instance, you might get $score = 50 + 5 \times hours_studied$.
5. Evaluation: You test this model on the unseen data and find an RMSE of 8, meaning the predictions are, on average, off by 8 points. The $R^2$ might be 0.75, indicating 75% of the score variance is explained by study hours.
6. Prediction: If a new student studies for 10 hours, you predict their score as $50 + 5 \times 10 = 100$.

This provides a foundational understanding of how to approach and implement linear regression.