Least Squares Regression

Least Squares Regression is used to model the effect of one or more predictor variables on a dependent variable.

It is widely used in predictive modeling and regression analysis. Least Squares Regression works by finding the optimal set of coefficients to multiply each predictor variable to estimate the dependent variable.

For example, suppose the gross national product of a country depends on its population size, the mean number of years spent in education, and the unemployment rate. Least Squares Regression determines the optimal weightings for these factors using training data. These weightings can then be used to estimate the gross national product of a new country for which the other variables are known.

Least Squares Regression procedures are often sensitive to outliers, which can significantly affect the model. It can be helpful to eliminate outliers from the training data if there is a theoretical basis for doing so. However, simply removing outliers introduces a dangerous bias into the learning calculation.

Regression is performed using matrix mathematics, which efficiently models the relationships between variable values. There is a spectrum of least squares regression procedures, ranging from simple methods with many constraints to complex methods with fewer constraints but requiring more training data.

Ordinary Least Squares Regression (OLSR) is the simplest form and works well with small training sets if all prerequisites are met. Weighted Least Squares Regression relaxes the homoscedasticity assumption by adjusting the effect of predictor variables based on their values. Generalized Least Squares Regression removes the assumptions of error independence and homoscedasticity by incorporating a matrix that expresses variable interactions. Non-linear Regression relaxes the linearity assumption, using iterative methods to approach optimal values.

Least Squares Regression is important for its simplicity and effectiveness in many predictive modeling tasks. Understanding its assumptions and limitations is crucial for successful application.