Summary
Highlights
Linear regression is a statistical technique used to model the relationship between an output variable and one or more input variables. It can be visualized as fitting a line through data points to make predictions on unseen data, assuming a linear relationship exists between the variables.
The familiar linear function y = mx + b is used, where 'y' is the output (dependent) variable and 'x' is the input (independent) variable. In statistics, 'm' and 'b' are often represented as beta1 and beta0, respectively. Beta1 controls the slope of the line, and beta0 (also known as bias in machine learning) controls the intercept. These are the coefficients that linear regression aims to solve for.
Linear regression can be extended to include multiple input variables (x1, x2, x3, etc.) each with their own beta coefficients (beta1, beta2, beta3). In these higher dimensions, the linear regression is visualized as a hyperplane.
To fit the line, residuals (differences between data points and the predicted line) are calculated. These residuals are squared to create 'squared errors'. The sum of these squared errors for a given line forms the 'loss function'. The goal of linear regression is to find the beta0 and beta1 coefficients that minimize this sum of squared error, using techniques like matrix decomposition or gradient descent.
To validate a linear regression model, practitioners often split the data into a training set (e.g., two-thirds) and a test set (e.g., one-third). The training set is used to fit the regression line, and the test set is used to validate its performance on new, unseen data. Metrics like R-squared, standard error of the estimate, prediction intervals, and statistical significance are used for evaluation.