Question 153 - MLS-C01 discussion

Linear least squares regression is a method of fitting a linear model to a set of data by minimizing the sum of squared errors between the observed and predicted values. The solution of the linear least squares problem can be obtained by solving the normal equations, which are given by

ATAx=ATb,

where A is the matrix of explanatory variables, b is the vector of response variables, and x is the vector of unknown coefficients.

However, if the matrix A has two features that are perfectly linearly dependent, then the matrix ATA will be singular, meaning that it does not have a unique inverse. This implies that the normal equations do not have a unique solution, and the linear least squares problem is ill-posed. In other words, there are infinitely many values of x that can satisfy the normal equations, and the linear model is not identifiable.

This can be an issue for the linear least squares regression model, as it can lead to instability, inconsistency, and poor generalization of the model. It can also cause numerical difficulties when trying to solve the normal equations using computational methods, such as matrix inversion or decomposition. Therefore, it is advisable to avoid or remove the linearly dependent features from the matrix A before applying the linear least squares regression model.

References:

Linear least squares (mathematics)

Linear Regression in Matrix Form

Singular Matrix Problem