Least Squares Regression, Explained: A Visual Guide with Code Examples for Beginners

Linear regression is a statistical method that predicts numerical values using a linear equation, modeling the relationship between a dependent variable and one or more independent variables. The most common approaches to linear regression are called "Least Squares Methods", which work by finding patterns in data by minimizing the squared differences between predictions and actual values. Ordinary Least Squares (OLS) is a fundamental approach to linear regression that finds the best-fitting line through data points by minimizing the sum of squared distances between each point and the line. The optimization goal is to find coefficients that minimize the sum of squared distances, which can be calculated using the normal equation. In the multidimensional case, the training process involves preparing the data matrix, computing the coefficients using the normal equation, and making predictions by multiplying new data points by the coefficients. Ridge regression is a variation of OLS that adds a penalty term to the objective function to discourage large coefficients, which can lead to overfitting. The penalty term is controlled by the lambda parameter, which determines how much to penalize large coefficients. The training process for Ridge regression is similar to OLS, but with a modification to the closed-form solution. The choice between OLS and Ridge depends on the data, with OLS suitable for well-behaved data and Ridge suitable for data with many features, multicollinearity, or signs of overfitting.