Least Squares and Linear Regression

Introduction

In statistics, linear regression is a powerful technique used to model the relationship between a dependent variable (Y) and one or more independent variables (X). It is one of the most commonly used methods for predictive analysis, especially in the field of machine learning. In this post, we will explore the basics of linear regression, with a focus on the least squares method.

Linear Least Squares

The goal of linear regression is to find the best-fitting line or curve that explains the relationship between the independent and dependent variables. The most common method for doing this is called linear least squares.

In linear least squares, we try to minimize the sum of squared errors between the predicted values and the actual values of the dependent variable. In other words, we try to find the line that best fits the data by minimizing the distance between the predicted values and the actual values.

To do this, we use the formula:

$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$

where $\beta$ is the vector of coefficients, $X$ is the matrix of independent variables, $Y$ is the vector of dependent variable values.

Regression to the Mean

Regression to the mean is a statistical phenomenon that is related to linear regression. It states that if a variable is extreme on its first measurement, it will tend to be closer to the average on its second measurement.

For example, suppose we measure the heights of a group of people and find that one person is extremely tall. According to the regression to the mean phenomenon, we would expect that person to be less tall on a second measurement.

This phenomenon is important to keep in mind when interpreting the results of linear regression. It means that extreme values in the independent variable may not be as influential on the dependent variable as they first appear.

Linear regression is a powerful tool for analyzing the relationship between independent and dependent variables. The least squares method is the most commonly used method for finding the best-fitting line, and regression to the mean is an important phenomenon to keep in mind when interpreting the results.

With a solid understanding of linear regression, you can apply this technique to a wide range of real-world problems, such as predicting customer churn, analyzing the impact of advertising campaigns, and forecasting sales revenue.